7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Research ArticleExact Solutions of Heat and Mass Transfer withMHD Flow in a Porous Medium under Time DependentShear Stress and Temperature

Arshad Khan1 Ilyas Khan2 Farhad Ali13 Asma Khalid14 and Sharidan Shafie1

983089 Department o Mathematical Sciences Faculty o Science Universiti eknologi Malaysia (UM) 983096983089983091983089983088 Johor Bahru Johor Malaysia983090College o Engineering Majmaah University Majmaah 983089983089983097983093983090 Saudi Arabia983091Department o Mathematics City University o Science and Inormation echnology Peshawar 983090983093983088983088983088 Pakistan983092Department o Mathematics SBK Womenrsquos University Quetta 983096983095983091983088983088 Pakistan

Correspondence should be addressed to Sharidan Sha1047297e sharidanutmmy

Received 983094 May 983090983088983089983092 Revised 983089983095 July 983090983088983089983092 Accepted 983090983093 July 983090983088983089983092

Academic Editor Saeed Islam

Copyright copy 983090983088983089983093 Arshad Khan et al Tis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Tis paper aims to study the in1047298uence o thermal radiation on unsteady magnetohyrdodynamic (MHD) natural convection 1047298ow o an optically thick 1047298uid over a vertical plate embedded in a porous medium with arbitrary shear stress Combined phenomenon o heat andmass transer is considered Closed-orm solutions in general orm are obtained by using the Laplace transorm techniqueTey are expressed in terms o exponential and complementary error unctions Velocity is expressed as a sum o thermal andmechanical parts Corresponding limiting solutions are also reduced rom the general solutions It is ound that the obtainedsolutions satisy all imposed initial and boundary conditions and reduce to some known solutions rom the literature as specialcases Analytical results or the pertinent 1047298ow parameters are drawn graphically and discussed in detail It is ound that the velocity pro1047297les o 1047298uid decrease with increasing shear stress Te magnetic parameter develops shear resistance which reduces the 1047298uidmotion whereas the inverse permeability parameter increases the 1047298uid 1047298ow

1 Introduction

Heat and mass transer process is observed in lots o practicalsituations or example evaporation and chemical reactionsas well as condensation Te industrial applications include

many transport processes where the simultaneous heat andmass transer occurs as a result o combined buoyancy effectso thermal diffusion and diffusion o chemical species Possi-bly this is because o the act that in many numbers o techni-cal transer processes the studyo mixed heat andmass trans-eris helpul Few attemptsin thisdirection are made by Singh[983089] Narahari [983090] Narahari and Nayan [983091] Narahari and Ishak [983092] Chaudhary and Jain [983093]Dasetal[983094] Soundalgekar et al[983095] and Muthucumaraswamy et al [983096 983097] A ew lateefforts inthe same area o research are presented in [983092ndash983097] Signi1047297cantconcern has been originated in the study o magnetic 1047297eldand the electrically conducting 1047298uids 1047298ow while medium isporous [983089983088] Te unsteady ree convection 1047298uid 1047298ows which

are incompressible and viscous near a porous in1047297nite platewith arbitrary time dependent heating plate are investigatedby oki and okis [983089983089] Te results o chemical reaction o vis-cous 1047298uid which are electrically conducting through a porousmedium in two-dimensional steady ree convection 1047298ow past

a vertical surace with slip 1047298ow region have been presented by Senapati et al [983089983090] MHD ree convection 1047298ow o an incom-pressible viscous 1047298uid near an oscillating plate embedded ina porous medium has been presented by Khan et al [983089983091]Tereore many researchers have studied ree convection1047298ow past a vertical plate with thermal radiation [983089983092ndash983089983094]

Moreover several research papers on ree convection1047298uid 1047298ows with different thermal conditions at the boundingplate which are continuous and well-de1047297ned at the wall areound However a number o problems seem with differentconditions at the wall Tereore its investigation understep change in wall temperature is meaningul Te physicalsigni1047297cance o this thought can be seen in the abrication o

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 975201 16 pageshttpdxdoiorg1011552015975201

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983090 Abstract and Applied Analysis

B0

gx

z

y

Cinfin

Tinfin

Porous medium

Momentum boundary layer

Termal boundary layer

Concentration boundary layer

u(0 t)

y =

f(t)

1038389 C(0 t) = Cw

T(0 t) = Tinfin + (Tw minus Tinfin) t

t0

0 lt t lt t0 T(0 t) = Tw t ge t0

F983145983143983157983154983141 983089 Physical con1047297guration o the problem

thin-1047297lm photovoltaic devices [983089983095] Whenever the conven-tional supposition o periodic outdoor conditions may leadto substantial errors in the case o a signi1047297cant temporary deviation o the temperature rom periodicity such as in

air conditioning periodic step changes in temperature areimportant [983089983096] Here some o recent and important contri-butions [983089983096ndash983090983092] are presented

Fluid 1047298ow past an in1047297nite plate is o much importancedue to its large practical applications Such motion is due tomany effects such as motions due to wall shear stress Closed-orm results o the problems with shear stress on the wallare difficult thereore a very rare research is ound in theliterature Slip velocity depends on the shear stress linearlythis idea was presented by Navier [983090983093] Free convection 1047298ow near a vertical plate that applies arbitrary shear stress to the1047298uid was investigated by Fetecau et al [983090983094] However as yet

no research has been presented earlier in the literature whichmainly ocuses on the ree convection conjugate 1047298ow withthermal diffusion while taking arbitrary shear stress alongramped wall temperature

Tereore exact solutions or MHD conjugate 1047298ow o a viscous 1047298uid past a vertical plate that applies arbitrary shearstress to the 1047298uid are presented in this paper Exact solutionso the initial and boundary value problems that govern the1047298ow are obtained by using Laplace transorm techniqueFrom general solutions some o special and limiting cases arederived Te results or velocity 1047297eld the temperature 1047297eldand concentration 1047297eld are shown graphically and discussedor different embedded parameters

2 Mathematical Formulation

Consider the unsteady MHD ree convection 1047298ow o anincompressible viscous 1047298uid over an in1047297nite vertical plate

Te geometry o the problem is presented in Figure 983089 Teplate is along the -axis and the 1038389-axis is assumed normal toit Te plate and the 1047298uid are at stationary positions with theconstant temperature 1103925infin and concentration 907317infin Te 1047298uidexperiences shear stress () by the plate afer = 0+ In themeantime the plate temperature is aroused or let down to1103925infin+(1103925minus1103925infin)(0) when le 0 and thereafer or gt 0 iskept at constant temperature1103925 and concentration is arousedto907317 Te radiationis taken in the energy equation Howeverthe radiative heat 1047298ux is assumed negligible in -directionWe suppose that the 1047298uid 1047298ow is laminar grey absorbing-emitting radiation but the medium is with no scattering Fur-thermore we suppose that the 1047298uid is electrically conductingHence we take the ollowing Maxwell equations

div B = 0Curl E = minus B Curl B = 1038389J

(983089)

In the above equations B E and 1038389 are the magnetic 1047297eldelectric 1047297eld intensity and the magnetic permeability o the1047298uid respectively By using Ohmrsquos law the current density Jis given as

J

= (E

+V

timesB

) (983090)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983091

where is the electrical conductivity o the 1047298uid Further wemake the ollowing assumptions

(i) 1038389 are constants throughout the 1047298ow 1047297eld

(ii) B is perpendicular to V

(iii) Te induced magnetic 1047297eld b is negligible comparedwith the imposed magnetic 1047297eld B0

(iv) Te magnetic Reynolds number is small

(v) Te electric 1047297eld is zero

Tereore the linearized orm o the electromagneticbody orce [983090983095] is

1 JtimesB = 1048616VtimesB0983081 timesB0 = minus 2

0V (983091)

Using Boussinesqrsquos approximation and neglecting the viscousdissipation the equations governing the 1047298ow are given by [983090]

= ]

210383892 + 1103925 10486161103925minus1103925infin983081+907317 1048616907317minus907317infin983081

minus ]minus 2

0 1038389 gt 0(983092)

907317 1103925 = 2110392510383892 minus 1038389 1038389 gt 0 (983093)

907317 = 2

90731710383892 +1 2

110392510383892 1038389 gt 0 (983094)

where 907317 1103925 ] 907317 1103925 907317 0 1 and are the 1047298uid velocity in -direction the 1047298uidconcentration the 1047298uid temperature its kinematic viscositythe gravitational acceleration the constant density the masstranser coefficient the heat transer coefficient the 1047298uidelectric conductivity the heat capacity the applied magnetic1047297eld the thermal conductivity the radiative heat 1047298ux massdiffusivity thermal diffusivity and the permeability o theporous medium

Te corresponding initial and boundary conditions are

10486161038389 0983081 = 0110392510486161038389 0983081 = 1103925infin90731710486161038389 0983081 = 907317infin

forall1038389 ⩾ 0 (0 )1038389 = () 907317 (0 ) = 907317

gt 0

1103925 (0 ) = 1103925infin + 10486161103925 minus1103925infin983081 0 0 lt lt 01103925 (0 ) = 1103925 ge 0 (infin ) = 0

1103925 (infin ) = 1103925infin907317 (infin ) = 907317infin gt 0

(983095)

Te radiation heat 1047298ux under Rosseland approximation oroptically thick 1047298uid [983090983096] is given by

= minus 4lowast3

11039254

1038389 (983096)

where

lowast and

are the Stean-Boltzman constant and the

mean absorption coefficient We can see rom (983096) that theradiation term is nonlinear Recently David Maxim Gururajand Anjali Devi [983090983097] used nonlinear radiation effects andstudied MHD boundary layer 1047298ow with orced convectionpast a nonlinearly stretching surace with variable tempera-ture Tereore we ollow David Maxim Gururaj and AnjaliDevi [983090983097] and assume that the temperature differences withinthe 1047298ow are sufficiently small that is the difference betweenthe 1047298uid temperature and the ree stream temperature isnegligible so that (983096) can be linearized by expanding 1103925 intothe aylor series about 1103925infin which afer neglecting higherorder terms takes the orm

11039254

asymp 411039253

infin1103925minus 311039254

infin (983097)

Introducing (983093) (983096) and (983097) we get

Pr 1103925 = ] 10486161+983081 2110392510383892

1038389 gt 0 (983089983088)

where Pr ] and are de1047297ned by

Pr = 907317 ] =

= 1611039253

infin3

(983089983089)

aking the nondimensional variables

lowast = radic 0]

1103925lowast = 1103925 minus 1103925infin1103925 minus 1103925infin 907317lowast = 907317 minus 907317infin

907317

minus 907317infin

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983092 Abstract and Applied Analysis

1038389lowast = 1038389991770]0 lowast = 0

lowast

1048616lowast

983081 = 0

() (983089983090)

by eliminating the star notations into (983092) (983094) and (983089983088) weobtain

= 210383892 +Gr1103925+Gm907317minusminus

Pre1047296 1103925 =

2110392510383892

907317 =

1

Sc

29073171038389

2

+Sr211039251038389

2

(983089983091)

where Pre1047296 = Pr(1+) is the effective Prandtl number [983090983096Equation (10)] and

Gr = 1103925 10486161103925 minus 1103925infin983081 ]

30

Gm = 1048616907317 minus 907317infin983081 ]

3

0

= 2

00

Sc = ] = ]0 0 = ]2

0

Sr = 1 10486161103925 minus 1103925infin9830811048616907317 minus 907317infin983081 ]

(983089983092)

are the Grasho number modi1047297ed Grasho number magneticparameter Schmidt number the inverse permeability param-

eter orthe porous medium the characteristic time andSoretnumber respectively

Te nondimensional initial and boundary conditions are

10486161038389 0983081 = 0110392510486161038389 0983081 = 090731710486161038389 0983081 = 0

forall1038389 ge 0

1038389

=0

= ()

1103925 (0 ) = 0 lt le 1

1103925 (0 ) = 1 gt 1

907317 (0

) = 1

907317 (infin ) = 01103925 (infin ) = 0 (infin ) = 0

gt 0(983089983093)

3 Solution of the Problem

o solve (983089983091) under conditions (983089983093) by taking Laplace trans-orm technique we obtained

10486161038389983081 = 21048616103838998308110383892 +Gr110392510486161038389983081+Gm90731710486161038389983081

minus10486161038389983081minus10486161038389983081 (983089983094)

1103925 10486161038389 983081 = 1

Pre1047296 2110392510486161038389983081

10383892 (983089983095)

907317 10486161038389 983081 = 1

Sc 290731710486161038389983081

10383892 + Sr

211039251048616103838998308110383892

(983089983096)

with boundary conditions

907317 1048616infin 983081 = 0907317 10486160 983081 = 1 1103925 1048616infin 983081 = 0 1048616infin 983081 = 0

104861610383899830811038389=0

= 1048616983081

1103925 10486160 983081 = 1 minus minus2

(983089983097)

Solving (983089983095) using (983089983097) we get

1103925 10486161038389 983081 = 12 minusradicPre1047296 minus minus

2 minusradicPre1047296 (983090983088)

by inverse Laplace transorm giving

1103925 10486161038389983081 = 10486161038389983081minus 10486161038389 minus1

983081 ( minus1

) (983090983089)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983093

where

10486161038389 983081 = 983080Pre1047296 10383892

2 + 1048617 er 983080991770Pre1047296 1038389

21057306 1048617

minusradicPre1047296

1038389 exp983080minusPre1047296

10383892

4 1048617 1103925104861610383899830811038389

=0 = 2991770Pre1047296 1057306 8520081057306 minus1057306 minus 1( minus 1)852009

(983090983090)

which is the Nusselt number Error complementary errorunctions o Gauss [983091983088] are denoted by er (sdot) and er (sdot)

Solution o (983089983096) under boundary conditions (983089983097) yields

907317 10486161038389 983081 = 1minusradic Sc + 18 10486161 minus minus9830812 minus991770 Sc991770

minus 18 10486161 minus minus9830812 minusradic Pre1047296 991770 (983090983091)

by taking inverse Laplace transorm giving

907317 10486161038389 983081 = er 98308010383891057306 Sc

21057306 1048617+ 18983080983080+ Sc10383892

2 1048617 er

sdot 98308010383891057306 Sc

21057306 1048617minus 10383891057306 Sc1057306 1057306 minus2Sc41048617minus 18983080983080

+ Pre1047296

10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617 minus 1038389991770Pre1047296

1057306 1057306 sdot minus2Pre1047296 41048617+98313118983080983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617

minus 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)1048617983133 ( minus 1)

minus 98313118

983080983080minus1

+ Sc10383892

2

1048617er

983080 10383891057306 Sc

2

1057306 minus1

1048617minus 10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)1048617983133 ( minus 1)

(983090983092)

907317104861610383899830811038389

=0 = minus21057306 Sc minus 3181057306 Sc1057306

minus983131minus21057306 Sc minus 3181057306 Sc ( minus 1)991770 ( minus 1) 983133 ( minus 1)

(983090983093)

which is the corresponding mass transer rate also known asSherwood number

Te solution o (983089983094) under boundary conditions (983089983097)results in

10486161038389 983081 = 21057306 10486161 minus minus983081

2

1048616 minus 1

983081991770+1

minusradic+1

+ 41057306 1048616 minus 3983081991770 + 1

minusradic+1

minus 1048616983081991770 + 1

minusradic+1

+ 61057306 10486161 minus minus9830812 1048616 minus 5983081991770+1

minusradic+1

minus 71057306 10486161 minus minus9830812 1048616 minus 1983081991770+1

minusradic+1

minus 8 10486161 minus minus

9830812 1048616 minus 1983081 minusradicPre1047296

+ 11 10486161 minus minus9830812 1048616 minus 1983081

minusradic Pre1047296

minus 10 10486161 minus minus9830812 1048616 minus 5983081

minusradic Sc minus 9 1048616 minus 3983081 minusradic Sc

(983090983094)

which upon inverse Laplace transorm results in

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983090983095)

where (1038389) corresponds to convective part o velocity which is de1047297ned as

10486161038389 983081 = 1 +2 (983090983096)

1 = 2 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 minus1048667852059 2

10486161983081321057306 sdot intminus1

0

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

85205921

sdot intminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983094 Abstract and Applied Analysis

minus 7 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 +1048667

852059

7

10486161

98308132

1057306 sdotintminus10

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) minus 1048667

85205971

sdotintminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 6 int0

3(minus) er 8520089917703 ( minus )8520091048616398308132 minus 21057306 minus 1057306 3

sdot minus1minus24s1057306 minus1048667852059

610486163983081321057306

sdotintminus10

3(minus1minus)minus1minus24 er 8520089917703 ( minus 1 minus )8520091057306 1048669

852061

sdot ( minus 1) + 1048667852059 6

3sdotintminus1

0

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 4 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

+ 81 983080 + Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306

1048617

minus98313181 983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617983133 (

minus 1) + 983131811038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 93minusradic3Sc

23 er 98308010383891057306 Sc

21057306 minus99177031048617 + 93

sdot er 98308010383891057306 Sc

2

1057306 1048617minus 103minusradic Sc3

2

23

sdot er 98308010383891057306 Sc

21057306 minus99177031048617 minus 93+radic 3Sc

23sdot er 98308010383891057306 Sc

21057306 +99177031048617 (983090983097)

2 = 103 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 1minusradic Pre1047296 122

1

er 9830801038389991770Pre1047296

21057306 minus99177011048617

minus 10486168 minus 11983081 1+radic Pre1047296 12

21

er 9830801038389991770Pre1047296

2

1057306 +99177011048617

+ 98313110310383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 103

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 1023

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 1198308111038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

1

sdot er 9830801038389991770Pre1047296

2

1057306 1048617minus 98313110486168 minus 11983081

21

er 9830801038389991770Pre1047296

2

1057306 minus11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc322

3

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 103+radic Sc322

3

er 98308010383891057306 Sc

21057306 +99177031048617minus 98313110

3

983080minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

2

1057306 minus11048617983133 (

minus 1) + [10486168 minus 11983081 1(minus1)minusradic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus8606981 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 1(minus1)+radic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2

991770( minus1

) +8606981 ( minus 1)1048617] ( minus 1)

(983091983088)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983095

and (1038389) is mechanical part o velocity de1047297ned as

10486161038389 983081 = minus 11057306 int

0

( minus ) minus1minus241057306 (983091983089)

where

1 = 1

Pre1047296 minus 1

2 = Gr991770Pre1047296 Pre1047296 minus 1

3 = 1

Sc minus 1

4 = Gm1057306 Sc

Sc minus 1

6 = GmScSrPre1047296 1057306 ScSc minus 1

7 = GmScSrPre1047296 991770Pre1047296 Pre1047296 minus 1

8 = Gr

Pre1047296 minus 1

9 = Gm

Sc minus 1

10 = GmScSrPre1047296 Sc

minus1

11 = GmScSrPre1047296 Pre1047296 minus 1

12 =

Pre1047296 minus 1

13 = Sc minus 1

15 =

Pre1047296 minus 1

16

= Sc minus 1 18 = SrScPre1047296

1 = +19 = GmSc32SrPre1047296 1048616Pre1047296 minus Sc983081 (Sc minus 1)

20 = GmPr32e1047296

SrSc

1048616Pre1047296 minus Sc983081 (Sc minus 1) 21 = GmPre1047296 SrSc

1048616Pre1047296

minusSc

983081 (Sc

minus1

)

(983091983090)

4 Plate with Constant Temperature

Te solution o (983089983095) under boundary conditions (983089983097) orconstant temperature yields

1103925 10486161038389 983081 = er 9830801038389991770Pre1047296

21057306 1048617 1103925 (0 )1038389 = minus 991770Pre1047296 1057306

(983091983091)

Te solution o (983089983094) under boundary conditions (983089983097) orconstant temperature is

10486161038389 983081 = 21057306 1048616 minus 1983081 991770 + 1

minusradic+1

+ 41057306 1048616 minus 3983081991770+1

minusradic+1

minus 1048616983081991770+1

minusradic+1

+ 191057306 1048616 minus 3983081991770+1

minusradic +1minus 8 1048616 minus 1983081

minusradicPre1047296

minus 20

1057306 1048616 minus 1983081991770+1 minusradic+1

minus 21 1048616 minus 3983081 minusradic Sc + 21 1048616 minus 1983081

minusradic Pre1047296

minus 9 1048616 minus 3983081 minusradic Sc

(983091983092)

upon inverse Laplace transorm

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983091983093)

where

10486161038389 983081 = 3 +4 (983091983094)

3

= 2int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

+ 4int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983096 Abstract and Applied Analysis

+ 19 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

minus 20 int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

minus 213minusradic 3Sc

23 er 98308010383891057306 Sc21057306 minus 99177031048617

+ 213 er 98308010383891057306 Sc

21057306 1048617

minus 213+radic3Sc23 er 98308010383891057306 Sc

21057306 +99177031048617 (983091983095)

4

= 211minusradic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 minus99177011048617minus 211 er 9830801038389991770Pre1047296

21057306 1048617

+ 211+radic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 81minusradic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 minus99177011048617

+ 8

1er

9830801038389991770Pre1047296

21057306 1048617minus 81+radic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 93minusradic3Sc23 er 98308010383891057306 Sc

21057306 minus99177031048617+ 93 er 98308010383891057306 Sc

21057306 1048617

minus 93+radic 3Sc

23er

98308010383891057306 Sc

21057306 +9917703

1048617

(983091983096)

5 Limiting Cases

Here some limiting cases are presented

983093983089 Solution in the Absence o Porous Effects ( rarr 0 )From (983090983089) and (983090983092) it is seen that the temperature 1047297eldsand concentration 1047297elds are not affected by the inversepermeability parameter or the porous medium Hencethe velocities or both case o the plate are as

10486161038389 983081 =

10486161038389983081+

10486161038389983081 (983091983097)

where

10486161038389 983081 = 5 +65 = 2int

0

12(minus) er 85200899177012 ( minus )852009

104861612

98308132 minus 21057306 minus

1057306 12

sdot minusminus241057306 minus1048667852059

2104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059212

sdotintminus1

0 8520082

1057306 minus1

minus852009 minusminus24

1057306 1048669852061 ( minus 1)minus 7 int

0

12(minus) er 85200899177012 ( minus )85200910486161298308132 minus 21057306 minus 1057306 12

sdot minusminus241057306 +1048667852059

7104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )852009

1057306 1048669

852061sdot ( minus 1) minus1048667852059

712

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int0

13(minus) er 85200899177013 ( minus )85200910486161398308132 minus 21057306 minus 1057306 13

sdot minusminus24

1057306 minus1048667852059 6

104861613983081321057306 sdotintminus1

0

13(minus1minus)minusminus24 er 85200899177013 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059613

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983097

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13 + 812 983080 +

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617

minus 983131 812 983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296 21057306 minus 11048617983133 (

minus 1) + 983131 8121038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 913minusradic 13Sc

23 er 98308010383891057306 Sc

21057306 minus991770131048617 + 913

sdot er 98308010383891057306 Sc

21057306 1048617minus 1013minusradic Sc3

223

sdot er 98308010383891057306 Sc

21057306 minus991770131048617 minus 9

13+radic 13Sc

213sdot er 98308010383891057306 Sc

21057306 +991770131048617 6 = 1013 983080+

Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus98313110213

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 12minusradic Pre1047296 12

2212

er

9830801038389991770Pre1047296

21057306 minus99177012

1048617minus 10486168 minus 11983081 12+radic Pre1047296 12

2212

er 9830801038389991770Pre1047296

21057306 +991770121048617+9831311013

10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1013sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 102

13

er 98308010383891057306 Sc

21057306 1048617minus 10486168 minus 11983081

12

1038389991770Pre1047296 1057306

1057306 minus2Pre1047296 4 + 10486168 minus 11983081

212

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 11983081212

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [1013(minus1)minusradic Sc322

13

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus86069813 ( minus 1)1048617] ( minus 1)

minus 1013+radic Sc32

213

er 98308010383891057306 Sc

2

1057306 +991770131048617

minus9831311013 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus 1) + [10486168 minus 11983081 12(minus1)minusradic Pre1047296 1222

12

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069812 ( minus 1)1048617] ( minus 1)+ [10486168 minus 11983081 12(minus1)+radic Pre1047296 12

2212

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069812 ( minus 1)1048617] ( minus 1) (983092983088)

and or isothermal

10486161038389 983081 = 7 +87

= 2 int0

12(minus) er 86069812 ( minus ) minusminus24

1057306 1

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13

+ 19 int0

13(minus)er 86069813 ( minus ) minusminus24

1057306 13

minus 20 int012(minus) er 86069812 ( minus ) minusminus

2

41057306 12 minus 2113minusradic13Sc

213 er 98308010383891057306 Sc

21057306 minus991770131048617+ 2113 er 98308010383891057306 Sc

21057306 1048617

minus 2113+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617

8

= 2112minusradic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 minus991770121048617minus 2112 er 9830801038389991770Pre1047296

21057306 1048617

+ 2112+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 812minusradic 12Pre1047296 2

12

er 9830801038389991770Pre1047296

2

1057306 minus991770121048617

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

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Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 317

Abstract and Applied Analysis 983091

where is the electrical conductivity o the 1047298uid Further wemake the ollowing assumptions

(i) 1038389 are constants throughout the 1047298ow 1047297eld

(ii) B is perpendicular to V

(iii) Te induced magnetic 1047297eld b is negligible comparedwith the imposed magnetic 1047297eld B0

(iv) Te magnetic Reynolds number is small

(v) Te electric 1047297eld is zero

Tereore the linearized orm o the electromagneticbody orce [983090983095] is

1 JtimesB = 1048616VtimesB0983081 timesB0 = minus 2

0V (983091)

Using Boussinesqrsquos approximation and neglecting the viscousdissipation the equations governing the 1047298ow are given by [983090]

= ]

210383892 + 1103925 10486161103925minus1103925infin983081+907317 1048616907317minus907317infin983081

minus ]minus 2

0 1038389 gt 0(983092)

907317 1103925 = 2110392510383892 minus 1038389 1038389 gt 0 (983093)

907317 = 2

90731710383892 +1 2

110392510383892 1038389 gt 0 (983094)

where 907317 1103925 ] 907317 1103925 907317 0 1 and are the 1047298uid velocity in -direction the 1047298uidconcentration the 1047298uid temperature its kinematic viscositythe gravitational acceleration the constant density the masstranser coefficient the heat transer coefficient the 1047298uidelectric conductivity the heat capacity the applied magnetic1047297eld the thermal conductivity the radiative heat 1047298ux massdiffusivity thermal diffusivity and the permeability o theporous medium

Te corresponding initial and boundary conditions are

10486161038389 0983081 = 0110392510486161038389 0983081 = 1103925infin90731710486161038389 0983081 = 907317infin

forall1038389 ⩾ 0 (0 )1038389 = () 907317 (0 ) = 907317

gt 0

1103925 (0 ) = 1103925infin + 10486161103925 minus1103925infin983081 0 0 lt lt 01103925 (0 ) = 1103925 ge 0 (infin ) = 0

1103925 (infin ) = 1103925infin907317 (infin ) = 907317infin gt 0

(983095)

Te radiation heat 1047298ux under Rosseland approximation oroptically thick 1047298uid [983090983096] is given by

= minus 4lowast3

11039254

1038389 (983096)

where

lowast and

are the Stean-Boltzman constant and the

mean absorption coefficient We can see rom (983096) that theradiation term is nonlinear Recently David Maxim Gururajand Anjali Devi [983090983097] used nonlinear radiation effects andstudied MHD boundary layer 1047298ow with orced convectionpast a nonlinearly stretching surace with variable tempera-ture Tereore we ollow David Maxim Gururaj and AnjaliDevi [983090983097] and assume that the temperature differences withinthe 1047298ow are sufficiently small that is the difference betweenthe 1047298uid temperature and the ree stream temperature isnegligible so that (983096) can be linearized by expanding 1103925 intothe aylor series about 1103925infin which afer neglecting higherorder terms takes the orm

11039254

asymp 411039253

infin1103925minus 311039254

infin (983097)

Introducing (983093) (983096) and (983097) we get

Pr 1103925 = ] 10486161+983081 2110392510383892

1038389 gt 0 (983089983088)

where Pr ] and are de1047297ned by

Pr = 907317 ] =

= 1611039253

infin3

(983089983089)

aking the nondimensional variables

lowast = radic 0]

1103925lowast = 1103925 minus 1103925infin1103925 minus 1103925infin 907317lowast = 907317 minus 907317infin

907317

minus 907317infin

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 417

983092 Abstract and Applied Analysis

1038389lowast = 1038389991770]0 lowast = 0

lowast

1048616lowast

983081 = 0

() (983089983090)

by eliminating the star notations into (983092) (983094) and (983089983088) weobtain

= 210383892 +Gr1103925+Gm907317minusminus

Pre1047296 1103925 =

2110392510383892

907317 =

1

Sc

29073171038389

2

+Sr211039251038389

2

(983089983091)

where Pre1047296 = Pr(1+) is the effective Prandtl number [983090983096Equation (10)] and

Gr = 1103925 10486161103925 minus 1103925infin983081 ]

30

Gm = 1048616907317 minus 907317infin983081 ]

3

0

= 2

00

Sc = ] = ]0 0 = ]2

0

Sr = 1 10486161103925 minus 1103925infin9830811048616907317 minus 907317infin983081 ]

(983089983092)

are the Grasho number modi1047297ed Grasho number magneticparameter Schmidt number the inverse permeability param-

eter orthe porous medium the characteristic time andSoretnumber respectively

Te nondimensional initial and boundary conditions are

10486161038389 0983081 = 0110392510486161038389 0983081 = 090731710486161038389 0983081 = 0

forall1038389 ge 0

1038389

=0

= ()

1103925 (0 ) = 0 lt le 1

1103925 (0 ) = 1 gt 1

907317 (0

) = 1

907317 (infin ) = 01103925 (infin ) = 0 (infin ) = 0

gt 0(983089983093)

3 Solution of the Problem

o solve (983089983091) under conditions (983089983093) by taking Laplace trans-orm technique we obtained

10486161038389983081 = 21048616103838998308110383892 +Gr110392510486161038389983081+Gm90731710486161038389983081

minus10486161038389983081minus10486161038389983081 (983089983094)

1103925 10486161038389 983081 = 1

Pre1047296 2110392510486161038389983081

10383892 (983089983095)

907317 10486161038389 983081 = 1

Sc 290731710486161038389983081

10383892 + Sr

211039251048616103838998308110383892

(983089983096)

with boundary conditions

907317 1048616infin 983081 = 0907317 10486160 983081 = 1 1103925 1048616infin 983081 = 0 1048616infin 983081 = 0

104861610383899830811038389=0

= 1048616983081

1103925 10486160 983081 = 1 minus minus2

(983089983097)

Solving (983089983095) using (983089983097) we get

1103925 10486161038389 983081 = 12 minusradicPre1047296 minus minus

2 minusradicPre1047296 (983090983088)

by inverse Laplace transorm giving

1103925 10486161038389983081 = 10486161038389983081minus 10486161038389 minus1

983081 ( minus1

) (983090983089)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 517

Abstract and Applied Analysis 983093

where

10486161038389 983081 = 983080Pre1047296 10383892

2 + 1048617 er 983080991770Pre1047296 1038389

21057306 1048617

minusradicPre1047296

1038389 exp983080minusPre1047296

10383892

4 1048617 1103925104861610383899830811038389

=0 = 2991770Pre1047296 1057306 8520081057306 minus1057306 minus 1( minus 1)852009

(983090983090)

which is the Nusselt number Error complementary errorunctions o Gauss [983091983088] are denoted by er (sdot) and er (sdot)

Solution o (983089983096) under boundary conditions (983089983097) yields

907317 10486161038389 983081 = 1minusradic Sc + 18 10486161 minus minus9830812 minus991770 Sc991770

minus 18 10486161 minus minus9830812 minusradic Pre1047296 991770 (983090983091)

by taking inverse Laplace transorm giving

907317 10486161038389 983081 = er 98308010383891057306 Sc

21057306 1048617+ 18983080983080+ Sc10383892

2 1048617 er

sdot 98308010383891057306 Sc

21057306 1048617minus 10383891057306 Sc1057306 1057306 minus2Sc41048617minus 18983080983080

+ Pre1047296

10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617 minus 1038389991770Pre1047296

1057306 1057306 sdot minus2Pre1047296 41048617+98313118983080983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617

minus 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)1048617983133 ( minus 1)

minus 98313118

983080983080minus1

+ Sc10383892

2

1048617er

983080 10383891057306 Sc

2

1057306 minus1

1048617minus 10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)1048617983133 ( minus 1)

(983090983092)

907317104861610383899830811038389

=0 = minus21057306 Sc minus 3181057306 Sc1057306

minus983131minus21057306 Sc minus 3181057306 Sc ( minus 1)991770 ( minus 1) 983133 ( minus 1)

(983090983093)

which is the corresponding mass transer rate also known asSherwood number

Te solution o (983089983094) under boundary conditions (983089983097)results in

10486161038389 983081 = 21057306 10486161 minus minus983081

2

1048616 minus 1

983081991770+1

minusradic+1

+ 41057306 1048616 minus 3983081991770 + 1

minusradic+1

minus 1048616983081991770 + 1

minusradic+1

+ 61057306 10486161 minus minus9830812 1048616 minus 5983081991770+1

minusradic+1

minus 71057306 10486161 minus minus9830812 1048616 minus 1983081991770+1

minusradic+1

minus 8 10486161 minus minus

9830812 1048616 minus 1983081 minusradicPre1047296

+ 11 10486161 minus minus9830812 1048616 minus 1983081

minusradic Pre1047296

minus 10 10486161 minus minus9830812 1048616 minus 5983081

minusradic Sc minus 9 1048616 minus 3983081 minusradic Sc

(983090983094)

which upon inverse Laplace transorm results in

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983090983095)

where (1038389) corresponds to convective part o velocity which is de1047297ned as

10486161038389 983081 = 1 +2 (983090983096)

1 = 2 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 minus1048667852059 2

10486161983081321057306 sdot intminus1

0

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

85205921

sdot intminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 617

983094 Abstract and Applied Analysis

minus 7 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 +1048667

852059

7

10486161

98308132

1057306 sdotintminus10

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) minus 1048667

85205971

sdotintminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 6 int0

3(minus) er 8520089917703 ( minus )8520091048616398308132 minus 21057306 minus 1057306 3

sdot minus1minus24s1057306 minus1048667852059

610486163983081321057306

sdotintminus10

3(minus1minus)minus1minus24 er 8520089917703 ( minus 1 minus )8520091057306 1048669

852061

sdot ( minus 1) + 1048667852059 6

3sdotintminus1

0

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 4 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

+ 81 983080 + Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306

1048617

minus98313181 983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617983133 (

minus 1) + 983131811038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 93minusradic3Sc

23 er 98308010383891057306 Sc

21057306 minus99177031048617 + 93

sdot er 98308010383891057306 Sc

2

1057306 1048617minus 103minusradic Sc3

2

23

sdot er 98308010383891057306 Sc

21057306 minus99177031048617 minus 93+radic 3Sc

23sdot er 98308010383891057306 Sc

21057306 +99177031048617 (983090983097)

2 = 103 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 1minusradic Pre1047296 122

1

er 9830801038389991770Pre1047296

21057306 minus99177011048617

minus 10486168 minus 11983081 1+radic Pre1047296 12

21

er 9830801038389991770Pre1047296

2

1057306 +99177011048617

+ 98313110310383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 103

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 1023

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 1198308111038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

1

sdot er 9830801038389991770Pre1047296

2

1057306 1048617minus 98313110486168 minus 11983081

21

er 9830801038389991770Pre1047296

2

1057306 minus11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc322

3

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 103+radic Sc322

3

er 98308010383891057306 Sc

21057306 +99177031048617minus 98313110

3

983080minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

2

1057306 minus11048617983133 (

minus 1) + [10486168 minus 11983081 1(minus1)minusradic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus8606981 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 1(minus1)+radic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2

991770( minus1

) +8606981 ( minus 1)1048617] ( minus 1)

(983091983088)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983095

and (1038389) is mechanical part o velocity de1047297ned as

10486161038389 983081 = minus 11057306 int

0

( minus ) minus1minus241057306 (983091983089)

where

1 = 1

Pre1047296 minus 1

2 = Gr991770Pre1047296 Pre1047296 minus 1

3 = 1

Sc minus 1

4 = Gm1057306 Sc

Sc minus 1

6 = GmScSrPre1047296 1057306 ScSc minus 1

7 = GmScSrPre1047296 991770Pre1047296 Pre1047296 minus 1

8 = Gr

Pre1047296 minus 1

9 = Gm

Sc minus 1

10 = GmScSrPre1047296 Sc

minus1

11 = GmScSrPre1047296 Pre1047296 minus 1

12 =

Pre1047296 minus 1

13 = Sc minus 1

15 =

Pre1047296 minus 1

16

= Sc minus 1 18 = SrScPre1047296

1 = +19 = GmSc32SrPre1047296 1048616Pre1047296 minus Sc983081 (Sc minus 1)

20 = GmPr32e1047296

SrSc

1048616Pre1047296 minus Sc983081 (Sc minus 1) 21 = GmPre1047296 SrSc

1048616Pre1047296

minusSc

983081 (Sc

minus1

)

(983091983090)

4 Plate with Constant Temperature

Te solution o (983089983095) under boundary conditions (983089983097) orconstant temperature yields

1103925 10486161038389 983081 = er 9830801038389991770Pre1047296

21057306 1048617 1103925 (0 )1038389 = minus 991770Pre1047296 1057306

(983091983091)

Te solution o (983089983094) under boundary conditions (983089983097) orconstant temperature is

10486161038389 983081 = 21057306 1048616 minus 1983081 991770 + 1

minusradic+1

+ 41057306 1048616 minus 3983081991770+1

minusradic+1

minus 1048616983081991770+1

minusradic+1

+ 191057306 1048616 minus 3983081991770+1

minusradic +1minus 8 1048616 minus 1983081

minusradicPre1047296

minus 20

1057306 1048616 minus 1983081991770+1 minusradic+1

minus 21 1048616 minus 3983081 minusradic Sc + 21 1048616 minus 1983081

minusradic Pre1047296

minus 9 1048616 minus 3983081 minusradic Sc

(983091983092)

upon inverse Laplace transorm

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983091983093)

where

10486161038389 983081 = 3 +4 (983091983094)

3

= 2int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

+ 4int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983096 Abstract and Applied Analysis

+ 19 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

minus 20 int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

minus 213minusradic 3Sc

23 er 98308010383891057306 Sc21057306 minus 99177031048617

+ 213 er 98308010383891057306 Sc

21057306 1048617

minus 213+radic3Sc23 er 98308010383891057306 Sc

21057306 +99177031048617 (983091983095)

4

= 211minusradic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 minus99177011048617minus 211 er 9830801038389991770Pre1047296

21057306 1048617

+ 211+radic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 81minusradic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 minus99177011048617

+ 8

1er

9830801038389991770Pre1047296

21057306 1048617minus 81+radic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 93minusradic3Sc23 er 98308010383891057306 Sc

21057306 minus99177031048617+ 93 er 98308010383891057306 Sc

21057306 1048617

minus 93+radic 3Sc

23er

98308010383891057306 Sc

21057306 +9917703

1048617

(983091983096)

5 Limiting Cases

Here some limiting cases are presented

983093983089 Solution in the Absence o Porous Effects ( rarr 0 )From (983090983089) and (983090983092) it is seen that the temperature 1047297eldsand concentration 1047297elds are not affected by the inversepermeability parameter or the porous medium Hencethe velocities or both case o the plate are as

10486161038389 983081 =

10486161038389983081+

10486161038389983081 (983091983097)

where

10486161038389 983081 = 5 +65 = 2int

0

12(minus) er 85200899177012 ( minus )852009

104861612

98308132 minus 21057306 minus

1057306 12

sdot minusminus241057306 minus1048667852059

2104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059212

sdotintminus1

0 8520082

1057306 minus1

minus852009 minusminus24

1057306 1048669852061 ( minus 1)minus 7 int

0

12(minus) er 85200899177012 ( minus )85200910486161298308132 minus 21057306 minus 1057306 12

sdot minusminus241057306 +1048667852059

7104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )852009

1057306 1048669

852061sdot ( minus 1) minus1048667852059

712

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int0

13(minus) er 85200899177013 ( minus )85200910486161398308132 minus 21057306 minus 1057306 13

sdot minusminus24

1057306 minus1048667852059 6

104861613983081321057306 sdotintminus1

0

13(minus1minus)minusminus24 er 85200899177013 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059613

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983097

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13 + 812 983080 +

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617

minus 983131 812 983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296 21057306 minus 11048617983133 (

minus 1) + 983131 8121038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 913minusradic 13Sc

23 er 98308010383891057306 Sc

21057306 minus991770131048617 + 913

sdot er 98308010383891057306 Sc

21057306 1048617minus 1013minusradic Sc3

223

sdot er 98308010383891057306 Sc

21057306 minus991770131048617 minus 9

13+radic 13Sc

213sdot er 98308010383891057306 Sc

21057306 +991770131048617 6 = 1013 983080+

Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus98313110213

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 12minusradic Pre1047296 12

2212

er

9830801038389991770Pre1047296

21057306 minus99177012

1048617minus 10486168 minus 11983081 12+radic Pre1047296 12

2212

er 9830801038389991770Pre1047296

21057306 +991770121048617+9831311013

10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1013sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 102

13

er 98308010383891057306 Sc

21057306 1048617minus 10486168 minus 11983081

12

1038389991770Pre1047296 1057306

1057306 minus2Pre1047296 4 + 10486168 minus 11983081

212

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 11983081212

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [1013(minus1)minusradic Sc322

13

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus86069813 ( minus 1)1048617] ( minus 1)

minus 1013+radic Sc32

213

er 98308010383891057306 Sc

2

1057306 +991770131048617

minus9831311013 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus 1) + [10486168 minus 11983081 12(minus1)minusradic Pre1047296 1222

12

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069812 ( minus 1)1048617] ( minus 1)+ [10486168 minus 11983081 12(minus1)+radic Pre1047296 12

2212

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069812 ( minus 1)1048617] ( minus 1) (983092983088)

and or isothermal

10486161038389 983081 = 7 +87

= 2 int0

12(minus) er 86069812 ( minus ) minusminus24

1057306 1

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13

+ 19 int0

13(minus)er 86069813 ( minus ) minusminus24

1057306 13

minus 20 int012(minus) er 86069812 ( minus ) minusminus

2

41057306 12 minus 2113minusradic13Sc

213 er 98308010383891057306 Sc

21057306 minus991770131048617+ 2113 er 98308010383891057306 Sc

21057306 1048617

minus 2113+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617

8

= 2112minusradic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 minus991770121048617minus 2112 er 9830801038389991770Pre1047296

21057306 1048617

+ 2112+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 812minusradic 12Pre1047296 2

12

er 9830801038389991770Pre1047296

2

1057306 minus991770121048617

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

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Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 417

983092 Abstract and Applied Analysis

1038389lowast = 1038389991770]0 lowast = 0

lowast

1048616lowast

983081 = 0

() (983089983090)

by eliminating the star notations into (983092) (983094) and (983089983088) weobtain

= 210383892 +Gr1103925+Gm907317minusminus

Pre1047296 1103925 =

2110392510383892

907317 =

1

Sc

29073171038389

2

+Sr211039251038389

2

(983089983091)

where Pre1047296 = Pr(1+) is the effective Prandtl number [983090983096Equation (10)] and

Gr = 1103925 10486161103925 minus 1103925infin983081 ]

30

Gm = 1048616907317 minus 907317infin983081 ]

3

0

= 2

00

Sc = ] = ]0 0 = ]2

0

Sr = 1 10486161103925 minus 1103925infin9830811048616907317 minus 907317infin983081 ]

(983089983092)

are the Grasho number modi1047297ed Grasho number magneticparameter Schmidt number the inverse permeability param-

eter orthe porous medium the characteristic time andSoretnumber respectively

Te nondimensional initial and boundary conditions are

10486161038389 0983081 = 0110392510486161038389 0983081 = 090731710486161038389 0983081 = 0

forall1038389 ge 0

1038389

=0

= ()

1103925 (0 ) = 0 lt le 1

1103925 (0 ) = 1 gt 1

907317 (0

) = 1

907317 (infin ) = 01103925 (infin ) = 0 (infin ) = 0

gt 0(983089983093)

3 Solution of the Problem

o solve (983089983091) under conditions (983089983093) by taking Laplace trans-orm technique we obtained

10486161038389983081 = 21048616103838998308110383892 +Gr110392510486161038389983081+Gm90731710486161038389983081

minus10486161038389983081minus10486161038389983081 (983089983094)

1103925 10486161038389 983081 = 1

Pre1047296 2110392510486161038389983081

10383892 (983089983095)

907317 10486161038389 983081 = 1

Sc 290731710486161038389983081

10383892 + Sr

211039251048616103838998308110383892

(983089983096)

with boundary conditions

907317 1048616infin 983081 = 0907317 10486160 983081 = 1 1103925 1048616infin 983081 = 0 1048616infin 983081 = 0

104861610383899830811038389=0

= 1048616983081

1103925 10486160 983081 = 1 minus minus2

(983089983097)

Solving (983089983095) using (983089983097) we get

1103925 10486161038389 983081 = 12 minusradicPre1047296 minus minus

2 minusradicPre1047296 (983090983088)

by inverse Laplace transorm giving

1103925 10486161038389983081 = 10486161038389983081minus 10486161038389 minus1

983081 ( minus1

) (983090983089)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 517

Abstract and Applied Analysis 983093

where

10486161038389 983081 = 983080Pre1047296 10383892

2 + 1048617 er 983080991770Pre1047296 1038389

21057306 1048617

minusradicPre1047296

1038389 exp983080minusPre1047296

10383892

4 1048617 1103925104861610383899830811038389

=0 = 2991770Pre1047296 1057306 8520081057306 minus1057306 minus 1( minus 1)852009

(983090983090)

which is the Nusselt number Error complementary errorunctions o Gauss [983091983088] are denoted by er (sdot) and er (sdot)

Solution o (983089983096) under boundary conditions (983089983097) yields

907317 10486161038389 983081 = 1minusradic Sc + 18 10486161 minus minus9830812 minus991770 Sc991770

minus 18 10486161 minus minus9830812 minusradic Pre1047296 991770 (983090983091)

by taking inverse Laplace transorm giving

907317 10486161038389 983081 = er 98308010383891057306 Sc

21057306 1048617+ 18983080983080+ Sc10383892

2 1048617 er

sdot 98308010383891057306 Sc

21057306 1048617minus 10383891057306 Sc1057306 1057306 minus2Sc41048617minus 18983080983080

+ Pre1047296

10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617 minus 1038389991770Pre1047296

1057306 1057306 sdot minus2Pre1047296 41048617+98313118983080983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617

minus 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)1048617983133 ( minus 1)

minus 98313118

983080983080minus1

+ Sc10383892

2

1048617er

983080 10383891057306 Sc

2

1057306 minus1

1048617minus 10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)1048617983133 ( minus 1)

(983090983092)

907317104861610383899830811038389

=0 = minus21057306 Sc minus 3181057306 Sc1057306

minus983131minus21057306 Sc minus 3181057306 Sc ( minus 1)991770 ( minus 1) 983133 ( minus 1)

(983090983093)

which is the corresponding mass transer rate also known asSherwood number

Te solution o (983089983094) under boundary conditions (983089983097)results in

10486161038389 983081 = 21057306 10486161 minus minus983081

2

1048616 minus 1

983081991770+1

minusradic+1

+ 41057306 1048616 minus 3983081991770 + 1

minusradic+1

minus 1048616983081991770 + 1

minusradic+1

+ 61057306 10486161 minus minus9830812 1048616 minus 5983081991770+1

minusradic+1

minus 71057306 10486161 minus minus9830812 1048616 minus 1983081991770+1

minusradic+1

minus 8 10486161 minus minus

9830812 1048616 minus 1983081 minusradicPre1047296

+ 11 10486161 minus minus9830812 1048616 minus 1983081

minusradic Pre1047296

minus 10 10486161 minus minus9830812 1048616 minus 5983081

minusradic Sc minus 9 1048616 minus 3983081 minusradic Sc

(983090983094)

which upon inverse Laplace transorm results in

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983090983095)

where (1038389) corresponds to convective part o velocity which is de1047297ned as

10486161038389 983081 = 1 +2 (983090983096)

1 = 2 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 minus1048667852059 2

10486161983081321057306 sdot intminus1

0

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

85205921

sdot intminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 617

983094 Abstract and Applied Analysis

minus 7 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 +1048667

852059

7

10486161

98308132

1057306 sdotintminus10

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) minus 1048667

85205971

sdotintminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 6 int0

3(minus) er 8520089917703 ( minus )8520091048616398308132 minus 21057306 minus 1057306 3

sdot minus1minus24s1057306 minus1048667852059

610486163983081321057306

sdotintminus10

3(minus1minus)minus1minus24 er 8520089917703 ( minus 1 minus )8520091057306 1048669

852061

sdot ( minus 1) + 1048667852059 6

3sdotintminus1

0

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 4 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

+ 81 983080 + Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306

1048617

minus98313181 983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617983133 (

minus 1) + 983131811038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 93minusradic3Sc

23 er 98308010383891057306 Sc

21057306 minus99177031048617 + 93

sdot er 98308010383891057306 Sc

2

1057306 1048617minus 103minusradic Sc3

2

23

sdot er 98308010383891057306 Sc

21057306 minus99177031048617 minus 93+radic 3Sc

23sdot er 98308010383891057306 Sc

21057306 +99177031048617 (983090983097)

2 = 103 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 1minusradic Pre1047296 122

1

er 9830801038389991770Pre1047296

21057306 minus99177011048617

minus 10486168 minus 11983081 1+radic Pre1047296 12

21

er 9830801038389991770Pre1047296

2

1057306 +99177011048617

+ 98313110310383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 103

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 1023

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 1198308111038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

1

sdot er 9830801038389991770Pre1047296

2

1057306 1048617minus 98313110486168 minus 11983081

21

er 9830801038389991770Pre1047296

2

1057306 minus11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc322

3

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 103+radic Sc322

3

er 98308010383891057306 Sc

21057306 +99177031048617minus 98313110

3

983080minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

2

1057306 minus11048617983133 (

minus 1) + [10486168 minus 11983081 1(minus1)minusradic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus8606981 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 1(minus1)+radic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2

991770( minus1

) +8606981 ( minus 1)1048617] ( minus 1)

(983091983088)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983095

and (1038389) is mechanical part o velocity de1047297ned as

10486161038389 983081 = minus 11057306 int

0

( minus ) minus1minus241057306 (983091983089)

where

1 = 1

Pre1047296 minus 1

2 = Gr991770Pre1047296 Pre1047296 minus 1

3 = 1

Sc minus 1

4 = Gm1057306 Sc

Sc minus 1

6 = GmScSrPre1047296 1057306 ScSc minus 1

7 = GmScSrPre1047296 991770Pre1047296 Pre1047296 minus 1

8 = Gr

Pre1047296 minus 1

9 = Gm

Sc minus 1

10 = GmScSrPre1047296 Sc

minus1

11 = GmScSrPre1047296 Pre1047296 minus 1

12 =

Pre1047296 minus 1

13 = Sc minus 1

15 =

Pre1047296 minus 1

16

= Sc minus 1 18 = SrScPre1047296

1 = +19 = GmSc32SrPre1047296 1048616Pre1047296 minus Sc983081 (Sc minus 1)

20 = GmPr32e1047296

SrSc

1048616Pre1047296 minus Sc983081 (Sc minus 1) 21 = GmPre1047296 SrSc

1048616Pre1047296

minusSc

983081 (Sc

minus1

)

(983091983090)

4 Plate with Constant Temperature

Te solution o (983089983095) under boundary conditions (983089983097) orconstant temperature yields

1103925 10486161038389 983081 = er 9830801038389991770Pre1047296

21057306 1048617 1103925 (0 )1038389 = minus 991770Pre1047296 1057306

(983091983091)

Te solution o (983089983094) under boundary conditions (983089983097) orconstant temperature is

10486161038389 983081 = 21057306 1048616 minus 1983081 991770 + 1

minusradic+1

+ 41057306 1048616 minus 3983081991770+1

minusradic+1

minus 1048616983081991770+1

minusradic+1

+ 191057306 1048616 minus 3983081991770+1

minusradic +1minus 8 1048616 minus 1983081

minusradicPre1047296

minus 20

1057306 1048616 minus 1983081991770+1 minusradic+1

minus 21 1048616 minus 3983081 minusradic Sc + 21 1048616 minus 1983081

minusradic Pre1047296

minus 9 1048616 minus 3983081 minusradic Sc

(983091983092)

upon inverse Laplace transorm

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983091983093)

where

10486161038389 983081 = 3 +4 (983091983094)

3

= 2int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

+ 4int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983096 Abstract and Applied Analysis

+ 19 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

minus 20 int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

minus 213minusradic 3Sc

23 er 98308010383891057306 Sc21057306 minus 99177031048617

+ 213 er 98308010383891057306 Sc

21057306 1048617

minus 213+radic3Sc23 er 98308010383891057306 Sc

21057306 +99177031048617 (983091983095)

4

= 211minusradic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 minus99177011048617minus 211 er 9830801038389991770Pre1047296

21057306 1048617

+ 211+radic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 81minusradic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 minus99177011048617

+ 8

1er

9830801038389991770Pre1047296

21057306 1048617minus 81+radic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 93minusradic3Sc23 er 98308010383891057306 Sc

21057306 minus99177031048617+ 93 er 98308010383891057306 Sc

21057306 1048617

minus 93+radic 3Sc

23er

98308010383891057306 Sc

21057306 +9917703

1048617

(983091983096)

5 Limiting Cases

Here some limiting cases are presented

983093983089 Solution in the Absence o Porous Effects ( rarr 0 )From (983090983089) and (983090983092) it is seen that the temperature 1047297eldsand concentration 1047297elds are not affected by the inversepermeability parameter or the porous medium Hencethe velocities or both case o the plate are as

10486161038389 983081 =

10486161038389983081+

10486161038389983081 (983091983097)

where

10486161038389 983081 = 5 +65 = 2int

0

12(minus) er 85200899177012 ( minus )852009

104861612

98308132 minus 21057306 minus

1057306 12

sdot minusminus241057306 minus1048667852059

2104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059212

sdotintminus1

0 8520082

1057306 minus1

minus852009 minusminus24

1057306 1048669852061 ( minus 1)minus 7 int

0

12(minus) er 85200899177012 ( minus )85200910486161298308132 minus 21057306 minus 1057306 12

sdot minusminus241057306 +1048667852059

7104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )852009

1057306 1048669

852061sdot ( minus 1) minus1048667852059

712

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int0

13(minus) er 85200899177013 ( minus )85200910486161398308132 minus 21057306 minus 1057306 13

sdot minusminus24

1057306 minus1048667852059 6

104861613983081321057306 sdotintminus1

0

13(minus1minus)minusminus24 er 85200899177013 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059613

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983097

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13 + 812 983080 +

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617

minus 983131 812 983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296 21057306 minus 11048617983133 (

minus 1) + 983131 8121038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 913minusradic 13Sc

23 er 98308010383891057306 Sc

21057306 minus991770131048617 + 913

sdot er 98308010383891057306 Sc

21057306 1048617minus 1013minusradic Sc3

223

sdot er 98308010383891057306 Sc

21057306 minus991770131048617 minus 9

13+radic 13Sc

213sdot er 98308010383891057306 Sc

21057306 +991770131048617 6 = 1013 983080+

Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus98313110213

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 12minusradic Pre1047296 12

2212

er

9830801038389991770Pre1047296

21057306 minus99177012

1048617minus 10486168 minus 11983081 12+radic Pre1047296 12

2212

er 9830801038389991770Pre1047296

21057306 +991770121048617+9831311013

10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1013sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 102

13

er 98308010383891057306 Sc

21057306 1048617minus 10486168 minus 11983081

12

1038389991770Pre1047296 1057306

1057306 minus2Pre1047296 4 + 10486168 minus 11983081

212

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 11983081212

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [1013(minus1)minusradic Sc322

13

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus86069813 ( minus 1)1048617] ( minus 1)

minus 1013+radic Sc32

213

er 98308010383891057306 Sc

2

1057306 +991770131048617

minus9831311013 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus 1) + [10486168 minus 11983081 12(minus1)minusradic Pre1047296 1222

12

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069812 ( minus 1)1048617] ( minus 1)+ [10486168 minus 11983081 12(minus1)+radic Pre1047296 12

2212

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069812 ( minus 1)1048617] ( minus 1) (983092983088)

and or isothermal

10486161038389 983081 = 7 +87

= 2 int0

12(minus) er 86069812 ( minus ) minusminus24

1057306 1

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13

+ 19 int0

13(minus)er 86069813 ( minus ) minusminus24

1057306 13

minus 20 int012(minus) er 86069812 ( minus ) minusminus

2

41057306 12 minus 2113minusradic13Sc

213 er 98308010383891057306 Sc

21057306 minus991770131048617+ 2113 er 98308010383891057306 Sc

21057306 1048617

minus 2113+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617

8

= 2112minusradic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 minus991770121048617minus 2112 er 9830801038389991770Pre1047296

21057306 1048617

+ 2112+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 812minusradic 12Pre1047296 2

12

er 9830801038389991770Pre1047296

2

1057306 minus991770121048617

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1317

Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 517

Abstract and Applied Analysis 983093

where

10486161038389 983081 = 983080Pre1047296 10383892

2 + 1048617 er 983080991770Pre1047296 1038389

21057306 1048617

minusradicPre1047296

1038389 exp983080minusPre1047296

10383892

4 1048617 1103925104861610383899830811038389

=0 = 2991770Pre1047296 1057306 8520081057306 minus1057306 minus 1( minus 1)852009

(983090983090)

which is the Nusselt number Error complementary errorunctions o Gauss [983091983088] are denoted by er (sdot) and er (sdot)

Solution o (983089983096) under boundary conditions (983089983097) yields

907317 10486161038389 983081 = 1minusradic Sc + 18 10486161 minus minus9830812 minus991770 Sc991770

minus 18 10486161 minus minus9830812 minusradic Pre1047296 991770 (983090983091)

by taking inverse Laplace transorm giving

907317 10486161038389 983081 = er 98308010383891057306 Sc

21057306 1048617+ 18983080983080+ Sc10383892

2 1048617 er

sdot 98308010383891057306 Sc

21057306 1048617minus 10383891057306 Sc1057306 1057306 minus2Sc41048617minus 18983080983080

+ Pre1047296

10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617 minus 1038389991770Pre1047296

1057306 1057306 sdot minus2Pre1047296 41048617+98313118983080983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617

minus 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)1048617983133 ( minus 1)

minus 98313118

983080983080minus1

+ Sc10383892

2

1048617er

983080 10383891057306 Sc

2

1057306 minus1

1048617minus 10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)1048617983133 ( minus 1)

(983090983092)

907317104861610383899830811038389

=0 = minus21057306 Sc minus 3181057306 Sc1057306

minus983131minus21057306 Sc minus 3181057306 Sc ( minus 1)991770 ( minus 1) 983133 ( minus 1)

(983090983093)

which is the corresponding mass transer rate also known asSherwood number

Te solution o (983089983094) under boundary conditions (983089983097)results in

10486161038389 983081 = 21057306 10486161 minus minus983081

2

1048616 minus 1

983081991770+1

minusradic+1

+ 41057306 1048616 minus 3983081991770 + 1

minusradic+1

minus 1048616983081991770 + 1

minusradic+1

+ 61057306 10486161 minus minus9830812 1048616 minus 5983081991770+1

minusradic+1

minus 71057306 10486161 minus minus9830812 1048616 minus 1983081991770+1

minusradic+1

minus 8 10486161 minus minus

9830812 1048616 minus 1983081 minusradicPre1047296

+ 11 10486161 minus minus9830812 1048616 minus 1983081

minusradic Pre1047296

minus 10 10486161 minus minus9830812 1048616 minus 5983081

minusradic Sc minus 9 1048616 minus 3983081 minusradic Sc

(983090983094)

which upon inverse Laplace transorm results in

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983090983095)

where (1038389) corresponds to convective part o velocity which is de1047297ned as

10486161038389 983081 = 1 +2 (983090983096)

1 = 2 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 minus1048667852059 2

10486161983081321057306 sdot intminus1

0

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

85205921

sdot intminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 617

983094 Abstract and Applied Analysis

minus 7 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 +1048667

852059

7

10486161

98308132

1057306 sdotintminus10

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) minus 1048667

85205971

sdotintminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 6 int0

3(minus) er 8520089917703 ( minus )8520091048616398308132 minus 21057306 minus 1057306 3

sdot minus1minus24s1057306 minus1048667852059

610486163983081321057306

sdotintminus10

3(minus1minus)minus1minus24 er 8520089917703 ( minus 1 minus )8520091057306 1048669

852061

sdot ( minus 1) + 1048667852059 6

3sdotintminus1

0

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 4 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

+ 81 983080 + Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306

1048617

minus98313181 983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617983133 (

minus 1) + 983131811038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 93minusradic3Sc

23 er 98308010383891057306 Sc

21057306 minus99177031048617 + 93

sdot er 98308010383891057306 Sc

2

1057306 1048617minus 103minusradic Sc3

2

23

sdot er 98308010383891057306 Sc

21057306 minus99177031048617 minus 93+radic 3Sc

23sdot er 98308010383891057306 Sc

21057306 +99177031048617 (983090983097)

2 = 103 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 1minusradic Pre1047296 122

1

er 9830801038389991770Pre1047296

21057306 minus99177011048617

minus 10486168 minus 11983081 1+radic Pre1047296 12

21

er 9830801038389991770Pre1047296

2

1057306 +99177011048617

+ 98313110310383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 103

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 1023

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 1198308111038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

1

sdot er 9830801038389991770Pre1047296

2

1057306 1048617minus 98313110486168 minus 11983081

21

er 9830801038389991770Pre1047296

2

1057306 minus11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc322

3

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 103+radic Sc322

3

er 98308010383891057306 Sc

21057306 +99177031048617minus 98313110

3

983080minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

2

1057306 minus11048617983133 (

minus 1) + [10486168 minus 11983081 1(minus1)minusradic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus8606981 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 1(minus1)+radic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2

991770( minus1

) +8606981 ( minus 1)1048617] ( minus 1)

(983091983088)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 717

Abstract and Applied Analysis 983095

and (1038389) is mechanical part o velocity de1047297ned as

10486161038389 983081 = minus 11057306 int

0

( minus ) minus1minus241057306 (983091983089)

where

1 = 1

Pre1047296 minus 1

2 = Gr991770Pre1047296 Pre1047296 minus 1

3 = 1

Sc minus 1

4 = Gm1057306 Sc

Sc minus 1

6 = GmScSrPre1047296 1057306 ScSc minus 1

7 = GmScSrPre1047296 991770Pre1047296 Pre1047296 minus 1

8 = Gr

Pre1047296 minus 1

9 = Gm

Sc minus 1

10 = GmScSrPre1047296 Sc

minus1

11 = GmScSrPre1047296 Pre1047296 minus 1

12 =

Pre1047296 minus 1

13 = Sc minus 1

15 =

Pre1047296 minus 1

16

= Sc minus 1 18 = SrScPre1047296

1 = +19 = GmSc32SrPre1047296 1048616Pre1047296 minus Sc983081 (Sc minus 1)

20 = GmPr32e1047296

SrSc

1048616Pre1047296 minus Sc983081 (Sc minus 1) 21 = GmPre1047296 SrSc

1048616Pre1047296

minusSc

983081 (Sc

minus1

)

(983091983090)

4 Plate with Constant Temperature

Te solution o (983089983095) under boundary conditions (983089983097) orconstant temperature yields

1103925 10486161038389 983081 = er 9830801038389991770Pre1047296

21057306 1048617 1103925 (0 )1038389 = minus 991770Pre1047296 1057306

(983091983091)

Te solution o (983089983094) under boundary conditions (983089983097) orconstant temperature is

10486161038389 983081 = 21057306 1048616 minus 1983081 991770 + 1

minusradic+1

+ 41057306 1048616 minus 3983081991770+1

minusradic+1

minus 1048616983081991770+1

minusradic+1

+ 191057306 1048616 minus 3983081991770+1

minusradic +1minus 8 1048616 minus 1983081

minusradicPre1047296

minus 20

1057306 1048616 minus 1983081991770+1 minusradic+1

minus 21 1048616 minus 3983081 minusradic Sc + 21 1048616 minus 1983081

minusradic Pre1047296

minus 9 1048616 minus 3983081 minusradic Sc

(983091983092)

upon inverse Laplace transorm

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983091983093)

where

10486161038389 983081 = 3 +4 (983091983094)

3

= 2int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

+ 4int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983096 Abstract and Applied Analysis

+ 19 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

minus 20 int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

minus 213minusradic 3Sc

23 er 98308010383891057306 Sc21057306 minus 99177031048617

+ 213 er 98308010383891057306 Sc

21057306 1048617

minus 213+radic3Sc23 er 98308010383891057306 Sc

21057306 +99177031048617 (983091983095)

4

= 211minusradic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 minus99177011048617minus 211 er 9830801038389991770Pre1047296

21057306 1048617

+ 211+radic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 81minusradic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 minus99177011048617

+ 8

1er

9830801038389991770Pre1047296

21057306 1048617minus 81+radic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 93minusradic3Sc23 er 98308010383891057306 Sc

21057306 minus99177031048617+ 93 er 98308010383891057306 Sc

21057306 1048617

minus 93+radic 3Sc

23er

98308010383891057306 Sc

21057306 +9917703

1048617

(983091983096)

5 Limiting Cases

Here some limiting cases are presented

983093983089 Solution in the Absence o Porous Effects ( rarr 0 )From (983090983089) and (983090983092) it is seen that the temperature 1047297eldsand concentration 1047297elds are not affected by the inversepermeability parameter or the porous medium Hencethe velocities or both case o the plate are as

10486161038389 983081 =

10486161038389983081+

10486161038389983081 (983091983097)

where

10486161038389 983081 = 5 +65 = 2int

0

12(minus) er 85200899177012 ( minus )852009

104861612

98308132 minus 21057306 minus

1057306 12

sdot minusminus241057306 minus1048667852059

2104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059212

sdotintminus1

0 8520082

1057306 minus1

minus852009 minusminus24

1057306 1048669852061 ( minus 1)minus 7 int

0

12(minus) er 85200899177012 ( minus )85200910486161298308132 minus 21057306 minus 1057306 12

sdot minusminus241057306 +1048667852059

7104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )852009

1057306 1048669

852061sdot ( minus 1) minus1048667852059

712

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int0

13(minus) er 85200899177013 ( minus )85200910486161398308132 minus 21057306 minus 1057306 13

sdot minusminus24

1057306 minus1048667852059 6

104861613983081321057306 sdotintminus1

0

13(minus1minus)minusminus24 er 85200899177013 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059613

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983097

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13 + 812 983080 +

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617

minus 983131 812 983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296 21057306 minus 11048617983133 (

minus 1) + 983131 8121038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 913minusradic 13Sc

23 er 98308010383891057306 Sc

21057306 minus991770131048617 + 913

sdot er 98308010383891057306 Sc

21057306 1048617minus 1013minusradic Sc3

223

sdot er 98308010383891057306 Sc

21057306 minus991770131048617 minus 9

13+radic 13Sc

213sdot er 98308010383891057306 Sc

21057306 +991770131048617 6 = 1013 983080+

Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus98313110213

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 12minusradic Pre1047296 12

2212

er

9830801038389991770Pre1047296

21057306 minus99177012

1048617minus 10486168 minus 11983081 12+radic Pre1047296 12

2212

er 9830801038389991770Pre1047296

21057306 +991770121048617+9831311013

10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1013sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 102

13

er 98308010383891057306 Sc

21057306 1048617minus 10486168 minus 11983081

12

1038389991770Pre1047296 1057306

1057306 minus2Pre1047296 4 + 10486168 minus 11983081

212

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 11983081212

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [1013(minus1)minusradic Sc322

13

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus86069813 ( minus 1)1048617] ( minus 1)

minus 1013+radic Sc32

213

er 98308010383891057306 Sc

2

1057306 +991770131048617

minus9831311013 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus 1) + [10486168 minus 11983081 12(minus1)minusradic Pre1047296 1222

12

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069812 ( minus 1)1048617] ( minus 1)+ [10486168 minus 11983081 12(minus1)+radic Pre1047296 12

2212

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069812 ( minus 1)1048617] ( minus 1) (983092983088)

and or isothermal

10486161038389 983081 = 7 +87

= 2 int0

12(minus) er 86069812 ( minus ) minusminus24

1057306 1

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13

+ 19 int0

13(minus)er 86069813 ( minus ) minusminus24

1057306 13

minus 20 int012(minus) er 86069812 ( minus ) minusminus

2

41057306 12 minus 2113minusradic13Sc

213 er 98308010383891057306 Sc

21057306 minus991770131048617+ 2113 er 98308010383891057306 Sc

21057306 1048617

minus 2113+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617

8

= 2112minusradic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 minus991770121048617minus 2112 er 9830801038389991770Pre1047296

21057306 1048617

+ 2112+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 812minusradic 12Pre1047296 2

12

er 9830801038389991770Pre1047296

2

1057306 minus991770121048617

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

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Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

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Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 617

983094 Abstract and Applied Analysis

minus 7 int0

1(minus) er 8520089917701 ( minus )8520091048616198308132 minus 21057306 minus 1057306 1

sdot minus1minus24

1057306 +1048667

852059

7

10486161

98308132

1057306 sdotintminus10

1(minus1minus)minus1minus24 er 8520089917701 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) minus 1048667

85205971

sdotintminus10

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 6 int0

3(minus) er 8520089917703 ( minus )8520091048616398308132 minus 21057306 minus 1057306 3

sdot minus1minus24s1057306 minus1048667852059

610486163983081321057306

sdotintminus10

3(minus1minus)minus1minus24 er 8520089917703 ( minus 1 minus )8520091057306 1048669

852061

sdot ( minus 1) + 1048667852059 6

3sdotintminus1

0

85200821057306 minus 1 minus852009 minus1minus241057306 1048669

852061 ( minus 1)

+ 4 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

+ 81 983080 + Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306

1048617

minus98313181 983080minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 minus 11048617983133 (

minus 1) + 983131811038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 93minusradic3Sc

23 er 98308010383891057306 Sc

21057306 minus99177031048617 + 93

sdot er 98308010383891057306 Sc

2

1057306 1048617minus 103minusradic Sc3

2

23

sdot er 98308010383891057306 Sc

21057306 minus99177031048617 minus 93+radic 3Sc

23sdot er 98308010383891057306 Sc

21057306 +99177031048617 (983090983097)

2 = 103 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 1minusradic Pre1047296 122

1

er 9830801038389991770Pre1047296

21057306 minus99177011048617

minus 10486168 minus 11983081 1+radic Pre1047296 12

21

er 9830801038389991770Pre1047296

2

1057306 +99177011048617

+ 98313110310383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 103

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 1023

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 1198308111038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

1

sdot er 9830801038389991770Pre1047296

2

1057306 1048617minus 98313110486168 minus 11983081

21

er 9830801038389991770Pre1047296

2

1057306 minus11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc322

3

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 103+radic Sc322

3

er 98308010383891057306 Sc

21057306 +99177031048617minus 98313110

3

983080minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

2

1057306 minus11048617983133 (

minus 1) + [10486168 minus 11983081 1(minus1)minusradic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus8606981 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 1(minus1)+radic Pre1047296 122

1

sdot er 983080 1038389991770Pre1047296

2

991770( minus1

) +8606981 ( minus 1)1048617] ( minus 1)

(983091983088)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 717

Abstract and Applied Analysis 983095

and (1038389) is mechanical part o velocity de1047297ned as

10486161038389 983081 = minus 11057306 int

0

( minus ) minus1minus241057306 (983091983089)

where

1 = 1

Pre1047296 minus 1

2 = Gr991770Pre1047296 Pre1047296 minus 1

3 = 1

Sc minus 1

4 = Gm1057306 Sc

Sc minus 1

6 = GmScSrPre1047296 1057306 ScSc minus 1

7 = GmScSrPre1047296 991770Pre1047296 Pre1047296 minus 1

8 = Gr

Pre1047296 minus 1

9 = Gm

Sc minus 1

10 = GmScSrPre1047296 Sc

minus1

11 = GmScSrPre1047296 Pre1047296 minus 1

12 =

Pre1047296 minus 1

13 = Sc minus 1

15 =

Pre1047296 minus 1

16

= Sc minus 1 18 = SrScPre1047296

1 = +19 = GmSc32SrPre1047296 1048616Pre1047296 minus Sc983081 (Sc minus 1)

20 = GmPr32e1047296

SrSc

1048616Pre1047296 minus Sc983081 (Sc minus 1) 21 = GmPre1047296 SrSc

1048616Pre1047296

minusSc

983081 (Sc

minus1

)

(983091983090)

4 Plate with Constant Temperature

Te solution o (983089983095) under boundary conditions (983089983097) orconstant temperature yields

1103925 10486161038389 983081 = er 9830801038389991770Pre1047296

21057306 1048617 1103925 (0 )1038389 = minus 991770Pre1047296 1057306

(983091983091)

Te solution o (983089983094) under boundary conditions (983089983097) orconstant temperature is

10486161038389 983081 = 21057306 1048616 minus 1983081 991770 + 1

minusradic+1

+ 41057306 1048616 minus 3983081991770+1

minusradic+1

minus 1048616983081991770+1

minusradic+1

+ 191057306 1048616 minus 3983081991770+1

minusradic +1minus 8 1048616 minus 1983081

minusradicPre1047296

minus 20

1057306 1048616 minus 1983081991770+1 minusradic+1

minus 21 1048616 minus 3983081 minusradic Sc + 21 1048616 minus 1983081

minusradic Pre1047296

minus 9 1048616 minus 3983081 minusradic Sc

(983091983092)

upon inverse Laplace transorm

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983091983093)

where

10486161038389 983081 = 3 +4 (983091983094)

3

= 2int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

+ 4int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 817

983096 Abstract and Applied Analysis

+ 19 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

minus 20 int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

minus 213minusradic 3Sc

23 er 98308010383891057306 Sc21057306 minus 99177031048617

+ 213 er 98308010383891057306 Sc

21057306 1048617

minus 213+radic3Sc23 er 98308010383891057306 Sc

21057306 +99177031048617 (983091983095)

4

= 211minusradic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 minus99177011048617minus 211 er 9830801038389991770Pre1047296

21057306 1048617

+ 211+radic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 81minusradic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 minus99177011048617

+ 8

1er

9830801038389991770Pre1047296

21057306 1048617minus 81+radic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 93minusradic3Sc23 er 98308010383891057306 Sc

21057306 minus99177031048617+ 93 er 98308010383891057306 Sc

21057306 1048617

minus 93+radic 3Sc

23er

98308010383891057306 Sc

21057306 +9917703

1048617

(983091983096)

5 Limiting Cases

Here some limiting cases are presented

983093983089 Solution in the Absence o Porous Effects ( rarr 0 )From (983090983089) and (983090983092) it is seen that the temperature 1047297eldsand concentration 1047297elds are not affected by the inversepermeability parameter or the porous medium Hencethe velocities or both case o the plate are as

10486161038389 983081 =

10486161038389983081+

10486161038389983081 (983091983097)

where

10486161038389 983081 = 5 +65 = 2int

0

12(minus) er 85200899177012 ( minus )852009

104861612

98308132 minus 21057306 minus

1057306 12

sdot minusminus241057306 minus1048667852059

2104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059212

sdotintminus1

0 8520082

1057306 minus1

minus852009 minusminus24

1057306 1048669852061 ( minus 1)minus 7 int

0

12(minus) er 85200899177012 ( minus )85200910486161298308132 minus 21057306 minus 1057306 12

sdot minusminus241057306 +1048667852059

7104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )852009

1057306 1048669

852061sdot ( minus 1) minus1048667852059

712

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int0

13(minus) er 85200899177013 ( minus )85200910486161398308132 minus 21057306 minus 1057306 13

sdot minusminus24

1057306 minus1048667852059 6

104861613983081321057306 sdotintminus1

0

13(minus1minus)minusminus24 er 85200899177013 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059613

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 917

Abstract and Applied Analysis 983097

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13 + 812 983080 +

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617

minus 983131 812 983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296 21057306 minus 11048617983133 (

minus 1) + 983131 8121038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 913minusradic 13Sc

23 er 98308010383891057306 Sc

21057306 minus991770131048617 + 913

sdot er 98308010383891057306 Sc

21057306 1048617minus 1013minusradic Sc3

223

sdot er 98308010383891057306 Sc

21057306 minus991770131048617 minus 9

13+radic 13Sc

213sdot er 98308010383891057306 Sc

21057306 +991770131048617 6 = 1013 983080+

Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus98313110213

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 12minusradic Pre1047296 12

2212

er

9830801038389991770Pre1047296

21057306 minus99177012

1048617minus 10486168 minus 11983081 12+radic Pre1047296 12

2212

er 9830801038389991770Pre1047296

21057306 +991770121048617+9831311013

10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1013sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 102

13

er 98308010383891057306 Sc

21057306 1048617minus 10486168 minus 11983081

12

1038389991770Pre1047296 1057306

1057306 minus2Pre1047296 4 + 10486168 minus 11983081

212

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 11983081212

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [1013(minus1)minusradic Sc322

13

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus86069813 ( minus 1)1048617] ( minus 1)

minus 1013+radic Sc32

213

er 98308010383891057306 Sc

2

1057306 +991770131048617

minus9831311013 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus 1) + [10486168 minus 11983081 12(minus1)minusradic Pre1047296 1222

12

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069812 ( minus 1)1048617] ( minus 1)+ [10486168 minus 11983081 12(minus1)+radic Pre1047296 12

2212

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069812 ( minus 1)1048617] ( minus 1) (983092983088)

and or isothermal

10486161038389 983081 = 7 +87

= 2 int0

12(minus) er 86069812 ( minus ) minusminus24

1057306 1

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13

+ 19 int0

13(minus)er 86069813 ( minus ) minusminus24

1057306 13

minus 20 int012(minus) er 86069812 ( minus ) minusminus

2

41057306 12 minus 2113minusradic13Sc

213 er 98308010383891057306 Sc

21057306 minus991770131048617+ 2113 er 98308010383891057306 Sc

21057306 1048617

minus 2113+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617

8

= 2112minusradic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 minus991770121048617minus 2112 er 9830801038389991770Pre1047296

21057306 1048617

+ 2112+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 812minusradic 12Pre1047296 2

12

er 9830801038389991770Pre1047296

2

1057306 minus991770121048617

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1017

983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1117

Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1217

983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 717

Abstract and Applied Analysis 983095

and (1038389) is mechanical part o velocity de1047297ned as

10486161038389 983081 = minus 11057306 int

0

( minus ) minus1minus241057306 (983091983089)

where

1 = 1

Pre1047296 minus 1

2 = Gr991770Pre1047296 Pre1047296 minus 1

3 = 1

Sc minus 1

4 = Gm1057306 Sc

Sc minus 1

6 = GmScSrPre1047296 1057306 ScSc minus 1

7 = GmScSrPre1047296 991770Pre1047296 Pre1047296 minus 1

8 = Gr

Pre1047296 minus 1

9 = Gm

Sc minus 1

10 = GmScSrPre1047296 Sc

minus1

11 = GmScSrPre1047296 Pre1047296 minus 1

12 =

Pre1047296 minus 1

13 = Sc minus 1

15 =

Pre1047296 minus 1

16

= Sc minus 1 18 = SrScPre1047296

1 = +19 = GmSc32SrPre1047296 1048616Pre1047296 minus Sc983081 (Sc minus 1)

20 = GmPr32e1047296

SrSc

1048616Pre1047296 minus Sc983081 (Sc minus 1) 21 = GmPre1047296 SrSc

1048616Pre1047296

minusSc

983081 (Sc

minus1

)

(983091983090)

4 Plate with Constant Temperature

Te solution o (983089983095) under boundary conditions (983089983097) orconstant temperature yields

1103925 10486161038389 983081 = er 9830801038389991770Pre1047296

21057306 1048617 1103925 (0 )1038389 = minus 991770Pre1047296 1057306

(983091983091)

Te solution o (983089983094) under boundary conditions (983089983097) orconstant temperature is

10486161038389 983081 = 21057306 1048616 minus 1983081 991770 + 1

minusradic+1

+ 41057306 1048616 minus 3983081991770+1

minusradic+1

minus 1048616983081991770+1

minusradic+1

+ 191057306 1048616 minus 3983081991770+1

minusradic +1minus 8 1048616 minus 1983081

minusradicPre1047296

minus 20

1057306 1048616 minus 1983081991770+1 minusradic+1

minus 21 1048616 minus 3983081 minusradic Sc + 21 1048616 minus 1983081

minusradic Pre1047296

minus 9 1048616 minus 3983081 minusradic Sc

(983091983092)

upon inverse Laplace transorm

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983091983093)

where

10486161038389 983081 = 3 +4 (983091983094)

3

= 2int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

+ 4int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 817

983096 Abstract and Applied Analysis

+ 19 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

minus 20 int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

minus 213minusradic 3Sc

23 er 98308010383891057306 Sc21057306 minus 99177031048617

+ 213 er 98308010383891057306 Sc

21057306 1048617

minus 213+radic3Sc23 er 98308010383891057306 Sc

21057306 +99177031048617 (983091983095)

4

= 211minusradic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 minus99177011048617minus 211 er 9830801038389991770Pre1047296

21057306 1048617

+ 211+radic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 81minusradic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 minus99177011048617

+ 8

1er

9830801038389991770Pre1047296

21057306 1048617minus 81+radic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 93minusradic3Sc23 er 98308010383891057306 Sc

21057306 minus99177031048617+ 93 er 98308010383891057306 Sc

21057306 1048617

minus 93+radic 3Sc

23er

98308010383891057306 Sc

21057306 +9917703

1048617

(983091983096)

5 Limiting Cases

Here some limiting cases are presented

983093983089 Solution in the Absence o Porous Effects ( rarr 0 )From (983090983089) and (983090983092) it is seen that the temperature 1047297eldsand concentration 1047297elds are not affected by the inversepermeability parameter or the porous medium Hencethe velocities or both case o the plate are as

10486161038389 983081 =

10486161038389983081+

10486161038389983081 (983091983097)

where

10486161038389 983081 = 5 +65 = 2int

0

12(minus) er 85200899177012 ( minus )852009

104861612

98308132 minus 21057306 minus

1057306 12

sdot minusminus241057306 minus1048667852059

2104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059212

sdotintminus1

0 8520082

1057306 minus1

minus852009 minusminus24

1057306 1048669852061 ( minus 1)minus 7 int

0

12(minus) er 85200899177012 ( minus )85200910486161298308132 minus 21057306 minus 1057306 12

sdot minusminus241057306 +1048667852059

7104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )852009

1057306 1048669

852061sdot ( minus 1) minus1048667852059

712

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int0

13(minus) er 85200899177013 ( minus )85200910486161398308132 minus 21057306 minus 1057306 13

sdot minusminus24

1057306 minus1048667852059 6

104861613983081321057306 sdotintminus1

0

13(minus1minus)minusminus24 er 85200899177013 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059613

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 917

Abstract and Applied Analysis 983097

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13 + 812 983080 +

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617

minus 983131 812 983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296 21057306 minus 11048617983133 (

minus 1) + 983131 8121038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 913minusradic 13Sc

23 er 98308010383891057306 Sc

21057306 minus991770131048617 + 913

sdot er 98308010383891057306 Sc

21057306 1048617minus 1013minusradic Sc3

223

sdot er 98308010383891057306 Sc

21057306 minus991770131048617 minus 9

13+radic 13Sc

213sdot er 98308010383891057306 Sc

21057306 +991770131048617 6 = 1013 983080+

Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus98313110213

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 12minusradic Pre1047296 12

2212

er

9830801038389991770Pre1047296

21057306 minus99177012

1048617minus 10486168 minus 11983081 12+radic Pre1047296 12

2212

er 9830801038389991770Pre1047296

21057306 +991770121048617+9831311013

10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1013sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 102

13

er 98308010383891057306 Sc

21057306 1048617minus 10486168 minus 11983081

12

1038389991770Pre1047296 1057306

1057306 minus2Pre1047296 4 + 10486168 minus 11983081

212

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 11983081212

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [1013(minus1)minusradic Sc322

13

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus86069813 ( minus 1)1048617] ( minus 1)

minus 1013+radic Sc32

213

er 98308010383891057306 Sc

2

1057306 +991770131048617

minus9831311013 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus 1) + [10486168 minus 11983081 12(minus1)minusradic Pre1047296 1222

12

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069812 ( minus 1)1048617] ( minus 1)+ [10486168 minus 11983081 12(minus1)+radic Pre1047296 12

2212

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069812 ( minus 1)1048617] ( minus 1) (983092983088)

and or isothermal

10486161038389 983081 = 7 +87

= 2 int0

12(minus) er 86069812 ( minus ) minusminus24

1057306 1

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13

+ 19 int0

13(minus)er 86069813 ( minus ) minusminus24

1057306 13

minus 20 int012(minus) er 86069812 ( minus ) minusminus

2

41057306 12 minus 2113minusradic13Sc

213 er 98308010383891057306 Sc

21057306 minus991770131048617+ 2113 er 98308010383891057306 Sc

21057306 1048617

minus 2113+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617

8

= 2112minusradic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 minus991770121048617minus 2112 er 9830801038389991770Pre1047296

21057306 1048617

+ 2112+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 812minusradic 12Pre1047296 2

12

er 9830801038389991770Pre1047296

2

1057306 minus991770121048617

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1017

983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1117

Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1217

983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1317

Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 817

983096 Abstract and Applied Analysis

+ 19 int0

3(minus) er 8606983 ( minus ) minus1minus24

1057306 3

minus 20 int0

1(minus) er 8606981 ( minus ) minus1minus24

1057306 1

minus 213minusradic 3Sc

23 er 98308010383891057306 Sc21057306 minus 99177031048617

+ 213 er 98308010383891057306 Sc

21057306 1048617

minus 213+radic3Sc23 er 98308010383891057306 Sc

21057306 +99177031048617 (983091983095)

4

= 211minusradic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 minus99177011048617minus 211 er 9830801038389991770Pre1047296

21057306 1048617

+ 211+radic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 81minusradic 1Pre1047296 21 er 9830801038389991770Pre1047296

21057306 minus99177011048617

+ 8

1er

9830801038389991770Pre1047296

21057306 1048617minus 81+radic 1Pre1047296

21 er 9830801038389991770Pre1047296

21057306 +99177011048617

minus 93minusradic3Sc23 er 98308010383891057306 Sc

21057306 minus99177031048617+ 93 er 98308010383891057306 Sc

21057306 1048617

minus 93+radic 3Sc

23er

98308010383891057306 Sc

21057306 +9917703

1048617

(983091983096)

5 Limiting Cases

Here some limiting cases are presented

983093983089 Solution in the Absence o Porous Effects ( rarr 0 )From (983090983089) and (983090983092) it is seen that the temperature 1047297eldsand concentration 1047297elds are not affected by the inversepermeability parameter or the porous medium Hencethe velocities or both case o the plate are as

10486161038389 983081 =

10486161038389983081+

10486161038389983081 (983091983097)

where

10486161038389 983081 = 5 +65 = 2int

0

12(minus) er 85200899177012 ( minus )852009

104861612

98308132 minus 21057306 minus

1057306 12

sdot minusminus241057306 minus1048667852059

2104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059212

sdotintminus1

0 8520082

1057306 minus1

minus852009 minusminus24

1057306 1048669852061 ( minus 1)minus 7 int

0

12(minus) er 85200899177012 ( minus )85200910486161298308132 minus 21057306 minus 1057306 12

sdot minusminus241057306 +1048667852059

7104861612983081321057306 sdotintminus1

0

12(minus1minus)minusminus24 er 85200899177012 ( minus 1 minus )852009

1057306 1048669

852061sdot ( minus 1) minus1048667852059

712

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int0

13(minus) er 85200899177013 ( minus )85200910486161398308132 minus 21057306 minus 1057306 13

sdot minusminus24

1057306 minus1048667852059 6

104861613983081321057306 sdotintminus1

0

13(minus1minus)minusminus24 er 85200899177013 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059613

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 917

Abstract and Applied Analysis 983097

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13 + 812 983080 +

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617

minus 983131 812 983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296 21057306 minus 11048617983133 (

minus 1) + 983131 8121038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 913minusradic 13Sc

23 er 98308010383891057306 Sc

21057306 minus991770131048617 + 913

sdot er 98308010383891057306 Sc

21057306 1048617minus 1013minusradic Sc3

223

sdot er 98308010383891057306 Sc

21057306 minus991770131048617 minus 9

13+radic 13Sc

213sdot er 98308010383891057306 Sc

21057306 +991770131048617 6 = 1013 983080+

Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus98313110213

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 12minusradic Pre1047296 12

2212

er

9830801038389991770Pre1047296

21057306 minus99177012

1048617minus 10486168 minus 11983081 12+radic Pre1047296 12

2212

er 9830801038389991770Pre1047296

21057306 +991770121048617+9831311013

10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1013sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 102

13

er 98308010383891057306 Sc

21057306 1048617minus 10486168 minus 11983081

12

1038389991770Pre1047296 1057306

1057306 minus2Pre1047296 4 + 10486168 minus 11983081

212

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 11983081212

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [1013(minus1)minusradic Sc322

13

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus86069813 ( minus 1)1048617] ( minus 1)

minus 1013+radic Sc32

213

er 98308010383891057306 Sc

2

1057306 +991770131048617

minus9831311013 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus 1) + [10486168 minus 11983081 12(minus1)minusradic Pre1047296 1222

12

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069812 ( minus 1)1048617] ( minus 1)+ [10486168 minus 11983081 12(minus1)+radic Pre1047296 12

2212

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069812 ( minus 1)1048617] ( minus 1) (983092983088)

and or isothermal

10486161038389 983081 = 7 +87

= 2 int0

12(minus) er 86069812 ( minus ) minusminus24

1057306 1

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13

+ 19 int0

13(minus)er 86069813 ( minus ) minusminus24

1057306 13

minus 20 int012(minus) er 86069812 ( minus ) minusminus

2

41057306 12 minus 2113minusradic13Sc

213 er 98308010383891057306 Sc

21057306 minus991770131048617+ 2113 er 98308010383891057306 Sc

21057306 1048617

minus 2113+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617

8

= 2112minusradic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 minus991770121048617minus 2112 er 9830801038389991770Pre1047296

21057306 1048617

+ 2112+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 812minusradic 12Pre1047296 2

12

er 9830801038389991770Pre1047296

2

1057306 minus991770121048617

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1017

983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1117

Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1217

983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

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Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 917

Abstract and Applied Analysis 983097

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13 + 812 983080 +

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617

minus 983131 812 983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296 21057306 minus 11048617983133 (

minus 1) + 983131 8121038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 913minusradic 13Sc

23 er 98308010383891057306 Sc

21057306 minus991770131048617 + 913

sdot er 98308010383891057306 Sc

21057306 1048617minus 1013minusradic Sc3

223

sdot er 98308010383891057306 Sc

21057306 minus991770131048617 minus 9

13+radic 13Sc

213sdot er 98308010383891057306 Sc

21057306 +991770131048617 6 = 1013 983080+

Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus98313110213

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)

minus 10486168 minus 11983081 12minusradic Pre1047296 12

2212

er

9830801038389991770Pre1047296

21057306 minus99177012

1048617minus 10486168 minus 11983081 12+radic Pre1047296 12

2212

er 9830801038389991770Pre1047296

21057306 +991770121048617+9831311013

10383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1013sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 102

13

er 98308010383891057306 Sc

21057306 1048617minus 10486168 minus 11983081

12

1038389991770Pre1047296 1057306

1057306 minus2Pre1047296 4 + 10486168 minus 11983081

212

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 11983081212

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [1013(minus1)minusradic Sc322

13

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus86069813 ( minus 1)1048617] ( minus 1)

minus 1013+radic Sc32

213

er 98308010383891057306 Sc

2

1057306 +991770131048617

minus9831311013 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus 1) + [10486168 minus 11983081 12(minus1)minusradic Pre1047296 1222

12

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069812 ( minus 1)1048617] ( minus 1)+ [10486168 minus 11983081 12(minus1)+radic Pre1047296 12

2212

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069812 ( minus 1)1048617] ( minus 1) (983092983088)

and or isothermal

10486161038389 983081 = 7 +87

= 2 int0

12(minus) er 86069812 ( minus ) minusminus24

1057306 1

+ 4 int0

13(minus) er 86069813 ( minus ) minusminus24

1057306 13

+ 19 int0

13(minus)er 86069813 ( minus ) minusminus24

1057306 13

minus 20 int012(minus) er 86069812 ( minus ) minusminus

2

41057306 12 minus 2113minusradic13Sc

213 er 98308010383891057306 Sc

21057306 minus991770131048617+ 2113 er 98308010383891057306 Sc

21057306 1048617

minus 2113+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617

8

= 2112minusradic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 minus991770121048617minus 2112 er 9830801038389991770Pre1047296

21057306 1048617

+ 2112+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 812minusradic 12Pre1047296 2

12

er 9830801038389991770Pre1047296

2

1057306 minus991770121048617

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1017

983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1117

Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1217

983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1317

Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1017

983089983088 Abstract and Applied Analysis

+ 812 er 9830801038389991770Pre1047296

21057306 1048617

minus 812+radic 12Pre1047296 212 er 9830801038389991770Pre1047296

21057306 +991770121048617

minus 913minusradic 13Sc

213 er 98308010383891057306 Sc21057306 minus991770131048617

+ 913 er 98308010383891057306 Sc

21057306 1048617

minus 913+radic13Sc213 er 98308010383891057306 Sc

21057306 +991770131048617 (983092983089)

983093983090 Solutions in theAbsence o Free Convection Consider thatthe 1047298uid 1047298ow is due to bounding plate and the corresponding

Gr and Gm are zero In this case the 1047298uid motion is only by the mechanical part o velocities given by (983091983089)

983093983091 Solutions in the Absence o Mechanical Effects Let ussuppose that the in1047297nite plate is motionless at every time thatis () is zero or all values o Mechanical parts areequivalently zero in both cases o the plates Tereore themotion in the 1047298uid is a result o the ree convection which iscaused due to the buoyancy orces Hence the 1047298uid velocitiesare only representedby their convective parts obtained in (983090983096)and (983091983094)

983093983092 Solution in the Absence o Magnetic Parameter (

rarr 0 )

From (983090983089) and (983090983092) the temperature 1047297elds and concentration1047297elds are not affected by and the velocities in the absenceo are given by

10486161038389 983081 = 10486161038389983081+ 10486161038389983081 (983092983090)

where

10486161038389 983081 = 9 +109 = 2 int

0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus24

1057306 minus1048667852059 2

104861615983081321057306 sdot intminus1

0

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) + 1048667

852059215

sdot intminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061

( minus 1)

minus 7 int0

15(minus) er 85200899177015 ( minus )85200910486161598308132 minus 21057306 minus 1057306 15

sdot minusminus241057306 +1048667

852059

7

104861615

98308132

1057306 sdotintminus10

15(minus1minus)minusminus24 er 85200899177015 ( minus 1 minus )8520091057306 1048669852061

sdot ( minus 1) minus1048667852059

715

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669

852061 ( minus 1)

+ 6 int

0 16(minus) er

85200899177016

( minus )85200910486161698308132 minus 2

1057306 minus 1057306 16 sdot minusminus24s1057306 minus1048667

8520596104861616983081321057306

sdotintminus10

16(minus1minus)minusminus24 er 85200899177016 ( minus 1 minus )8520091057306 1048669

852061sdot ( minus 1) +1048667

852059

616

sdotintminus10

85200821057306 minus 1 minus852009 minusminus241057306 1048669852061 ( minus 1)

+ 4 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16 + 815 983080+

Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

21057306 1048617minus983131 8

15

983080 minus 1+ Pre1047296 10383892

2 1048617 er 9830801038389991770Pre1047296

2

1057306 minus11048617983133 (

minus 1) + 983131 815 1038389991770Pre1047296 1057306 minus 11057306 minus2Pre1047296 4(minus1)983133 ( minus 1)

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus991770161048617 + 916

sdot er 98308010383891057306 Sc

21057306 1048617minus 1016minusradic Sc16

2216

sdot er 98308010383891057306 Sc

2

1057306 minus991770161048617 minus 916+radic 16Sc2

16

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1117

Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1217

983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1317

Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1117

Abstract and Applied Analysis 983089983089

sdot er 98308010383891057306 Sc

21057306 +991770161048617

10 = 1016 983080 + Sc10383892

2 1048617 er 98308010383891057306 Sc

21057306 1048617minus 9831311023

sdot er 983080 10383891057306 Sc

21057306 minus 11048617983133 ( minus 1)minus 10486168 minus 11983081 15minusradic Pre1047296 15

2215

er 9830801038389991770Pre1047296

21057306 minus991770151048617

minus 10486168 minus 11983081 1+radic Pre1047296 122

15

er 9830801038389991770Pre1047296

21057306 +99177011048617

+983131101610383891057306 Sc1057306 minus 11057306 minus2Sc4(minus1)983133 ( minus 1) minus 1016

sdot 10383891057306 Sc1057306 1057306 minus2Sc4 + 10216

er 98308010383891057306 Sc

21057306 1048617

minus 10486168 minus 11983081151038389991770Pre1047296 1057306 1057306 minus2Pre1047296 4 + 10486168 minus 119830812

15

sdot er 9830801038389991770Pre1047296

21057306 1048617minus98313110486168 minus 119830812

15

er 9830801038389991770Pre1047296

21057306 minus 11048617983133

sdot ( minus 1) + [103(minus1)minusradic Sc32

216

sdot er 983080 10383891057306 Sc

2991770( minus 1) minus8606983 ( minus 1)1048617] ( minus 1)

minus 1016+radic Sc1622

16

er 98308010383891057306 Sc

21057306 + 991770161048617

minus 9831311016 983080 minus 1+ Sc10383892

2 1048617 er 983080 10383891057306 Sc

21057306 minus 11048617983133 (

minus1

) + [10486168 minus 11983081 15(minus1)minusradic Pre1047296 15

22

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) minus86069815 ( minus 1)1048617] ( minus 1)

+ [10486168 minus 11983081 15(minus1)+radic Pre1047296 1522

15

sdot er 983080 1038389991770Pre1047296

2991770( minus 1) +86069815 ( minus 1)1048617] ( minus 1) (983092983091)

and or isothermal

10486161038389 983081 = 11 +1211

= 2int

0 15(minus)

er 86069815 ( minus ) minusminus24

1057306 15 + 4 int

0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

+ 19 int0

16(minus) er 86069816 ( minus ) minusminus24

1057306 16

minus 20 int0

15(minus) er 86069815 ( minus ) minusminus24

1057306 15

minus 21

16minusradic 16Sc

216 er 98308010383891057306 Sc

21057306 minus991770161048617+ 2116 er 98308010383891057306 Sc

21057306 1048617

minus 2116+radic 16Sc216 er 98308010383891057306 Sc

21057306 +991770161048617 12

= 2115minusradic 15Pre1047296

2

15

er

9830801038389991770Pre1047296

21057306 minus99177015

1048617minus 2115 er 9830801038389991770Pre1047296

21057306 1048617

+ 2115+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 815minusradic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 minus991770151048617

+ 815

er 9830801038389991770Pre1047296

2

1057306 1048617

minus 815+radic15Pre1047296 215 er 9830801038389991770Pre1047296

21057306 +991770151048617

minus 916minusradic 16Sc216 er 98308010383891057306 Sc

21057306 minus 991770161048617

+ 916 er 98308010383891057306 Sc

21057306 1048617

minus 916+radic 16Sc2

16

er 98308010383891057306 Sc

2

1057306 + 991770161048617

(983092983092)

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1217

983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1317

Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1217

983089983090 Abstract and Applied Analysis

6 Special Cases

Te obtained velocities in Section 983091 are more general Hencesome special cases o the present results are presented Inorder to learn much more about the physical importance o the problem technical relevance o these cases is ound inliterature

983094983089 Case-I () = () Consider () = () where and (sdot) are a dimensionless constant and the unit stepunction Te in1047297nite vertical plate applies a constant shearstress to the 1047298uid which is observed afer time = 0 Tereis no change seen in convective part o the velocity while themechanical part has been changed as

10486161038389 983081 = minus 1057306 int

0

minus24minus11057306 (983092983093)

equivalently

10486161038389 983081 = minus 9917701

minusradic 1 + 21057306 intinfin991770 minus242minus12 (983092983094)

or = 0 = 0 Moreover i we take = 0 (983092983093) reducesto the orm

10486161038389 983081 = minus 860698 minus1057306 + 21057306 int

infin

991770 minus242minus2 (983092983095)

which is equivalent to [983091983088 Equation (28)] with the correction

o 860698Moreover in the absence o both

and

(983092983093) is

10486161038389 983081 = minus 1057306 int

0

minus241057306 (983092983096)

983094983090 Case-II () = sin() We take () = sin() inwhich an oscillating shear stress to the 1047298uid is applied by theplate where denotes the dimensionless requency o theshear stress Te convective part o velocity remains the samelike the 1047297rst case however the mechanical part changed andtakes the orm

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1

1057306 (983092983097)

Furthermore it can be written as a sum o the steady-stateand transient solutions

10486161038389 983081 = 10486161038389 983081 + 10486161038389 983081 (983093983088)

where

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus11057306

10486161038389 983081 =

1057306 intinfin

sin ( minus) minus24minus1

1057306

(983093983089)

By taking = 0 the steady-state component reduces to [983091983088Equation (35)]

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus24minus1057306 (983093983090)

In addition when = 0 (983093983090) results in

10486161038389 983081 = minus 1057306 int

0

sin (minus) minus241057306 (983093983091)

which can be written in simpli1047297ed orm as

10486161038389983081= 1057306 exp minus1038389radic

2 cosminus1038389radic

2 +

4 (983093983092)

equivalent to [983090983094 Equation (33)]

7 Graphical Results and Discussion

In this section the obtained solutions are numerically studiedor determining the effects o several involved parameterssuch as magnetic parameter the inverse permeability parameter or the porous medium effective Prandtlnumber Pre1047296 Grasho number Gr modi1047297ed Grasho numberGm Schmidt number Sc shear stress and Soret numberSr It is depicted rom Figure 983090 that with increasing valueso Gr the velocity pro1047297les increase Physically this scenariois important due to the act that an increase in Gr givesrise to buoyancy effects which results in more induced1047298ows In Figure 983091 the velocity pro1047297les or different values o modi1047297ed Grasho number Gm are displayed It can be seenthat the velocity and boundary layer thickness decrease aswith increasing distance rom the leading edge Howeverthe velocity and boundary layer thickness increase withincreasing values o Gm In Figure 983092 the velocity pro1047297les areshown or different values o Schmidt number Sc Here the

values o Sc are chosen as 983088983090983090 983088983094983088 and 983088983097983094 to representthe presence o species by hydrogen water vapor and carbondioxide respectively It is observed that the velocity decreaseswith increasing Schmidt number Physically this reers to thephenomenon that increasing Schmidt number implies thedominance o the viscous orces over the diffusional effectsAs a result the 1047298ow will be thereore decelerated with a

rise in Schmidt number Te velocity pro1047297les or different values o magnetic parameter are presented in Figure 983093Magnetic 1047297eld ranges rom 983088 to 983090 Velocity is decreasingwith increasing values o Physically it is true because therictional orce is directly proportional to thereore 1047298uid1047298ow tends to resist and decrease in velocity is observed It isurther observed that in the absence o the magnetic 1047297eldthe MHD effect approaches zero and hydrodynamic 1047298ow is observed Te effects o inverse permeability parameter on the velocity pro1047297les are presented in Figure 983094 It isound that velocity decreases with increasing Tis resultis in good agreement with the previous study [983091983088] Figure 983091Te effects o the wall shear stress

are shown in Figure 983095

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1317

Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1317

Abstract and Applied Analysis 983089983091

0 2 4 6

05

1

15

u ( y

t ) Gr = 05 08 1

y

F983145983143983157983154983141 983090 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gr when the plate applies constantshear stress = minus025

0 2 4 6

05

1

15

u ( y

t )

Gm = 05 08 11

y

F983145983143983157983154983141 983091 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o Gm when the plate appliesconstant shear stress = minus025

0 1 2 3 4

05

1

15

u ( y

t )

y

022 060 096Sc =

F983145983143983157983154983141 983092 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 0 7 = 12 and different values o Sc when the plate appliesconstant shear stress = minus025

Te velocity o 1047298uid is ound to decrease with increasing Te in1047298uence o the effective Prandtl number Pre1047296 on velocity pro1047297les are presented in Figure 983096 It is observed that the

velocity is a decreasing unction with respect to Pre1047296 Tesegraphical results are in accordance with [983091983088] Figure 983090 Te

velocity pro1047297les or different values o Soret number Sr areshown in Figure 983097 It is noticed that thevelocityincreases with

0 2 4 6

1

2

u ( y

t )

y

M = 0 1 2

F983145983143983157983154983141 983093 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 07 = 12 and different values o when the plate appliesconstant shear stress

= minus0

25

increasing values o Sr Te temperature variations against1038389 or various values o effective Prandtl number are shownin Figure 983089983088 Te signi1047297cant decrease o the temperature isound as a result o an increase o the effective Prandtlnumber Te 1047298uid temperature is maximum at the boundary while it has minimum value as ar rom the plate Teconcentration pro1047297les or variuos values o Schmidt numberSc are drawn in Figure 983089983089 It is clear rom this 1047297gure that theconcentration pro1047297les and the concentration boundary layerthickness decrease with increasing values o Sc Physically itis valid as molecular diffusivity decreases while increasing the

value Sc

8 Conclusions

Te aim o this research is to examine the unsteady MHDree convection 1047298ow o an incompressible Newtonian 1047298uidover an in1047297nite plate Te ramped wall temperature thermaldiffusion and wall shear stress to the 1047298uid are taken Closed-orm solutions or velocity 1047297eld temperature 1047297eld andconcentration 1047297eld are obtained using the Laplace transormtechnique Te results are shown in terms o the comple-mentary error unction Te initial and boundary conditionsare satis1047297ed by the obtained results and different parameters

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1417

983089983092 Abstract and Applied Analysis

0 2 4 6

05

1

15

y

u ( y

t ) Kp =1 2 3

F983145983143983157983154983141 983094 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07) = 12 and different values o when the plate applies constantshear stress = minus025

0 2 4 6

1

2

f = minus1 minus2 minus3

y

u ( y

t )

F983145983143983157983154983141 983095 Velocity pro1047297les or Pre1047296 = 0350 ( = 1 Pr = 07)

= 0

7

= 1

2 and different values o constant shear stress

0 5 10 15

05

1

15

y

Preff = 00075 0350 350 u ( y

t )

F983145983143983157983154983141 983096 Velocity pro1047297les or = 07 = 09 and different valueso Pre1047296 when the plate applies constant shear stress = minus025

are plotted Te 1047298uid velocity (1038389 ) is written as a sum o two components that is convective and mechanical Te

velocity solution is urther written as a sum o the steady-stateand transient solutions (1038389) respectively (1038389) in thesecond special case in which the plate applies an oscillatingshear stressto the1047298uidMagneticparameter slows the1047298uid1047298ow

0 2 4 6

05

1

y

Sr = 001 09 2 u ( y

t )

F983145983143983157983154983141 983097 Velocity pro1047297les or = 07 = 09 and differentvalueso Sr when the plate applies constant shear stress = minus025

0 2 4 6 8 10

02

04

06

08

y

0116 0175 035Preff = T ( y

t )

F983145983143983157983154983141 983089983088 emperature pro1047297le or = 12 and different values o Pre1047296

0 2 4 6 8 10

05

1

022 060 096Sc =

y

C

( y

t )

F983145983143983157983154983141 983089983089 Concentration pro1047297les or = 12 and different values o Sc

Conflict of Interests

Te authors o this paper do not have any con1047298icto interests

Acknowledgments

Te authors would like to acknowledge MOE and ResearchManagement Centre UM or the 1047297nancial support throughVotes nos 983088983092H983090983095 and 983092F983090983093983093 or this research

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1517

Abstract and Applied Analysis 983089983093

References

[983089] J Singh ldquoFlow o elasto-viscous 1047298uid past an accelerated porousplaterdquo Astrophysics and Space Science vol 983089983088983094 no 983089 pp 983092983089ndash983092983093983089983097983096983092

[983090] M Narahari ldquoEffects o thermal radiation and ree convectioncurrents on the unsteady Couette 1047298ow between two vertical

parallel plates with constant heat 1047298ux at one boundaryrdquo WSEASransactions on Heat and Mass ranser vol 983093 no 983089 pp 983090983089ndash983091983088983090983088983089983088

[983091] M Narahari and M Y Nayan ldquoFree convection 1047298ow past animpulsively started in1047297nite vertical plate with Newtonian heat-ing in the presence o thermal radiation and mass diffusionrdquourkish Journal o Engineering and Environmental Sciences vol983091983093 no 983091 pp 983089983096983095ndash983089983097983096 983090983088983089983089

[983092] M Narahari and A Ishak ldquoRadiation effects on ree convection1047298ow near a moving vertical plate with newtonian heatingrdquo Journal o Applied Sciences vol 983089983089 no 983095 pp 983089983088983097983094ndash983089983089983088983092 983090983088983089983089

[983093] R C Chaudhary and P Jain ldquoUnsteady ree convectionboundary-layer 1047298ow past an impulsively started vertical surace

with Newtonian heatingrdquo Romanian Journal o Physics vol 983093983089p 983097983089983089 983090983088983088983094

[983094] U N Das R Deka and V M Soundalgekar ldquoEffects o masstranser on 1047298ow past an impulsively started in1047297nite vertical platewith constant heat 1047298ux and chemical reactionrdquo Forschung imIngenieurwesen vol 983094983088 no 983089983088 pp 983090983096983092ndash983090983096983095 983089983097983097983092

[983095] V M Soundalgekar N S Birajdar andV K Darwhekar ldquoMass-transer effects on the 1047298ow past an impulsively started in1047297nite vertical plate with variable temperature or constant heat 1047298uxrdquo Astrophysics and Space Science vol 983089983088983088 no 983089-983090 pp 983089983093983097ndash983089983094983092983089983097983096983092

[983096] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoOn 1047298ow and heat transer o a viscous incompressible 1047298uid past

an impulsively started vertical isothermal platerdquo International Journal o Termal Sciences vol 983092983088 no 983091 pp 983090983097983095ndash983091983088983090 983090983088983088983089

[983097] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transer effects on 1047298ow past an impulsively started vertical platerdquo Acta Mechanica vol 983089983092983094 no 983089-983090 pp 983089ndash983096 983090983088983088983089

[983089983088] S S Das S R Biswal U K ripathy and P Das ldquoMasstranser effects on unsteady hydromagnetic convective 1047298ow pasta vertical porous plate in a porous medium with heat sourcerdquo Journal o Applied Fluid Mechanics vol 983092no 983092pp983097983089ndash983089983088983088983090983088983089983089

[983089983089] C J oki and J N okis ldquoExact solutions or the unsteady ree convection 1047298ows on a porous plate with time-dependentheatingrdquo Zeitschrif ur Angewandte Mathematik und Mechanik vol 983096983095 no 983089 pp 983092ndash983089983091 983090983088983088983095

[983089983090] N Senapati R K Dhal and K Das ldquoEffects o chemicalreaction on ree convection MHD 1047298ow through porous mediumbounded by vertical surace with slip 1047298ow regionrdquo American Journal o Computational and Applied Mathematics vol 983090 pp983089983090983092ndash983089983091983093 983090983088983089983090

[983089983091] I Khan K Fakhar and S Sha1047297e ldquoMagnetohydrodynamic reeconvection 1047298ow past an oscillating plate embedded in a porousmediumrdquo Journal o the Physical Society o Japan vol 983096983088 no 983089983088Article ID 983089983088983092983092983088983089 983090983088983089983089

[983089983092] A Khan I Khan F Ali S Ulhaq and S Sha1047297e ldquoEffects o wall shear stress on unsteady MHD conjugate 1047298ow in a porousmedium with ramped wall temperaturerdquo PLoS ONE vol 983097 no983091 Article ID e983097983088983090983096983088 983090983088983089983092

[983089983093] A Hussanan Z Ismail I Khan A G Hussein and S Sha1047297eldquoUnsteady boundary layer MHD ree convection 1047298ow in aporous medium with constant mass diffusion and Newtonianheatingrdquo Te European Physical Journal Plus vol 983089983090983097 article 983092983094983090983088983089983092

[983089983094] R N Barik G C Dash and M Kar ldquoFree convection heatand mass transer MHD 1047298ow in a vertical porous channel inthe presence o chemical reactionrdquo Journal o Fluids vol 983090983088983089983091Article ID 983090983097983095983092983097983091 983089983092 pages 983090983088983089983091

[983089983095] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heat and Mass ranser vol 983092983089 no 983093 pp 983092983093983097ndash983092983094983092 983090983088983088983093

[983089983096] M Narahari O A Beg and S K Ghosh ldquoMathematicalmodelling o mass transer and ree convection current effectson unsteady viscous 1047298ow with ramped wall temperaturerdquo World Journal o Mechanics vol 983089 pp 983089983095983094ndash983089983096983092 983090983088983089983089

[983089983097] V Rajesh ldquoChemical reaction and radiation effects on thetransient MHD ree convection 1047298ow o dissipative 1047298uid past anIn1047297nite vertical porous plate with ramped wall temperaturerdquoChemical Industry amp ChemicalEngineering Quarterly vol 983089983095 no

983090 pp 983089983096983097ndash983089983097983096 983090983088983089983089[983090983088] R R Patra S Das R N Jana and S K Ghosh ldquoransient

approach to radiative heat transer ree convection 1047298ow withramped wall temperaturerdquo Journal o Applied Fluid Mechanics vol 983093 no 983090 pp 983097ndash983089983091 983090983088983089983090

[983090983089] G S Seth G K Mahatoo and S Sarkar ldquoEffects o Hallcurrent and rotation on MHD natural convection 1047298ow pastan impulsively moving vertical plate with ramped temperaturein the presence o thermal diffusion with heat absorptionrdquoInternational Journal o Energy amp echnology vol 983093 pp 983089ndash983089983090983090983088983089983091

[983090983090] Samiulhaq I Khan F Ali and S Sha1047297e ldquoMHD ree convection1047298ow in a porous medium with thermal diffusion and ramped

wall temperaturerdquo Journal o the Physical Society o Japan vol983096983089 no 983092 Article ID 983088983092983092983092983088983089 983090983088983089983090

[983090983091] N Ghara S Das S L Maji and R N Jana ldquoEffect o radiationon MHD ree convection 1047298ow past an impulsively moving vertical plate with ramped wall temperaturerdquo American Journal o Scienti1047297c and Industrial Research vol 983091 no 983094 pp 983091983095983094ndash983091983096983094983090983088983089983090

[983090983092] S Das C Mandal and R N Jana ldquoEffects o radiation onunsteady Couette 1047298ow between two vertical parallel plates withramped wall temperaturerdquo International Journal o Computer Applications vol 983091983097 pp 983091983092ndash983092983090 983090983088983089983090

[983090983093] C L M H Navier ldquoSur les lois dee mouvement des 1047298uidsrdquo M emoires de lrsquoAcad emie Royale des Sciences de lrsquoInstitut deFrance vol 983094 pp 983091983096983097ndash983092983092983088 983089983096983090983095

[983090983094] C Fetecau C Fetecau and M Rana ldquoGeneral solutions orthe unsteady 1047298ow o second-grade 1047298uids over an in1047297niteplate that applies arbitrary shear to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983094no 983089983090 pp 983095983093983091ndash983095983093983097 983090983088983089983089

[983090983095] Hayat I Khan R Ellahi and C Fetecau ldquoSome MHD 1047298owso a second grade 1047298uid through the porous mediumrdquo Journal o Porous Media vol 983089983089 no 983092 pp 983091983096983097ndash983092983088983088 983090983088983088983096

[983090983096] E Magyari and A Pantokratoras ldquoNote on the effect o thermalradiation in the linearized Rosseland approximation on theheat transer characteristics o various boundary layer 1047298owsrdquoInternational Communications in Heat and Mass ranser vol983091983096 no 983093 pp 983093983093983092ndash983093983093983094 983090983088983089983089

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1617

983089983094 Abstract and Applied Analysis

[983090983097] A David Maxim Gururaj and S P Anjali Devi ldquoMHD bound-ary layer 1047298ow with orced convection past a nonlinearly stretch-ing surace with variable temperature and nonlinear radiationeffectsrdquo International Journal o Development Research vol 983092pp 983095983093ndash983096983088 983090983088983089983092

[983091983088] C Fetecau M Rana and C Fetecau ldquoRadiative and porouseffects on ree convection 1047298ow near a vertical plate that appliesshear stress to the 1047298uidrdquo Zeitschrif ur NaturorschungmdashSection A Journal o Physical Sciences vol 983094983096 no 983089-983090 pp 983089983091983088ndash983089983091983096 983090983088983089983091

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e

7232019 Exact Solutions of Heat and Mass Transfer With MHD Flow in a Porous Medium Under Time Dependent Shear Strehellip

httpslidepdfcomreaderfullexact-solutions-of-heat-and-mass-transfer-with-mhd-flow-in-a-porous-medium 1717

C o p y r i g h t o f A b s t r a c t amp A p p l i e d A n a l y s i s i s t h e p r o p e r t y o f H i n d a w i P u b l i s h i n g C o r p o r a t i o n

a n d i t s c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t

t h e c o p y r i g h t h o l d e r s e x p r e s s w r i t t e n p e r m i s s i o n H o w e v e r u s e r s m a y p r i n t d o w n l o a d o r

e m a i l a r t i c l e s f o r i n d i v i d u a l u s e