Research ArticleInfluence of Thermal Radiation on Unsteady MHD FreeConvection Flow of Jeffrey Fluid over a Vertical Plate withRamped Wall Temperature

Nor Athirah Mohd Zin1 Ilyas Khan2 and Sharidan Shafie1

1Department of Mathematical Sciences Faculty of Science Universiti Teknologi Malaysia 81310 Johor Bahru Malaysia2College of Engineering Majmaah University PO Box 66 Majmaah 11952 Saudi Arabia

Correspondence should be addressed to Ilyas Khan isaidmuedusa

Received 20 September 2015 Revised 22 December 2015 Accepted 31 December 2015

Academic Editor Mohsen Torabi

Copyright copy 2016 Nor Athirah Mohd Zin et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Influence of thermal radiation on unsteady magnetohydrodynamic (MHD) free convection flow of Jeffrey fluid over a vertical platewith ramped wall temperature is studied The Laplace transform technique is used to obtain the analytical solutions Expressionsfor skin friction and Nusselt number are also obtained Results of velocity and temperature distributions are shown graphically forembedded parameters such as Jeffrey fluid parameter 120582 Prandtl number Pr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 It is observed that the amplitude of velocity and temperature profile for isothermal arealways higher than ramped wall temperature

1 Introduction

Investigation of non-Newtonian fluids has become verypopular in this research area due to their wide range inindustrial and technologies applications such as the plasticmanufacture performance of lubricants food processing ormovement of biological fluids Various models have beensuggested to predict and describe the rheological behaviorof non-Newtonian fluids The Jeffrey fluid model is oneof the subfamily of non-Newtonian fluids which has beengiven much attention in the present problem This fluid is arelatively simple viscoelastic non-Newtonian fluidmodel thatexhibits both relaxation and retardation effects [1]

Hayat and Mustafa [2] discussed the effect of the thermalradiation on the unsteady mixed convection flow of Jeffreyfluid past a porous vertical stretching surface analyticallyusing homotopy analysis method Hayat et al [3] extendedthe previous idea to examine the flow of an incompressibleJeffrey fluid over a stretching surface in the presence ofpower heat flux and heat source Shehzad et al [4] obtainedhomotopy solutions for magnetohydrodynamic radiativeflowof an incompressible Jeffrey fluid over a linearly stretched

surface In another investigation Shehzad et al [5] discussedthe three-dimensional hydromagnetic flow of Jeffrey fluidwith nanoparticles where the effects of thermal radiationand internal heat generation are considered Hussain et al[6] conducted heat and mass transfer analysis of two-dimensional hydromagnetic flow of an incompressible Jeffreynanofluid over an exponentially stretching surface in thepresence of thermal radiation viscous dissipation Brownianmotion and thermophoresis effects Very recently the influ-ence of melting heat transfer and thermal radiation onMHDstagnation point flow of an electrically conducting Jeffreyfluid over a stretching sheet with partial surface slip has beenanalyzed numerically by Das et al [7] with the help of Runge-Kutta-Fehlberg method

Besides that Jeffrey fluid also has been studied inperistalsis known as major mechanism of fluid transportespecially in biological system for example urine trans-port from kidney to bladder through ureter movementof chime in the gastrointestinal tract the movement ofspermatozoa in the ducts efferent of the male reproductivetract ovum in the female fallopian tube the locomotion

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6257071 12 pageshttpdxdoiorg10115520166257071

2 Mathematical Problems in Engineering

of some worms and transport of lymph in the lymphaticvessels [8] Abd-Alla et al [9] investigated magnetic fieldand gravity effects on peristaltic transport of a Jeffrey fluidin an asymmetric channel Quite recently Abd-Alla andAbo-Dahab [10] analyzed the influence of magnetic fieldand rotation effects on peristaltic transport of a Jeffrey fluidin asymmetric channel Akram and Nadeem [11] providedexact and closed form Adomian solutions for the peristalticmotion of two-dimensional Jeffrey fluid in an asymmetricchannel under the effects of induced magnetic field and heattransfer

Moreover the influence of heat transfer on MHD oscil-latory flow of Jeffrey fluid in a channel has been investigatedby Kavita et al [12] Idowu et al [13] studied the effects ofheat and mass transfer on unsteady MHD oscillatory flowof Jeffrey fluid in a horizontal channel with chemical reac-tion and solved analytically using perturbation techniqueAli and Asghar [14] presented analytical solution for two-dimensional oscillatory flow inside a rectangular channelfor Jeffrey fluid with small suction They used perturbationWentzel-Kramers-Brillouin and variation of parameter asthe methodologies to solve this complex problem Detaileddiscussions on oscillatory flow of Jeffrey fluid also can befound in Khan [15] and Hayat et al [1 16]

On the other hand the steady boundary layer flow of aJeffrey fluid over a shrinking sheet is highlighted by Nadeemet al [17] Hamad et al [18] presented a study of thermaljump effects on boundary layer flow of a Jeffrey fluid nearthe stagnation point on a stretchingshrinking sheet withvariable thermal conductivity and solved numerically usingfinite difference method Hayat et al [19] considered two-dimensional stagnation point flow of Jeffrey fluid over a con-vectively heated stretching sheet constructed by homotopyanalysis method Other than that Qasim [20] observed thecombined effect of heat and mass transfer in Jeffrey fluidover a stretching sheet in the presence of heat sourceheatsink where the obtained exact solutions are derived by powerseries method using Kummerrsquos confluent hypergeometricfunctions Dalir [21] investigated entropy generation for thesteady two-dimensional laminar forced convection flow andheat transfer of an incompressible Jeffrey non-Newtonianfluid over a linearly stretching impermeable and isothermalsheet numerically by Kellerrsquos box method

Diverse contributions dealing with boundary layer flowover stretching sheet in various situations are well docu-mented in Dalir et al [22] and Hayat et al [23] Hayatet al [24] analyzed the effects of Newtonian heating magne-tohydrodynamics (MHD) in a flow of a Jeffrey fluid over aradially stretching surface In the same year Hayat et al [25]carried out the study of boundary layer stretched flow of aJeffrey fluid subject to the convective boundary conditionShehzad et al [26] analytically discussed magnetohydrody-namics (MHD) three-dimensional flow of Jeffrey fluid in thepresence of Newtonian heating Farooq et al [27] examinedthe combined effects of Joule and Newtonian heating inmagnetohydrodynamic (MHD) flow of Jeffrey fluid over astretching cylinder with heat sourcesink solved analyticallyby homotopy analysis method (HAM) whereas Khan [28]recently implemented the Laplace transform technique to

find the exact solutions for unsteady free convection flow ofa Jeffrey fluid past an infinite isothermal vertical plate

In view of the above analysis so far no study has beenmade to examine the effect of ramped wall temperaturein a Jeffrey fluid flow Thus our main focus is to discussthe influence of thermal radiation on unsteady MHD freeconvection flow of Jeffrey fluid over a vertical plate withramped wall temperature The Laplace transform techniqueis applied in this study in order to find the analytical solutionsfor velocity and temperature profiles Expressions of skinfriction and Nusselt number are also given Graphical resultsare provided and discussed for embedded parameters

2 Mathematical Formulation of the Problem

The unsteady incompressible flow is described by the follow-ing governing equations

nabla sdot V = 0

120588 [120597V120597119905

+ (V sdot nabla)V] = divT + J times B + 120588g

curlE = minus120597B120597119905

curlB = 120583119898J

nabla sdot B = 0

(1)

In the above equations V indicates the velocity field T isthe Cauchy stress tensor 120588 is the fluid density J is the currentdensity B = 119861

0+ 119887 is the total magnetic field where 119861

0and

119887 are the applied and induced magnetic fields respectively gis the gravitational force which is downward in 119909-directionE is the total electric field current and 120583

119898is the magnetic

permeability The constitutive equations for Jeffrey fluid aregiven by

T = minus119901I + S

S =120583

1 + 1205821

[A1+ 1205822(120597A1

120597119905+ (V sdot nabla)A

1)]

(2)

in which minus119901I is the indeterminate part of the stress S isthe extra tensor 120583 is the dynamic viscosity and 120582

1and 120582

2

are the material parameters of the Jeffrey fluid known as theratio of relaxation to retardation times and retardation timerespectively Notice that if 120582

1= 1205822= 0 it will lead to the

expression of an incompressible viscous fluid The Rivlin-Ericksen tensor A

1 is defined by

A1= nablaV + (nablaV)119879 (3)

For unidirectional flow we assume that velocity and stressfields are in the form of

V = V (119910 119905) = 119906 (119910 119905) i

S = S (119910 119905) (4)

where 119906 is the 119909-component of velocity V and i is the unitvector along the 119909-direction of the Cartesian coordinate

Mathematical Problems in Engineering 3

Thermalboundary

layer

Momentumboundary

layer

T(infin t) = Tinfin

x

y

g

0

B0

T(0t)=Tinfin

+(T

wminusTinfin)tt

0T

(0t)=Twt

get 0

Figure 1 Schematic diagram for the flow of Jeffrey fluid past avertical plate

system The stress field satisfies the condition S(119910 0) = 0

which gives the following results

119878119909119909= 119878119910119910= 119878119911119911= 119878119909119911= 119878119911119909= 119878119910119911= 119878119911119910= 0

119878119909119910=

120583

1 + 1205821

(1 + 1205822

120597

120597119905)120597119906

120597119910

(5)

where 119878119909119910

is the nontrivial tangential stressConsider the effect of thermal radiation on unsteady free

convection flow of Jeffrey fluid near the vertical plate situatedin the (119909 119911) plane of a Cartesian coordinate system of 119909 119910and 119911 as presented in Figure 1 It is assumed that at time119905 le 0 both plate and fluid are at rest at constant temperature119879infin At time 119905 gt 0 the temperature of the plate is raised or

lowered to119879infin+(119879119908minus119879infin)(1199051199050)when 119905 le 119905

0 and the constant

temperature 119879119908is maintained at 119905 gt 119905

0

Under the above assumption and applying the usualBoussinesq approximation themomentum and energy equa-tions for unsteady free convection flow of Jeffrey fluid past aninfinite vertical plate are given by

120588120597119906

120597119905=

120583

1 + 1205821

(1 + 1205822

120597

120597119905)1205972

119906

1205971199102+ 120588119892120573 (119879 minus 119879

infin)

minus 1205901198612

0119906

(6)

120588119888119901

120597119879

120597119905= 119896

1205972

119879

1205971199102minus120597119902119903

120597119910 (7)

with appropriate initial and boundary conditions

119906 (119910 0) = 0

119879 (119910 0) = 119879infin

119910 gt 0

119906 (0 119905) = 0 119905 gt 0

119879 (0 119905) = 119879infin+ (119879119908minus 119879infin)119905

1199050

0 lt 119905 lt 1199050

119879 (0 119905) = 119879119908 119905 ge 119905

0

119906 (infin 119905) = 0

119879 (infin 119905) = 119879infin

119905 gt 0

(8)

where 119906 is the fluid velocity in 119909-direction 119879 is the tempera-ture 120588 is the constant density of the fluid 119892 is the accelerationdue to gravity 119888

119901is the specific heat capacity 119896 is the thermal

conductivity119879119908is the wall temperature119879

infinis the free stream

temperature and 119902119903is the radiation heat flux respectively

Using Rosseland approximation the radiation heat flux canbe written as

119902119903= minus

4120590lowast

31198961

1205971198794

120597119910 (9)

where 120590lowast denotes the Stefan-Boltzmann and 1198961is the absorp-

tion coefficient We assume that the temperature differenceswithin the flow are sufficiently small such that 119879

4can be

expanded inTaylor seriesHence expanding1198794about119879

infinand

neglecting higher order terms we get 1198794 asymp 41198793

infin119879 minus 3119879

4

infin

Therefore (7) is simplified to

120588119888119901

120597119879

120597119905= 119896(1 +

16120590lowast

1198793

infin

31198961198961

)1205972

119879

1205971199102 (10)

Introducing the dimensionless variables

119906lowast

=119906

1198800

120578lowast

=1199101198800

120592

120591lowast

=1199051198802

0

120592

120579 =119879 minus 119879infin

119879119908minus 119879infin

(11)

into (6) and (10) yields the following dimensionless equations(lowast notations are dropped for simplicity)

120597119906

120597120591=

1

1 + 1205821

(1 + 120582120597

120597120591)1205972

119906

1205971205782minusHa 119906 + Gr 120579

Pr120597120579120597120591

= (1 + Rd) 1205972

120579

1205971205782

(12)

4 Mathematical Problems in Engineering

where 120582 is the dimensionless Jeffrey fluid parameter Ha isthe Hartmann number Gr is the Grashof number Pr is thePrandtl number and Rd is the radiation parameter which aredenoted as follows

120582 =12058221198802

0

120592

Ha =1205901198612

0120592

12058811988020

Gr =119892120573120592 (119879

119908minus 119879infin)

11988030

Pr =120583119888119901

119896

Rd =16120590lowast

1198793

infin

31198961198961

(13)

Also initial and boundary conditions (8) become

119906 (120578 0) = 0

120579 (120578 0) = 0

120578 gt 0

119906 (0 120591) = 0 120591 gt 0

120579 (0 120591) = 120591 0 lt 120591 le 1

120579 (0 120591) = 1 120591 gt 1

119906 (infin 120591) = 0

120579 (infin 120591) = 0

120591 gt 0

(14)

3 Solution of the Problem

Applying the Laplace transform to (12) and (14) yields thefollowing equations in the transformed (120578 119902)-plane

1198892

119906 (120578 119902)

1198891205782minus (

(1 + 1205821) (Ha + 119902)

1 + 120582119902)119906 (120578 119902)

= minusGr((1 + 120582

1)

1 + 120582119902) 120579 (120578 119902)

(15)

(1 + Rd)1198892

120579 (120578 119902)

1198891205782minus Pr 119902120579 (120578 119902) = 0 (16)

and initial and boundary conditions are transformed asfollows

119906 (0 119902) = 0

120579 (0 119902) =1 minus exp (minus119902)

1199022

119906 (infin 119902) = 0

120579 (infin 119902) = 0

(17)

where 119906(120578 119902) and 120579(120578 119902) are Laplace transforms of 119906(120578 120591) and120579(120578 120591)

Equation (16) subject to boundary conditions (17) has thefollowing solution

120579 (120578 119902) =1 minus exp (minus119902)

1199022exp(minus120578radic

Pr 1199021 + Rd

) (18)

Denote

1205791(120578 120591) = 119871

minus1

1

1199022exp(minus120578radic Pr

1 + Rdradic119902)

= (1205782

2

Pr1 + Rd

+ 120591) erfc(120578

2radic120591radic

Pr1 + Rd

)

minus120578

radic120587radic

Pr 1205911 + Rd

exp(minus1205782

4120591

Pr1 + Rd

)

(19)

And using the second shift property we obtain the followingsolution for temperature distribution

120579 (120578 120591) = 1205791(120578 120591) minus 120579

1(120578 120591 minus 1)119867 (120591 minus 1) (20)

where 119867(sdot) is Heaviside function and erfc(sdot) denote thecomplementary error function

Meanwhile the solution of (15) under boundary condi-tions (17) results in

119906 (120578 119902) = [1 minus exp (minus119902)] 1199063(119902) sdot 119906

4(120578 119902) (21)

where

1199063(119902) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821

(119902 + 1198981)2

minus 1198982

2

(22)

1199064(120578 119902) =

1

1199022

exp(minus120578radic(1 + 120582

1) (Ha + 119902)

1 + 120582119902)

minus exp(minus120578radicPr 1199021 + Rd

)

(23)

Mathematical Problems in Engineering 5

with

1198981=Pr minus (1 + 120582

1) (1 + Rd)

2Pr 120582

1198982=radic((Pr minus (1 + 120582

1) (1 + Rd)) Pr 120582)2 + 4 (1 + 120582

1) (1 + Rd)HaPr 120582

4

(24)

The inverse Laplace transform of (22) leads to the followingexpressions

1199063(120591) = 119871

minus1

1199063(119902)

=Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

119890minus1198981120591 sinh (119898

2120591)

(25)

In order to determine the inverse Laplace transform of1199064(120578 120591) of the function 119906

4(120578 119902) we split (23) in the following

forms

1198601(120578 119902) =

1

1199022exp(minus120578radic

(1 + 1205821) (Ha + 119902)

1 + 120582119902) (26)

1198602(120578 119902) =

1

1199022exp(minus120578radic

Pr 1199021 + Rd

) (27)

Using the inversion formula of compound function and theinverse Laplace formula into (26) implies

1198601(120578 120591) = 119871

minus1

1198601(120578 119902) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904

(28)

and the inverse Laplace transform of (27) is given by

1198602(120578 120591) = 119871

minus1

1198602(120578 119902)

= (120591 +1205782

2

Pr1 + Rd

) erfc(120578

2radic

Pr120591 (1 + Rd)

)

minus 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(29)

Thus

1199064(120578 120591) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904 minus (120591 +1205782

2

Pr1 + Rd

)

sdot erfc(120578

2radic

Pr120591 (1 + Rd)

)

+ 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(30)

where 1198961= 120582(1 + 120582

1) 1198962= 1(1 + 120582

1) 119887 = 119896

2minus 1198961Ha and

1198681(sdot) is modified Bessel function of the first kind of order one

Consequently the expression for velocity distribution can bewritten as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (31)

where 1199061198771(120578 120591) can be found by using convolution theorem

1199061198771(120578 120591) = (119906

3lowast 1199064) (120591)

= int

120591

0

1199063(120591 minus 119901) 119906

4(120578 119901) 119889119901

(32)

in which

1199061198771(120578 120591) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

6 Mathematical Problems in Engineering

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

+Gr (1 + 120582

1)

1205821198982

radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(33)

4 Solution for an Isothermal Plate

The analytical solution of temperature and velocity distri-bution for the case of ramped wall temperature are shownin (20) and (31) It is good to compare these results withthose corresponding to the flow near a plate with uniformtemperature to see the effects of ramped wall temperature of

the plate on the fluid flowThe solution for temperature profilecan be obtained as

120579 (120578 120591) = erfc(120578

2radic120591radic

Pr1 + Rd

) (34)

and using the same way as before the expression of velocityprofile for case of isothermal is

119906 (120578 120591) =Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp(minus1198981(120591 minus 119901) minus

120578

radic1198961

) sinh (1198982(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

sdot int

120591

0

int

119901

0

int

infin

0

120578

2119906radic120587119906radic119887119906

119904

1

1198961

exp[minus(1198981(120591 minus 119901) +

1198962119904 + 119906

1198961

+1205782

4119906)] sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)

119889119906119889119904 119889119901

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp (minus1198981(120591 minus 119901)) sinh (119898

2(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(35)

5 Nusselt Number and Skin Friction

Theexpression ofNusselt numberNu and skin friction119865(120591)for both cases ramped wall temperature and isothermalis discussed in this section The Nusselt number and skinfriction for ramped wall temperature can be written as

Nu = minus 120597120579

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

=2

radic120587radic

Pr1 + Rd

[radic120591 minus radic120591 minus 1119867 (120591 minus 1)]

119865 (120578 120591) =1

1 + 1205821

(1 + 120582120597

120597120591)120597119906

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

= 1198651(120591) minus 119865

1(120591 minus 1)119867 (120591 minus 1)

(36)

where

1198651(120591) = minus

Gr (1 + Rd)Prradic1198961

int

120591

0

119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901

+ 2Grradic1 + Rd120587Pr

int

120591

0

radic119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901 +

Gr (1 + Rd)21198961radic120587Pr

sdot int

120591

0

int

119901

0

int

infin

0

(119901 minus 119904)

119906radic119887

119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]

119889119906119889119904 119889119901

(37)

Mathematical Problems in Engineering 7

and Nusselt number and skin friction for isothermal aregiven by

Nu = 1

radic120587120591radic

Pr1 + Rd

119865 (120591) = minusGr (1 + Rd)

1198961Pr

int

120591

0

119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901 +

Gr (1 + Rd)21198961Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

1

119906radic119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119906 119889119904 119889119901

+ Grradic1 + Rd120587Pr

int

120591

0

119890minus1198981(120591minus119901)

radic119901[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901

(38)

6 Limiting Cases

In this section we discuss our present results and provideexact solutions for two special cases by taking suitableparameters equal to zero

61 Absence of MHD and Radiation Parameter By makingHartmann number Ha = 0 and radiation parameter Rd = 0

the obtained temperature profile (34) of isothermal caseimplies

120579 (120578 120591) = erfc(120578radicPr2radic120591

) (39)

which is quite in agreement with [28 equation (18)] Further-more by putting Ha = 0 and Rd = 0 in (35) it reduces to

119906 (120578 120591) =Gr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) 119889119901

+Gr (1 + 120582

1)

212058211989811198961Pr

radic1198962

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

1(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) erfc(

120578radicPr2radic119901

)119889119901

(40)

which is identical to the solution obtained by Khan [28equation (31)] In Figure 2 it is observed that the graph of(40) matches well (31) in [28] Hence this fact confirms theaccuracy of our results

62 The Case of 1205821= 0 (Second-Grade Fluid with MHD

and Radiation Effects) Interestingly by putting 1205821

= 0

the solutions in (31) and (35) can be reduced to second-grade fluid with the effects of radiation and MHD Thus thevelocity distribution for ramped wall temperature in (31) canbe written as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (41)

where

1199061198771(120578 120591) =

Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd)212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906119889119904 119889119901

8 Mathematical Problems in Engineering

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901 +Gr1205821198982

sdot radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(42)

and the velocity profile for isothermal case (35) reduces to

119906 (120578 120591) =Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+Gr (1 + Rd)21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(43)

However taking Ha = 0 Rd = 0 and Gr = 1 into (41) and(43) gives the well-known results

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (44)

where

1199061198771(120578 120591)

=1

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901

+1

212058211989611198982Prradic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782Pr2

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

+1

1205821198982radic120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+120578

2Pr4119901) sinh (1198982(120591 minus 119901)) 119889119901

(45)

119906 (120578 120591)

=1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+1

21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

(46)

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

of some worms and transport of lymph in the lymphaticvessels [8] Abd-Alla et al [9] investigated magnetic fieldand gravity effects on peristaltic transport of a Jeffrey fluidin an asymmetric channel Quite recently Abd-Alla andAbo-Dahab [10] analyzed the influence of magnetic fieldand rotation effects on peristaltic transport of a Jeffrey fluidin asymmetric channel Akram and Nadeem [11] providedexact and closed form Adomian solutions for the peristalticmotion of two-dimensional Jeffrey fluid in an asymmetricchannel under the effects of induced magnetic field and heattransfer

Moreover the influence of heat transfer on MHD oscil-latory flow of Jeffrey fluid in a channel has been investigatedby Kavita et al [12] Idowu et al [13] studied the effects ofheat and mass transfer on unsteady MHD oscillatory flowof Jeffrey fluid in a horizontal channel with chemical reac-tion and solved analytically using perturbation techniqueAli and Asghar [14] presented analytical solution for two-dimensional oscillatory flow inside a rectangular channelfor Jeffrey fluid with small suction They used perturbationWentzel-Kramers-Brillouin and variation of parameter asthe methodologies to solve this complex problem Detaileddiscussions on oscillatory flow of Jeffrey fluid also can befound in Khan [15] and Hayat et al [1 16]

On the other hand the steady boundary layer flow of aJeffrey fluid over a shrinking sheet is highlighted by Nadeemet al [17] Hamad et al [18] presented a study of thermaljump effects on boundary layer flow of a Jeffrey fluid nearthe stagnation point on a stretchingshrinking sheet withvariable thermal conductivity and solved numerically usingfinite difference method Hayat et al [19] considered two-dimensional stagnation point flow of Jeffrey fluid over a con-vectively heated stretching sheet constructed by homotopyanalysis method Other than that Qasim [20] observed thecombined effect of heat and mass transfer in Jeffrey fluidover a stretching sheet in the presence of heat sourceheatsink where the obtained exact solutions are derived by powerseries method using Kummerrsquos confluent hypergeometricfunctions Dalir [21] investigated entropy generation for thesteady two-dimensional laminar forced convection flow andheat transfer of an incompressible Jeffrey non-Newtonianfluid over a linearly stretching impermeable and isothermalsheet numerically by Kellerrsquos box method

Diverse contributions dealing with boundary layer flowover stretching sheet in various situations are well docu-mented in Dalir et al [22] and Hayat et al [23] Hayatet al [24] analyzed the effects of Newtonian heating magne-tohydrodynamics (MHD) in a flow of a Jeffrey fluid over aradially stretching surface In the same year Hayat et al [25]carried out the study of boundary layer stretched flow of aJeffrey fluid subject to the convective boundary conditionShehzad et al [26] analytically discussed magnetohydrody-namics (MHD) three-dimensional flow of Jeffrey fluid in thepresence of Newtonian heating Farooq et al [27] examinedthe combined effects of Joule and Newtonian heating inmagnetohydrodynamic (MHD) flow of Jeffrey fluid over astretching cylinder with heat sourcesink solved analyticallyby homotopy analysis method (HAM) whereas Khan [28]recently implemented the Laplace transform technique to

find the exact solutions for unsteady free convection flow ofa Jeffrey fluid past an infinite isothermal vertical plate

In view of the above analysis so far no study has beenmade to examine the effect of ramped wall temperaturein a Jeffrey fluid flow Thus our main focus is to discussthe influence of thermal radiation on unsteady MHD freeconvection flow of Jeffrey fluid over a vertical plate withramped wall temperature The Laplace transform techniqueis applied in this study in order to find the analytical solutionsfor velocity and temperature profiles Expressions of skinfriction and Nusselt number are also given Graphical resultsare provided and discussed for embedded parameters

2 Mathematical Formulation of the Problem

The unsteady incompressible flow is described by the follow-ing governing equations

nabla sdot V = 0

120588 [120597V120597119905

+ (V sdot nabla)V] = divT + J times B + 120588g

curlE = minus120597B120597119905

curlB = 120583119898J

nabla sdot B = 0

(1)

In the above equations V indicates the velocity field T isthe Cauchy stress tensor 120588 is the fluid density J is the currentdensity B = 119861

0+ 119887 is the total magnetic field where 119861

0and

119887 are the applied and induced magnetic fields respectively gis the gravitational force which is downward in 119909-directionE is the total electric field current and 120583

119898is the magnetic

permeability The constitutive equations for Jeffrey fluid aregiven by

T = minus119901I + S

S =120583

1 + 1205821

[A1+ 1205822(120597A1

120597119905+ (V sdot nabla)A

1)]

(2)

in which minus119901I is the indeterminate part of the stress S isthe extra tensor 120583 is the dynamic viscosity and 120582

1and 120582

2

are the material parameters of the Jeffrey fluid known as theratio of relaxation to retardation times and retardation timerespectively Notice that if 120582

1= 1205822= 0 it will lead to the

expression of an incompressible viscous fluid The Rivlin-Ericksen tensor A

1 is defined by

A1= nablaV + (nablaV)119879 (3)

For unidirectional flow we assume that velocity and stressfields are in the form of

V = V (119910 119905) = 119906 (119910 119905) i

S = S (119910 119905) (4)

where 119906 is the 119909-component of velocity V and i is the unitvector along the 119909-direction of the Cartesian coordinate

Mathematical Problems in Engineering 3

Thermalboundary

layer

Momentumboundary

layer

T(infin t) = Tinfin

x

y

g

0

B0

T(0t)=Tinfin

+(T

wminusTinfin)tt

0T

(0t)=Twt

get 0

Figure 1 Schematic diagram for the flow of Jeffrey fluid past avertical plate

system The stress field satisfies the condition S(119910 0) = 0

which gives the following results

119878119909119909= 119878119910119910= 119878119911119911= 119878119909119911= 119878119911119909= 119878119910119911= 119878119911119910= 0

119878119909119910=

120583

1 + 1205821

(1 + 1205822

120597

120597119905)120597119906

120597119910

(5)

where 119878119909119910

is the nontrivial tangential stressConsider the effect of thermal radiation on unsteady free

convection flow of Jeffrey fluid near the vertical plate situatedin the (119909 119911) plane of a Cartesian coordinate system of 119909 119910and 119911 as presented in Figure 1 It is assumed that at time119905 le 0 both plate and fluid are at rest at constant temperature119879infin At time 119905 gt 0 the temperature of the plate is raised or

lowered to119879infin+(119879119908minus119879infin)(1199051199050)when 119905 le 119905

0 and the constant

temperature 119879119908is maintained at 119905 gt 119905

0

Under the above assumption and applying the usualBoussinesq approximation themomentum and energy equa-tions for unsteady free convection flow of Jeffrey fluid past aninfinite vertical plate are given by

120588120597119906

120597119905=

120583

1 + 1205821

(1 + 1205822

120597

120597119905)1205972

119906

1205971199102+ 120588119892120573 (119879 minus 119879

infin)

minus 1205901198612

0119906

(6)

120588119888119901

120597119879

120597119905= 119896

1205972

119879

1205971199102minus120597119902119903

120597119910 (7)

with appropriate initial and boundary conditions

119906 (119910 0) = 0

119879 (119910 0) = 119879infin

119910 gt 0

119906 (0 119905) = 0 119905 gt 0

119879 (0 119905) = 119879infin+ (119879119908minus 119879infin)119905

1199050

0 lt 119905 lt 1199050

119879 (0 119905) = 119879119908 119905 ge 119905

0

119906 (infin 119905) = 0

119879 (infin 119905) = 119879infin

119905 gt 0

(8)

where 119906 is the fluid velocity in 119909-direction 119879 is the tempera-ture 120588 is the constant density of the fluid 119892 is the accelerationdue to gravity 119888

119901is the specific heat capacity 119896 is the thermal

conductivity119879119908is the wall temperature119879

infinis the free stream

temperature and 119902119903is the radiation heat flux respectively

Using Rosseland approximation the radiation heat flux canbe written as

119902119903= minus

4120590lowast

31198961

1205971198794

120597119910 (9)

where 120590lowast denotes the Stefan-Boltzmann and 1198961is the absorp-

tion coefficient We assume that the temperature differenceswithin the flow are sufficiently small such that 119879

4can be

expanded inTaylor seriesHence expanding1198794about119879

infinand

neglecting higher order terms we get 1198794 asymp 41198793

infin119879 minus 3119879

4

infin

Therefore (7) is simplified to

120588119888119901

120597119879

120597119905= 119896(1 +

16120590lowast

1198793

infin

31198961198961

)1205972

119879

1205971199102 (10)

Introducing the dimensionless variables

119906lowast

=119906

1198800

120578lowast

=1199101198800

120592

120591lowast

=1199051198802

0

120592

120579 =119879 minus 119879infin

119879119908minus 119879infin

(11)

into (6) and (10) yields the following dimensionless equations(lowast notations are dropped for simplicity)

120597119906

120597120591=

1

1 + 1205821

(1 + 120582120597

120597120591)1205972

119906

1205971205782minusHa 119906 + Gr 120579

Pr120597120579120597120591

= (1 + Rd) 1205972

120579

1205971205782

(12)

4 Mathematical Problems in Engineering

where 120582 is the dimensionless Jeffrey fluid parameter Ha isthe Hartmann number Gr is the Grashof number Pr is thePrandtl number and Rd is the radiation parameter which aredenoted as follows

120582 =12058221198802

0

120592

Ha =1205901198612

0120592

12058811988020

Gr =119892120573120592 (119879

119908minus 119879infin)

11988030

Pr =120583119888119901

119896

Rd =16120590lowast

1198793

infin

31198961198961

(13)

Also initial and boundary conditions (8) become

119906 (120578 0) = 0

120579 (120578 0) = 0

120578 gt 0

119906 (0 120591) = 0 120591 gt 0

120579 (0 120591) = 120591 0 lt 120591 le 1

120579 (0 120591) = 1 120591 gt 1

119906 (infin 120591) = 0

120579 (infin 120591) = 0

120591 gt 0

(14)

3 Solution of the Problem

Applying the Laplace transform to (12) and (14) yields thefollowing equations in the transformed (120578 119902)-plane

1198892

119906 (120578 119902)

1198891205782minus (

(1 + 1205821) (Ha + 119902)

1 + 120582119902)119906 (120578 119902)

= minusGr((1 + 120582

1)

1 + 120582119902) 120579 (120578 119902)

(15)

(1 + Rd)1198892

120579 (120578 119902)

1198891205782minus Pr 119902120579 (120578 119902) = 0 (16)

and initial and boundary conditions are transformed asfollows

119906 (0 119902) = 0

120579 (0 119902) =1 minus exp (minus119902)

1199022

119906 (infin 119902) = 0

120579 (infin 119902) = 0

(17)

where 119906(120578 119902) and 120579(120578 119902) are Laplace transforms of 119906(120578 120591) and120579(120578 120591)

Equation (16) subject to boundary conditions (17) has thefollowing solution

120579 (120578 119902) =1 minus exp (minus119902)

1199022exp(minus120578radic

Pr 1199021 + Rd

) (18)

Denote

1205791(120578 120591) = 119871

minus1

1

1199022exp(minus120578radic Pr

1 + Rdradic119902)

= (1205782

2

Pr1 + Rd

+ 120591) erfc(120578

2radic120591radic

Pr1 + Rd

)

minus120578

radic120587radic

Pr 1205911 + Rd

exp(minus1205782

4120591

Pr1 + Rd

)

(19)

And using the second shift property we obtain the followingsolution for temperature distribution

120579 (120578 120591) = 1205791(120578 120591) minus 120579

1(120578 120591 minus 1)119867 (120591 minus 1) (20)

where 119867(sdot) is Heaviside function and erfc(sdot) denote thecomplementary error function

Meanwhile the solution of (15) under boundary condi-tions (17) results in

119906 (120578 119902) = [1 minus exp (minus119902)] 1199063(119902) sdot 119906

4(120578 119902) (21)

where

1199063(119902) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821

(119902 + 1198981)2

minus 1198982

2

(22)

1199064(120578 119902) =

1

1199022

exp(minus120578radic(1 + 120582

1) (Ha + 119902)

1 + 120582119902)

minus exp(minus120578radicPr 1199021 + Rd

)

(23)

Mathematical Problems in Engineering 5

with

1198981=Pr minus (1 + 120582

1) (1 + Rd)

2Pr 120582

1198982=radic((Pr minus (1 + 120582

1) (1 + Rd)) Pr 120582)2 + 4 (1 + 120582

1) (1 + Rd)HaPr 120582

4

(24)

The inverse Laplace transform of (22) leads to the followingexpressions

1199063(120591) = 119871

minus1

1199063(119902)

=Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

119890minus1198981120591 sinh (119898

2120591)

(25)

In order to determine the inverse Laplace transform of1199064(120578 120591) of the function 119906

4(120578 119902) we split (23) in the following

forms

1198601(120578 119902) =

1

1199022exp(minus120578radic

(1 + 1205821) (Ha + 119902)

1 + 120582119902) (26)

1198602(120578 119902) =

1

1199022exp(minus120578radic

Pr 1199021 + Rd

) (27)

Using the inversion formula of compound function and theinverse Laplace formula into (26) implies

1198601(120578 120591) = 119871

minus1

1198601(120578 119902) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904

(28)

and the inverse Laplace transform of (27) is given by

1198602(120578 120591) = 119871

minus1

1198602(120578 119902)

= (120591 +1205782

2

Pr1 + Rd

) erfc(120578

2radic

Pr120591 (1 + Rd)

)

minus 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(29)

Thus

1199064(120578 120591) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904 minus (120591 +1205782

2

Pr1 + Rd

)

sdot erfc(120578

2radic

Pr120591 (1 + Rd)

)

+ 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(30)

where 1198961= 120582(1 + 120582

1) 1198962= 1(1 + 120582

1) 119887 = 119896

2minus 1198961Ha and

1198681(sdot) is modified Bessel function of the first kind of order one

Consequently the expression for velocity distribution can bewritten as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (31)

where 1199061198771(120578 120591) can be found by using convolution theorem

1199061198771(120578 120591) = (119906

3lowast 1199064) (120591)

= int

120591

0

1199063(120591 minus 119901) 119906

4(120578 119901) 119889119901

(32)

in which

1199061198771(120578 120591) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

6 Mathematical Problems in Engineering

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

+Gr (1 + 120582

1)

1205821198982

radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(33)

4 Solution for an Isothermal Plate

The analytical solution of temperature and velocity distri-bution for the case of ramped wall temperature are shownin (20) and (31) It is good to compare these results withthose corresponding to the flow near a plate with uniformtemperature to see the effects of ramped wall temperature of

the plate on the fluid flowThe solution for temperature profilecan be obtained as

120579 (120578 120591) = erfc(120578

2radic120591radic

Pr1 + Rd

) (34)

and using the same way as before the expression of velocityprofile for case of isothermal is

119906 (120578 120591) =Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp(minus1198981(120591 minus 119901) minus

120578

radic1198961

) sinh (1198982(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

sdot int

120591

0

int

119901

0

int

infin

0

120578

2119906radic120587119906radic119887119906

119904

1

1198961

exp[minus(1198981(120591 minus 119901) +

1198962119904 + 119906

1198961

+1205782

4119906)] sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)

119889119906119889119904 119889119901

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp (minus1198981(120591 minus 119901)) sinh (119898

2(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(35)

5 Nusselt Number and Skin Friction

Theexpression ofNusselt numberNu and skin friction119865(120591)for both cases ramped wall temperature and isothermalis discussed in this section The Nusselt number and skinfriction for ramped wall temperature can be written as

Nu = minus 120597120579

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

=2

radic120587radic

Pr1 + Rd

[radic120591 minus radic120591 minus 1119867 (120591 minus 1)]

119865 (120578 120591) =1

1 + 1205821

(1 + 120582120597

120597120591)120597119906

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

= 1198651(120591) minus 119865

1(120591 minus 1)119867 (120591 minus 1)

(36)

where

1198651(120591) = minus

Gr (1 + Rd)Prradic1198961

int

120591

0

119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901

+ 2Grradic1 + Rd120587Pr

int

120591

0

radic119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901 +

Gr (1 + Rd)21198961radic120587Pr

sdot int

120591

0

int

119901

0

int

infin

0

(119901 minus 119904)

119906radic119887

119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]

119889119906119889119904 119889119901

(37)

Mathematical Problems in Engineering 7

and Nusselt number and skin friction for isothermal aregiven by

Nu = 1

radic120587120591radic

Pr1 + Rd

119865 (120591) = minusGr (1 + Rd)

1198961Pr

int

120591

0

119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901 +

Gr (1 + Rd)21198961Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

1

119906radic119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119906 119889119904 119889119901

+ Grradic1 + Rd120587Pr

int

120591

0

119890minus1198981(120591minus119901)

radic119901[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901

(38)

6 Limiting Cases

In this section we discuss our present results and provideexact solutions for two special cases by taking suitableparameters equal to zero

61 Absence of MHD and Radiation Parameter By makingHartmann number Ha = 0 and radiation parameter Rd = 0

the obtained temperature profile (34) of isothermal caseimplies

120579 (120578 120591) = erfc(120578radicPr2radic120591

) (39)

which is quite in agreement with [28 equation (18)] Further-more by putting Ha = 0 and Rd = 0 in (35) it reduces to

119906 (120578 120591) =Gr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) 119889119901

+Gr (1 + 120582

1)

212058211989811198961Pr

radic1198962

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

1(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) erfc(

120578radicPr2radic119901

)119889119901

(40)

which is identical to the solution obtained by Khan [28equation (31)] In Figure 2 it is observed that the graph of(40) matches well (31) in [28] Hence this fact confirms theaccuracy of our results

62 The Case of 1205821= 0 (Second-Grade Fluid with MHD

and Radiation Effects) Interestingly by putting 1205821

= 0

the solutions in (31) and (35) can be reduced to second-grade fluid with the effects of radiation and MHD Thus thevelocity distribution for ramped wall temperature in (31) canbe written as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (41)

where

1199061198771(120578 120591) =

Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd)212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906119889119904 119889119901

8 Mathematical Problems in Engineering

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901 +Gr1205821198982

sdot radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(42)

and the velocity profile for isothermal case (35) reduces to

119906 (120578 120591) =Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+Gr (1 + Rd)21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(43)

However taking Ha = 0 Rd = 0 and Gr = 1 into (41) and(43) gives the well-known results

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (44)

where

1199061198771(120578 120591)

=1

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901

+1

212058211989611198982Prradic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782Pr2

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

+1

1205821198982radic120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+120578

2Pr4119901) sinh (1198982(120591 minus 119901)) 119889119901

(45)

119906 (120578 120591)

=1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+1

21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

(46)

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Thermalboundary

layer

Momentumboundary

layer

T(infin t) = Tinfin

x

y

g

0

B0

T(0t)=Tinfin

+(T

wminusTinfin)tt

0T

(0t)=Twt

get 0

Figure 1 Schematic diagram for the flow of Jeffrey fluid past avertical plate

system The stress field satisfies the condition S(119910 0) = 0

which gives the following results

119878119909119909= 119878119910119910= 119878119911119911= 119878119909119911= 119878119911119909= 119878119910119911= 119878119911119910= 0

119878119909119910=

120583

1 + 1205821

(1 + 1205822

120597

120597119905)120597119906

120597119910

(5)

where 119878119909119910

is the nontrivial tangential stressConsider the effect of thermal radiation on unsteady free

convection flow of Jeffrey fluid near the vertical plate situatedin the (119909 119911) plane of a Cartesian coordinate system of 119909 119910and 119911 as presented in Figure 1 It is assumed that at time119905 le 0 both plate and fluid are at rest at constant temperature119879infin At time 119905 gt 0 the temperature of the plate is raised or

lowered to119879infin+(119879119908minus119879infin)(1199051199050)when 119905 le 119905

0 and the constant

temperature 119879119908is maintained at 119905 gt 119905

0

Under the above assumption and applying the usualBoussinesq approximation themomentum and energy equa-tions for unsteady free convection flow of Jeffrey fluid past aninfinite vertical plate are given by

120588120597119906

120597119905=

120583

1 + 1205821

(1 + 1205822

120597

120597119905)1205972

119906

1205971199102+ 120588119892120573 (119879 minus 119879

infin)

minus 1205901198612

0119906

(6)

120588119888119901

120597119879

120597119905= 119896

1205972

119879

1205971199102minus120597119902119903

120597119910 (7)

with appropriate initial and boundary conditions

119906 (119910 0) = 0

119879 (119910 0) = 119879infin

119910 gt 0

119906 (0 119905) = 0 119905 gt 0

119879 (0 119905) = 119879infin+ (119879119908minus 119879infin)119905

1199050

0 lt 119905 lt 1199050

119879 (0 119905) = 119879119908 119905 ge 119905

0

119906 (infin 119905) = 0

119879 (infin 119905) = 119879infin

119905 gt 0

(8)

where 119906 is the fluid velocity in 119909-direction 119879 is the tempera-ture 120588 is the constant density of the fluid 119892 is the accelerationdue to gravity 119888

119901is the specific heat capacity 119896 is the thermal

conductivity119879119908is the wall temperature119879

infinis the free stream

temperature and 119902119903is the radiation heat flux respectively

Using Rosseland approximation the radiation heat flux canbe written as

119902119903= minus

4120590lowast

31198961

1205971198794

120597119910 (9)

where 120590lowast denotes the Stefan-Boltzmann and 1198961is the absorp-

tion coefficient We assume that the temperature differenceswithin the flow are sufficiently small such that 119879

4can be

expanded inTaylor seriesHence expanding1198794about119879

infinand

neglecting higher order terms we get 1198794 asymp 41198793

infin119879 minus 3119879

4

infin

Therefore (7) is simplified to

120588119888119901

120597119879

120597119905= 119896(1 +

16120590lowast

1198793

infin

31198961198961

)1205972

119879

1205971199102 (10)

Introducing the dimensionless variables

119906lowast

=119906

1198800

120578lowast

=1199101198800

120592

120591lowast

=1199051198802

0

120592

120579 =119879 minus 119879infin

119879119908minus 119879infin

(11)

into (6) and (10) yields the following dimensionless equations(lowast notations are dropped for simplicity)

120597119906

120597120591=

1

1 + 1205821

(1 + 120582120597

120597120591)1205972

119906

1205971205782minusHa 119906 + Gr 120579

Pr120597120579120597120591

= (1 + Rd) 1205972

120579

1205971205782

(12)

4 Mathematical Problems in Engineering

where 120582 is the dimensionless Jeffrey fluid parameter Ha isthe Hartmann number Gr is the Grashof number Pr is thePrandtl number and Rd is the radiation parameter which aredenoted as follows

120582 =12058221198802

0

120592

Ha =1205901198612

0120592

12058811988020

Gr =119892120573120592 (119879

119908minus 119879infin)

11988030

Pr =120583119888119901

119896

Rd =16120590lowast

1198793

infin

31198961198961

(13)

Also initial and boundary conditions (8) become

119906 (120578 0) = 0

120579 (120578 0) = 0

120578 gt 0

119906 (0 120591) = 0 120591 gt 0

120579 (0 120591) = 120591 0 lt 120591 le 1

120579 (0 120591) = 1 120591 gt 1

119906 (infin 120591) = 0

120579 (infin 120591) = 0

120591 gt 0

(14)

3 Solution of the Problem

Applying the Laplace transform to (12) and (14) yields thefollowing equations in the transformed (120578 119902)-plane

1198892

119906 (120578 119902)

1198891205782minus (

(1 + 1205821) (Ha + 119902)

1 + 120582119902)119906 (120578 119902)

= minusGr((1 + 120582

1)

1 + 120582119902) 120579 (120578 119902)

(15)

(1 + Rd)1198892

120579 (120578 119902)

1198891205782minus Pr 119902120579 (120578 119902) = 0 (16)

and initial and boundary conditions are transformed asfollows

119906 (0 119902) = 0

120579 (0 119902) =1 minus exp (minus119902)

1199022

119906 (infin 119902) = 0

120579 (infin 119902) = 0

(17)

where 119906(120578 119902) and 120579(120578 119902) are Laplace transforms of 119906(120578 120591) and120579(120578 120591)

Equation (16) subject to boundary conditions (17) has thefollowing solution

120579 (120578 119902) =1 minus exp (minus119902)

1199022exp(minus120578radic

Pr 1199021 + Rd

) (18)

Denote

1205791(120578 120591) = 119871

minus1

1

1199022exp(minus120578radic Pr

1 + Rdradic119902)

= (1205782

2

Pr1 + Rd

+ 120591) erfc(120578

2radic120591radic

Pr1 + Rd

)

minus120578

radic120587radic

Pr 1205911 + Rd

exp(minus1205782

4120591

Pr1 + Rd

)

(19)

And using the second shift property we obtain the followingsolution for temperature distribution

120579 (120578 120591) = 1205791(120578 120591) minus 120579

1(120578 120591 minus 1)119867 (120591 minus 1) (20)

where 119867(sdot) is Heaviside function and erfc(sdot) denote thecomplementary error function

Meanwhile the solution of (15) under boundary condi-tions (17) results in

119906 (120578 119902) = [1 minus exp (minus119902)] 1199063(119902) sdot 119906

4(120578 119902) (21)

where

1199063(119902) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821

(119902 + 1198981)2

minus 1198982

2

(22)

1199064(120578 119902) =

1

1199022

exp(minus120578radic(1 + 120582

1) (Ha + 119902)

1 + 120582119902)

minus exp(minus120578radicPr 1199021 + Rd

)

(23)

Mathematical Problems in Engineering 5

with

1198981=Pr minus (1 + 120582

1) (1 + Rd)

2Pr 120582

1198982=radic((Pr minus (1 + 120582

1) (1 + Rd)) Pr 120582)2 + 4 (1 + 120582

1) (1 + Rd)HaPr 120582

4

(24)

The inverse Laplace transform of (22) leads to the followingexpressions

1199063(120591) = 119871

minus1

1199063(119902)

=Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

119890minus1198981120591 sinh (119898

2120591)

(25)

In order to determine the inverse Laplace transform of1199064(120578 120591) of the function 119906

4(120578 119902) we split (23) in the following

forms

1198601(120578 119902) =

1

1199022exp(minus120578radic

(1 + 1205821) (Ha + 119902)

1 + 120582119902) (26)

1198602(120578 119902) =

1

1199022exp(minus120578radic

Pr 1199021 + Rd

) (27)

Using the inversion formula of compound function and theinverse Laplace formula into (26) implies

1198601(120578 120591) = 119871

minus1

1198601(120578 119902) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904

(28)

and the inverse Laplace transform of (27) is given by

1198602(120578 120591) = 119871

minus1

1198602(120578 119902)

= (120591 +1205782

2

Pr1 + Rd

) erfc(120578

2radic

Pr120591 (1 + Rd)

)

minus 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(29)

Thus

1199064(120578 120591) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904 minus (120591 +1205782

2

Pr1 + Rd

)

sdot erfc(120578

2radic

Pr120591 (1 + Rd)

)

+ 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(30)

where 1198961= 120582(1 + 120582

1) 1198962= 1(1 + 120582

1) 119887 = 119896

2minus 1198961Ha and

1198681(sdot) is modified Bessel function of the first kind of order one

Consequently the expression for velocity distribution can bewritten as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (31)

where 1199061198771(120578 120591) can be found by using convolution theorem

1199061198771(120578 120591) = (119906

3lowast 1199064) (120591)

= int

120591

0

1199063(120591 minus 119901) 119906

4(120578 119901) 119889119901

(32)

in which

1199061198771(120578 120591) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

6 Mathematical Problems in Engineering

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

+Gr (1 + 120582

1)

1205821198982

radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(33)

4 Solution for an Isothermal Plate

The analytical solution of temperature and velocity distri-bution for the case of ramped wall temperature are shownin (20) and (31) It is good to compare these results withthose corresponding to the flow near a plate with uniformtemperature to see the effects of ramped wall temperature of

the plate on the fluid flowThe solution for temperature profilecan be obtained as

120579 (120578 120591) = erfc(120578

2radic120591radic

Pr1 + Rd

) (34)

and using the same way as before the expression of velocityprofile for case of isothermal is

119906 (120578 120591) =Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp(minus1198981(120591 minus 119901) minus

120578

radic1198961

) sinh (1198982(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

sdot int

120591

0

int

119901

0

int

infin

0

120578

2119906radic120587119906radic119887119906

119904

1

1198961

exp[minus(1198981(120591 minus 119901) +

1198962119904 + 119906

1198961

+1205782

4119906)] sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)

119889119906119889119904 119889119901

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp (minus1198981(120591 minus 119901)) sinh (119898

2(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(35)

5 Nusselt Number and Skin Friction

Theexpression ofNusselt numberNu and skin friction119865(120591)for both cases ramped wall temperature and isothermalis discussed in this section The Nusselt number and skinfriction for ramped wall temperature can be written as

Nu = minus 120597120579

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

=2

radic120587radic

Pr1 + Rd

[radic120591 minus radic120591 minus 1119867 (120591 minus 1)]

119865 (120578 120591) =1

1 + 1205821

(1 + 120582120597

120597120591)120597119906

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

= 1198651(120591) minus 119865

1(120591 minus 1)119867 (120591 minus 1)

(36)

where

1198651(120591) = minus

Gr (1 + Rd)Prradic1198961

int

120591

0

119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901

+ 2Grradic1 + Rd120587Pr

int

120591

0

radic119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901 +

Gr (1 + Rd)21198961radic120587Pr

sdot int

120591

0

int

119901

0

int

infin

0

(119901 minus 119904)

119906radic119887

119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]

119889119906119889119904 119889119901

(37)

Mathematical Problems in Engineering 7

and Nusselt number and skin friction for isothermal aregiven by

Nu = 1

radic120587120591radic

Pr1 + Rd

119865 (120591) = minusGr (1 + Rd)

1198961Pr

int

120591

0

119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901 +

Gr (1 + Rd)21198961Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

1

119906radic119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119906 119889119904 119889119901

+ Grradic1 + Rd120587Pr

int

120591

0

119890minus1198981(120591minus119901)

radic119901[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901

(38)

6 Limiting Cases

In this section we discuss our present results and provideexact solutions for two special cases by taking suitableparameters equal to zero

61 Absence of MHD and Radiation Parameter By makingHartmann number Ha = 0 and radiation parameter Rd = 0

the obtained temperature profile (34) of isothermal caseimplies

120579 (120578 120591) = erfc(120578radicPr2radic120591

) (39)

which is quite in agreement with [28 equation (18)] Further-more by putting Ha = 0 and Rd = 0 in (35) it reduces to

119906 (120578 120591) =Gr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) 119889119901

+Gr (1 + 120582

1)

212058211989811198961Pr

radic1198962

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

1(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) erfc(

120578radicPr2radic119901

)119889119901

(40)

which is identical to the solution obtained by Khan [28equation (31)] In Figure 2 it is observed that the graph of(40) matches well (31) in [28] Hence this fact confirms theaccuracy of our results

62 The Case of 1205821= 0 (Second-Grade Fluid with MHD

and Radiation Effects) Interestingly by putting 1205821

= 0

the solutions in (31) and (35) can be reduced to second-grade fluid with the effects of radiation and MHD Thus thevelocity distribution for ramped wall temperature in (31) canbe written as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (41)

where

1199061198771(120578 120591) =

Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd)212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906119889119904 119889119901

8 Mathematical Problems in Engineering

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901 +Gr1205821198982

sdot radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(42)

and the velocity profile for isothermal case (35) reduces to

119906 (120578 120591) =Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+Gr (1 + Rd)21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(43)

However taking Ha = 0 Rd = 0 and Gr = 1 into (41) and(43) gives the well-known results

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (44)

where

1199061198771(120578 120591)

=1

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901

+1

212058211989611198982Prradic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782Pr2

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

+1

1205821198982radic120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+120578

2Pr4119901) sinh (1198982(120591 minus 119901)) 119889119901

(45)

119906 (120578 120591)

=1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+1

21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

(46)

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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4 Mathematical Problems in Engineering

where 120582 is the dimensionless Jeffrey fluid parameter Ha isthe Hartmann number Gr is the Grashof number Pr is thePrandtl number and Rd is the radiation parameter which aredenoted as follows

120582 =12058221198802

0

120592

Ha =1205901198612

0120592

12058811988020

Gr =119892120573120592 (119879

119908minus 119879infin)

11988030

Pr =120583119888119901

119896

Rd =16120590lowast

1198793

infin

31198961198961

(13)

Also initial and boundary conditions (8) become

119906 (120578 0) = 0

120579 (120578 0) = 0

120578 gt 0

119906 (0 120591) = 0 120591 gt 0

120579 (0 120591) = 120591 0 lt 120591 le 1

120579 (0 120591) = 1 120591 gt 1

119906 (infin 120591) = 0

120579 (infin 120591) = 0

120591 gt 0

(14)

3 Solution of the Problem

Applying the Laplace transform to (12) and (14) yields thefollowing equations in the transformed (120578 119902)-plane

1198892

119906 (120578 119902)

1198891205782minus (

(1 + 1205821) (Ha + 119902)

1 + 120582119902)119906 (120578 119902)

= minusGr((1 + 120582

1)

1 + 120582119902) 120579 (120578 119902)

(15)

(1 + Rd)1198892

120579 (120578 119902)

1198891205782minus Pr 119902120579 (120578 119902) = 0 (16)

and initial and boundary conditions are transformed asfollows

119906 (0 119902) = 0

120579 (0 119902) =1 minus exp (minus119902)

1199022

119906 (infin 119902) = 0

120579 (infin 119902) = 0

(17)

where 119906(120578 119902) and 120579(120578 119902) are Laplace transforms of 119906(120578 120591) and120579(120578 120591)

Equation (16) subject to boundary conditions (17) has thefollowing solution

120579 (120578 119902) =1 minus exp (minus119902)

1199022exp(minus120578radic

Pr 1199021 + Rd

) (18)

Denote

1205791(120578 120591) = 119871

minus1

1

1199022exp(minus120578radic Pr

1 + Rdradic119902)

= (1205782

2

Pr1 + Rd

+ 120591) erfc(120578

2radic120591radic

Pr1 + Rd

)

minus120578

radic120587radic

Pr 1205911 + Rd

exp(minus1205782

4120591

Pr1 + Rd

)

(19)

And using the second shift property we obtain the followingsolution for temperature distribution

120579 (120578 120591) = 1205791(120578 120591) minus 120579

1(120578 120591 minus 1)119867 (120591 minus 1) (20)

where 119867(sdot) is Heaviside function and erfc(sdot) denote thecomplementary error function

Meanwhile the solution of (15) under boundary condi-tions (17) results in

119906 (120578 119902) = [1 minus exp (minus119902)] 1199063(119902) sdot 119906

4(120578 119902) (21)

where

1199063(119902) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821

(119902 + 1198981)2

minus 1198982

2

(22)

1199064(120578 119902) =

1

1199022

exp(minus120578radic(1 + 120582

1) (Ha + 119902)

1 + 120582119902)

minus exp(minus120578radicPr 1199021 + Rd

)

(23)

Mathematical Problems in Engineering 5

with

1198981=Pr minus (1 + 120582

1) (1 + Rd)

2Pr 120582

1198982=radic((Pr minus (1 + 120582

1) (1 + Rd)) Pr 120582)2 + 4 (1 + 120582

1) (1 + Rd)HaPr 120582

4

(24)

The inverse Laplace transform of (22) leads to the followingexpressions

1199063(120591) = 119871

minus1

1199063(119902)

=Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

119890minus1198981120591 sinh (119898

2120591)

(25)

In order to determine the inverse Laplace transform of1199064(120578 120591) of the function 119906

4(120578 119902) we split (23) in the following

forms

1198601(120578 119902) =

1

1199022exp(minus120578radic

(1 + 1205821) (Ha + 119902)

1 + 120582119902) (26)

1198602(120578 119902) =

1

1199022exp(minus120578radic

Pr 1199021 + Rd

) (27)

Using the inversion formula of compound function and theinverse Laplace formula into (26) implies

1198601(120578 120591) = 119871

minus1

1198601(120578 119902) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904

(28)

and the inverse Laplace transform of (27) is given by

1198602(120578 120591) = 119871

minus1

1198602(120578 119902)

= (120591 +1205782

2

Pr1 + Rd

) erfc(120578

2radic

Pr120591 (1 + Rd)

)

minus 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(29)

Thus

1199064(120578 120591) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904 minus (120591 +1205782

2

Pr1 + Rd

)

sdot erfc(120578

2radic

Pr120591 (1 + Rd)

)

+ 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(30)

where 1198961= 120582(1 + 120582

1) 1198962= 1(1 + 120582

1) 119887 = 119896

2minus 1198961Ha and

1198681(sdot) is modified Bessel function of the first kind of order one

Consequently the expression for velocity distribution can bewritten as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (31)

where 1199061198771(120578 120591) can be found by using convolution theorem

1199061198771(120578 120591) = (119906

3lowast 1199064) (120591)

= int

120591

0

1199063(120591 minus 119901) 119906

4(120578 119901) 119889119901

(32)

in which

1199061198771(120578 120591) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

6 Mathematical Problems in Engineering

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

+Gr (1 + 120582

1)

1205821198982

radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(33)

4 Solution for an Isothermal Plate

The analytical solution of temperature and velocity distri-bution for the case of ramped wall temperature are shownin (20) and (31) It is good to compare these results withthose corresponding to the flow near a plate with uniformtemperature to see the effects of ramped wall temperature of

the plate on the fluid flowThe solution for temperature profilecan be obtained as

120579 (120578 120591) = erfc(120578

2radic120591radic

Pr1 + Rd

) (34)

and using the same way as before the expression of velocityprofile for case of isothermal is

119906 (120578 120591) =Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp(minus1198981(120591 minus 119901) minus

120578

radic1198961

) sinh (1198982(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

sdot int

120591

0

int

119901

0

int

infin

0

120578

2119906radic120587119906radic119887119906

119904

1

1198961

exp[minus(1198981(120591 minus 119901) +

1198962119904 + 119906

1198961

+1205782

4119906)] sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)

119889119906119889119904 119889119901

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp (minus1198981(120591 minus 119901)) sinh (119898

2(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(35)

5 Nusselt Number and Skin Friction

Theexpression ofNusselt numberNu and skin friction119865(120591)for both cases ramped wall temperature and isothermalis discussed in this section The Nusselt number and skinfriction for ramped wall temperature can be written as

Nu = minus 120597120579

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

=2

radic120587radic

Pr1 + Rd

[radic120591 minus radic120591 minus 1119867 (120591 minus 1)]

119865 (120578 120591) =1

1 + 1205821

(1 + 120582120597

120597120591)120597119906

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

= 1198651(120591) minus 119865

1(120591 minus 1)119867 (120591 minus 1)

(36)

where

1198651(120591) = minus

Gr (1 + Rd)Prradic1198961

int

120591

0

119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901

+ 2Grradic1 + Rd120587Pr

int

120591

0

radic119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901 +

Gr (1 + Rd)21198961radic120587Pr

sdot int

120591

0

int

119901

0

int

infin

0

(119901 minus 119904)

119906radic119887

119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]

119889119906119889119904 119889119901

(37)

Mathematical Problems in Engineering 7

and Nusselt number and skin friction for isothermal aregiven by

Nu = 1

radic120587120591radic

Pr1 + Rd

119865 (120591) = minusGr (1 + Rd)

1198961Pr

int

120591

0

119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901 +

Gr (1 + Rd)21198961Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

1

119906radic119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119906 119889119904 119889119901

+ Grradic1 + Rd120587Pr

int

120591

0

119890minus1198981(120591minus119901)

radic119901[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901

(38)

6 Limiting Cases

In this section we discuss our present results and provideexact solutions for two special cases by taking suitableparameters equal to zero

61 Absence of MHD and Radiation Parameter By makingHartmann number Ha = 0 and radiation parameter Rd = 0

the obtained temperature profile (34) of isothermal caseimplies

120579 (120578 120591) = erfc(120578radicPr2radic120591

) (39)

which is quite in agreement with [28 equation (18)] Further-more by putting Ha = 0 and Rd = 0 in (35) it reduces to

119906 (120578 120591) =Gr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) 119889119901

+Gr (1 + 120582

1)

212058211989811198961Pr

radic1198962

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

1(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) erfc(

120578radicPr2radic119901

)119889119901

(40)

which is identical to the solution obtained by Khan [28equation (31)] In Figure 2 it is observed that the graph of(40) matches well (31) in [28] Hence this fact confirms theaccuracy of our results

62 The Case of 1205821= 0 (Second-Grade Fluid with MHD

and Radiation Effects) Interestingly by putting 1205821

= 0

the solutions in (31) and (35) can be reduced to second-grade fluid with the effects of radiation and MHD Thus thevelocity distribution for ramped wall temperature in (31) canbe written as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (41)

where

1199061198771(120578 120591) =

Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd)212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906119889119904 119889119901

8 Mathematical Problems in Engineering

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901 +Gr1205821198982

sdot radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(42)

and the velocity profile for isothermal case (35) reduces to

119906 (120578 120591) =Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+Gr (1 + Rd)21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(43)

However taking Ha = 0 Rd = 0 and Gr = 1 into (41) and(43) gives the well-known results

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (44)

where

1199061198771(120578 120591)

=1

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901

+1

212058211989611198982Prradic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782Pr2

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

+1

1205821198982radic120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+120578

2Pr4119901) sinh (1198982(120591 minus 119901)) 119889119901

(45)

119906 (120578 120591)

=1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+1

21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

(46)

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

with

1198981=Pr minus (1 + 120582

1) (1 + Rd)

2Pr 120582

1198982=radic((Pr minus (1 + 120582

1) (1 + Rd)) Pr 120582)2 + 4 (1 + 120582

1) (1 + Rd)HaPr 120582

4

(24)

The inverse Laplace transform of (22) leads to the followingexpressions

1199063(120591) = 119871

minus1

1199063(119902)

=Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

119890minus1198981120591 sinh (119898

2120591)

(25)

In order to determine the inverse Laplace transform of1199064(120578 120591) of the function 119906

4(120578 119902) we split (23) in the following

forms

1198601(120578 119902) =

1

1199022exp(minus120578radic

(1 + 1205821) (Ha + 119902)

1 + 120582119902) (26)

1198602(120578 119902) =

1

1199022exp(minus120578radic

Pr 1199021 + Rd

) (27)

Using the inversion formula of compound function and theinverse Laplace formula into (26) implies

1198601(120578 120591) = 119871

minus1

1198601(120578 119902) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904

(28)

and the inverse Laplace transform of (27) is given by

1198602(120578 120591) = 119871

minus1

1198602(120578 119902)

= (120591 +1205782

2

Pr1 + Rd

) erfc(120578

2radic

Pr120591 (1 + Rd)

)

minus 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(29)

Thus

1199064(120578 120591) = 120591119890

minus120578radic1198961 +

120578

21198961

sdot radic119887

120587int

120591

0

int

infin

0

(120591 minus 119904)

119906radic119906radic119906

119904119890minus((1198962119904+119906)119896

1+12057824119906)

1198681(2

1198962

sdot radic119887119906119904) 119889119906 119889119904 minus (120591 +1205782

2

Pr1 + Rd

)

sdot erfc(120578

2radic

Pr120591 (1 + Rd)

)

+ 120578radicPr 120591

120587 (1 + Rd)119890minus((12057824120591)(Pr(1+Rd)))

(30)

where 1198961= 120582(1 + 120582

1) 1198962= 1(1 + 120582

1) 119887 = 119896

2minus 1198961Ha and

1198681(sdot) is modified Bessel function of the first kind of order one

Consequently the expression for velocity distribution can bewritten as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (31)

where 1199061198771(120578 120591) can be found by using convolution theorem

1199061198771(120578 120591) = (119906

3lowast 1199064) (120591)

= int

120591

0

1199063(120591 minus 119901) 119906

4(120578 119901) 119889119901

(32)

in which

1199061198771(120578 120591) =

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

6 Mathematical Problems in Engineering

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

+Gr (1 + 120582

1)

1205821198982

radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(33)

4 Solution for an Isothermal Plate

The analytical solution of temperature and velocity distri-bution for the case of ramped wall temperature are shownin (20) and (31) It is good to compare these results withthose corresponding to the flow near a plate with uniformtemperature to see the effects of ramped wall temperature of

the plate on the fluid flowThe solution for temperature profilecan be obtained as

120579 (120578 120591) = erfc(120578

2radic120591radic

Pr1 + Rd

) (34)

and using the same way as before the expression of velocityprofile for case of isothermal is

119906 (120578 120591) =Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp(minus1198981(120591 minus 119901) minus

120578

radic1198961

) sinh (1198982(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

sdot int

120591

0

int

119901

0

int

infin

0

120578

2119906radic120587119906radic119887119906

119904

1

1198961

exp[minus(1198981(120591 minus 119901) +

1198962119904 + 119906

1198961

+1205782

4119906)] sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)

119889119906119889119904 119889119901

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp (minus1198981(120591 minus 119901)) sinh (119898

2(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(35)

5 Nusselt Number and Skin Friction

Theexpression ofNusselt numberNu and skin friction119865(120591)for both cases ramped wall temperature and isothermalis discussed in this section The Nusselt number and skinfriction for ramped wall temperature can be written as

Nu = minus 120597120579

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

=2

radic120587radic

Pr1 + Rd

[radic120591 minus radic120591 minus 1119867 (120591 minus 1)]

119865 (120578 120591) =1

1 + 1205821

(1 + 120582120597

120597120591)120597119906

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

= 1198651(120591) minus 119865

1(120591 minus 1)119867 (120591 minus 1)

(36)

where

1198651(120591) = minus

Gr (1 + Rd)Prradic1198961

int

120591

0

119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901

+ 2Grradic1 + Rd120587Pr

int

120591

0

radic119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901 +

Gr (1 + Rd)21198961radic120587Pr

sdot int

120591

0

int

119901

0

int

infin

0

(119901 minus 119904)

119906radic119887

119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]

119889119906119889119904 119889119901

(37)

Mathematical Problems in Engineering 7

and Nusselt number and skin friction for isothermal aregiven by

Nu = 1

radic120587120591radic

Pr1 + Rd

119865 (120591) = minusGr (1 + Rd)

1198961Pr

int

120591

0

119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901 +

Gr (1 + Rd)21198961Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

1

119906radic119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119906 119889119904 119889119901

+ Grradic1 + Rd120587Pr

int

120591

0

119890minus1198981(120591minus119901)

radic119901[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901

(38)

6 Limiting Cases

In this section we discuss our present results and provideexact solutions for two special cases by taking suitableparameters equal to zero

61 Absence of MHD and Radiation Parameter By makingHartmann number Ha = 0 and radiation parameter Rd = 0

the obtained temperature profile (34) of isothermal caseimplies

120579 (120578 120591) = erfc(120578radicPr2radic120591

) (39)

which is quite in agreement with [28 equation (18)] Further-more by putting Ha = 0 and Rd = 0 in (35) it reduces to

119906 (120578 120591) =Gr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) 119889119901

+Gr (1 + 120582

1)

212058211989811198961Pr

radic1198962

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

1(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) erfc(

120578radicPr2radic119901

)119889119901

(40)

which is identical to the solution obtained by Khan [28equation (31)] In Figure 2 it is observed that the graph of(40) matches well (31) in [28] Hence this fact confirms theaccuracy of our results

62 The Case of 1205821= 0 (Second-Grade Fluid with MHD

and Radiation Effects) Interestingly by putting 1205821

= 0

the solutions in (31) and (35) can be reduced to second-grade fluid with the effects of radiation and MHD Thus thevelocity distribution for ramped wall temperature in (31) canbe written as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (41)

where

1199061198771(120578 120591) =

Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd)212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906119889119904 119889119901

8 Mathematical Problems in Engineering

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901 +Gr1205821198982

sdot radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(42)

and the velocity profile for isothermal case (35) reduces to

119906 (120578 120591) =Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+Gr (1 + Rd)21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(43)

However taking Ha = 0 Rd = 0 and Gr = 1 into (41) and(43) gives the well-known results

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (44)

where

1199061198771(120578 120591)

=1

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901

+1

212058211989611198982Prradic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782Pr2

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

+1

1205821198982radic120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+120578

2Pr4119901) sinh (1198982(120591 minus 119901)) 119889119901

(45)

119906 (120578 120591)

=1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+1

21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

(46)

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

+Gr (1 + 120582

1)

1205821198982

radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(33)

4 Solution for an Isothermal Plate

The analytical solution of temperature and velocity distri-bution for the case of ramped wall temperature are shownin (20) and (31) It is good to compare these results withthose corresponding to the flow near a plate with uniformtemperature to see the effects of ramped wall temperature of

the plate on the fluid flowThe solution for temperature profilecan be obtained as

120579 (120578 120591) = erfc(120578

2radic120591radic

Pr1 + Rd

) (34)

and using the same way as before the expression of velocityprofile for case of isothermal is

119906 (120578 120591) =Gr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp(minus1198981(120591 minus 119901) minus

120578

radic1198961

) sinh (1198982(120591 minus 119901)) 119889119901 +

Gr (1 + Rd) (1 + 1205821)

Pr 1205821198982

sdot int

120591

0

int

119901

0

int

infin

0

120578

2119906radic120587119906radic119887119906

119904

1

1198961

exp[minus(1198981(120591 minus 119901) +

1198962119904 + 119906

1198961

+1205782

4119906)] sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)

119889119906119889119904 119889119901

minusGr (1 + Rd) (1 + 120582

1)

Pr 1205821198982

int

120591

0

exp (minus1198981(120591 minus 119901)) sinh (119898

2(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(35)

5 Nusselt Number and Skin Friction

Theexpression ofNusselt numberNu and skin friction119865(120591)for both cases ramped wall temperature and isothermalis discussed in this section The Nusselt number and skinfriction for ramped wall temperature can be written as

Nu = minus 120597120579

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

=2

radic120587radic

Pr1 + Rd

[radic120591 minus radic120591 minus 1119867 (120591 minus 1)]

119865 (120578 120591) =1

1 + 1205821

(1 + 120582120597

120597120591)120597119906

120597120578

10038161003816100381610038161003816100381610038161003816120578=0

= 1198651(120591) minus 119865

1(120591 minus 1)119867 (120591 minus 1)

(36)

where

1198651(120591) = minus

Gr (1 + Rd)Prradic1198961

int

120591

0

119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901

+ 2Grradic1 + Rd120587Pr

int

120591

0

radic119901119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901 +

Gr (1 + Rd)21198961radic120587Pr

sdot int

120591

0

int

119901

0

int

infin

0

(119901 minus 119904)

119906radic119887

119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]

119889119906119889119904 119889119901

(37)

Mathematical Problems in Engineering 7

and Nusselt number and skin friction for isothermal aregiven by

Nu = 1

radic120587120591radic

Pr1 + Rd

119865 (120591) = minusGr (1 + Rd)

1198961Pr

int

120591

0

119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901 +

Gr (1 + Rd)21198961Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

1

119906radic119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119906 119889119904 119889119901

+ Grradic1 + Rd120587Pr

int

120591

0

119890minus1198981(120591minus119901)

radic119901[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901

(38)

6 Limiting Cases

In this section we discuss our present results and provideexact solutions for two special cases by taking suitableparameters equal to zero

61 Absence of MHD and Radiation Parameter By makingHartmann number Ha = 0 and radiation parameter Rd = 0

the obtained temperature profile (34) of isothermal caseimplies

120579 (120578 120591) = erfc(120578radicPr2radic120591

) (39)

which is quite in agreement with [28 equation (18)] Further-more by putting Ha = 0 and Rd = 0 in (35) it reduces to

119906 (120578 120591) =Gr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) 119889119901

+Gr (1 + 120582

1)

212058211989811198961Pr

radic1198962

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

1(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) erfc(

120578radicPr2radic119901

)119889119901

(40)

which is identical to the solution obtained by Khan [28equation (31)] In Figure 2 it is observed that the graph of(40) matches well (31) in [28] Hence this fact confirms theaccuracy of our results

62 The Case of 1205821= 0 (Second-Grade Fluid with MHD

and Radiation Effects) Interestingly by putting 1205821

= 0

the solutions in (31) and (35) can be reduced to second-grade fluid with the effects of radiation and MHD Thus thevelocity distribution for ramped wall temperature in (31) canbe written as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (41)

where

1199061198771(120578 120591) =

Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd)212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906119889119904 119889119901

8 Mathematical Problems in Engineering

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901 +Gr1205821198982

sdot radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(42)

and the velocity profile for isothermal case (35) reduces to

119906 (120578 120591) =Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+Gr (1 + Rd)21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(43)

However taking Ha = 0 Rd = 0 and Gr = 1 into (41) and(43) gives the well-known results

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (44)

where

1199061198771(120578 120591)

=1

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901

+1

212058211989611198982Prradic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782Pr2

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

+1

1205821198982radic120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+120578

2Pr4119901) sinh (1198982(120591 minus 119901)) 119889119901

(45)

119906 (120578 120591)

=1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+1

21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

(46)

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

and Nusselt number and skin friction for isothermal aregiven by

Nu = 1

radic120587120591radic

Pr1 + Rd

119865 (120591) = minusGr (1 + Rd)

1198961Pr

int

120591

0

119890minus1198981(120591minus119901)

[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119901 +

Gr (1 + Rd)21198961Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

1

119906radic119904119890minus((1198962119904+119906)119896

1+1198981(120591minus119901))

1198681(2

1198961

radic119887119906119904)[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))]119889119906 119889119904 119889119901

+ Grradic1 + Rd120587Pr

int

120591

0

119890minus1198981(120591minus119901)

radic119901[(1 minus 120582119898

1)

1205821198982

sinh (1198982(120591 minus 119901)) + cosh (119898

2(120591 minus 119901))] 119889119901

(38)

6 Limiting Cases

In this section we discuss our present results and provideexact solutions for two special cases by taking suitableparameters equal to zero

61 Absence of MHD and Radiation Parameter By makingHartmann number Ha = 0 and radiation parameter Rd = 0

the obtained temperature profile (34) of isothermal caseimplies

120579 (120578 120591) = erfc(120578radicPr2radic120591

) (39)

which is quite in agreement with [28 equation (18)] Further-more by putting Ha = 0 and Rd = 0 in (35) it reduces to

119906 (120578 120591) =Gr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) 119889119901

+Gr (1 + 120582

1)

212058211989811198961Pr

radic1198962

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

1(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + 120582

1)

Pr 1205821198981

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198981(120591 minus 119901)) erfc(

120578radicPr2radic119901

)119889119901

(40)

which is identical to the solution obtained by Khan [28equation (31)] In Figure 2 it is observed that the graph of(40) matches well (31) in [28] Hence this fact confirms theaccuracy of our results

62 The Case of 1205821= 0 (Second-Grade Fluid with MHD

and Radiation Effects) Interestingly by putting 1205821

= 0

the solutions in (31) and (35) can be reduced to second-grade fluid with the effects of radiation and MHD Thus thevelocity distribution for ramped wall temperature in (31) canbe written as

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (41)

where

1199061198771(120578 120591) =

Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901 +

Gr (1 + Rd)212058211989611198982Pr

sdot radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906119889119904 119889119901

8 Mathematical Problems in Engineering

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901 +Gr1205821198982

sdot radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(42)

and the velocity profile for isothermal case (35) reduces to

119906 (120578 120591) =Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+Gr (1 + Rd)21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(43)

However taking Ha = 0 Rd = 0 and Gr = 1 into (41) and(43) gives the well-known results

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (44)

where

1199061198771(120578 120591)

=1

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901

+1

212058211989611198982Prradic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782Pr2

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

+1

1205821198982radic120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+120578

2Pr4119901) sinh (1198982(120591 minus 119901)) 119889119901

(45)

119906 (120578 120591)

=1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+1

21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

(46)

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782

2

Pr1 + Rd

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901 +Gr1205821198982

sdot radic1 + Rd120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+(120578

24119901)(Pr(1+Rd))) sinh (119898

2(120591 minus 119901)) 119889119901

(42)

and the velocity profile for isothermal case (35) reduces to

119906 (120578 120591) =Gr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+Gr (1 + Rd)21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minusGr (1 + Rd)Pr 120582119898

2

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radic119901radic

Pr1 + Rd

)119889119901

(43)

However taking Ha = 0 Rd = 0 and Gr = 1 into (41) and(43) gives the well-known results

119906 (120578 120591) = 1199061198771(120578 120591) minus 119906

1198771(120578 120591 minus 1)119867 (120591 minus 1) (44)

where

1199061198771(120578 120591)

=1

Pr 1205821198982

int

120591

0

119901119890minus(1198981(120591minus119901)+120578radic119896

1) sinh (119898

2(120591 minus 119901)) 119889119901

+1

212058211989611198982Prradic119887

120587int

120591

0

int

119901

0

int

infin

0

120578 (119901 minus 119904)

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)

(119901 +1205782Pr2

) sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

+1

1205821198982radic120587Pr

int

120591

0

120578radic119901119890minus(1198981(120591minus119901)+120578

2Pr4119901) sinh (1198982(120591 minus 119901)) 119889119901

(45)

119906 (120578 120591)

=1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) 119889119901

+1

21198961Pr 120582119898

2

radic119887

120587int

120591

0

int

119901

0

int

infin

0

120578

119906radic119904119890minus(1198981(120591minus119901)+(119896

2119904+119906)119896

1+12057824119906) sinh (119898

2(120591 minus 119901)) 119868

1(2

1198961

radic119887119906119904)119889119906 119889119904 119889119901

minus1

Pr 1205821198982

int

120591

0

119890minus1198981(120591minus119901)minus120578radic119896

1 sinh (1198982(120591 minus 119901)) erfc(

120578

2radicPr119901)119889119901

(46)

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

2 4 60120578

0

005

01

u(120578120591)

Khan [28] (31)Equation (40) when Ha = Rd = 0

Equation (40) when Ha = Rd = 1

Equation (40) when Ha = Rd = 2

Figure 2 Comparison of velocity 119906(120578 120591) in (40) with (31) obtainedby Khan [28]

0

005

01

015

02

u(120578120591)

2 4 60120578

Samiulhaq et al [29] (11)Second-grade fluid (46) when Rd = Ha = 0

Second-grade fluid (43) when Rd = Ha = 05

Jeffrey fluid (35) when Rd = Ha = 05

Figure 3 Comparison of velocity 119906(120578 120591) for isothermal in (46) with(11) obtained by Samiulhaq et al [29]

Equation (46) is identical to the published results obtained bySamiulhaq et al [29 equation (11)] Furthermore in Figure 3it is showed that the graph of (46) matches well (11) in [29]Hence we can say the results are found to be in excellentagreement

7 Results and Discussions

Numerical computation of analytical expressions (20) (31)(34) and (35) of the present problem has been carriedout for various values of embedded parameters namelydimensionless Jeffrey fluid parameter 120582 Prandtl numberPr Grashof number Gr Hartmann number Ha radiationparameter Rd and dimensionless time 120591 The velocity and

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

120582 = 08 10 20

Figure 4 Velocity profile for different values of 120582 when Pr = 071Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

005

01u(120578120591)

Pr = 071 10 70

2 4 60120578

Figure 5 Velocity profile for different values of Pr when 120582 = 1Rd = 2 Gr = 1 Ha = 2 and 120591 = 05

temperature profiles are illustrated graphically in Figures 4ndash12 in order to give clear insight of physical problem of Jeffreyfluid

The influence of Jeffrey fluid parameter 120582 on velocityprofile is shown in Figure 4 It is found that the momen-tum boundary layer thickness increases as the Jeffrey fluidparameter effect is strengthened Hence the velocity profiledecreases due to this fact Moreover the fluid velocity forramped wall temperature is always lower compared to thecase of isothermal The variations of velocity profile for threedifferent values of Prandtl number Pr = 071 10 70 whichcorrespond to air electrolyte solution andwater respectivelyare plotted in Figure 5 It is noticed that the increase of Prmakes the fluid motion slower Physically it is justified dueto the fact that the fluid with high Prandtl number has highviscosity which makes the fluid thick As a result the velocitydecreases

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

IsothermalRamped

0

005

01

u(120578120591)

2 4 60120578

Rd = 00 05 10 20

Figure 6 Velocity profile for different values of Rd when 120582 = 1Pr = 071 Gr = 1 Ha = 2 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

2 4 60120578

Gr = 00 05 10 20

Figure 7 Velocity profile for different values of Gr when 120582 = 1Pr = 071 Rd = 2 Ha = 2 and 120591 = 05

Further the effect of radiation parameter Rd on thevelocity profile is depicted in Figure 6 The trend shows thatvelocity increases with increasing values of radiation parame-ter This is because when the intensity of radiation parameterincreasedwhich in turn increased the rate of energy transportto the fluids the bond holding the components of the fluidparticles is easily broken and thus decreases the viscosityThiswill make the fluid move faster which leads to increase ofthe velocity profileThe graphical results for Grashof numberare presented in Figure 7 Physically the Grashof number isdefined as the ratio of buoyancy forces to viscous forces Largevalues of Gr enhance the buoyancy forces which gives rise toan increase in the induced flow

The effect of Hartmann number Ha upon velocity 119906(120578 120591)is elucidated from Figure 8 As expected an increase in Hareduces the velocity profile This is because of the Lorentzforce which arises due to the application of magnetic field toan electrically conducting fluid and gives rise to a resistance

IsothermalRamped

2 4 60120578

0

005

01

u(120578120591)

Ha = 0 2 4 6

Figure 8 Velocity profile for different values of Ha when 120582 = 1Pr = 071 Rd = 2 Gr = 1 and 120591 = 05

IsothermalRamped

0

01

02

u(120578120591)

5 100120578

120591 = 08 14

Figure 9 Velocity profile for different values of 120591 when 120582 = 1 Pr =071 Rd = 2 Gr = 1 and Ha = 2

force Due to this force the motion of fluid flow in momen-tum boundary layer tends to retard On the other handFigure 9 shows the effect of dimensionless time 120591 towardsvelocity 119906(120578 120591) It can be seen from the figure that velocityis an increasing function of 120591

Figure 10 reveals that the temperature profile is reducedwhenPr is increasedThe explanation for such behavior lies inthe fact that Prandtl number is defined as the ratio ofmomen-tum diffusivity to thermal diffusivity Therefore at smallervalues of Pr fluids possess high thermal conductivity whichmakes the heat diffuse away from the heated surface morerapidly and faster compared to higher values of Prandtl num-ber Thus increase the boundary layer thickness and conse-quently decrease the temperature distribution The influenceof radiation Rd on temperature field is demonstrated inFigure 11 As anticipated an increase in Rd increases thetemperature profile since radiation parameter signifies therelative contribution of conduction heat transfer to thermal

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

IsothermalRamped

0

02

04

06

08

2 4 60120578

Pr = 071 10 70

120579(120578120591)

Figure 10 Temperature profile for different values of Pr when Rd =2 and 120591 = 05

IsothermalRamped

0

02

04

06

08

2 4 60120578

Rd = 00 05 10 20

120579(120578120591)

Figure 11 Temperature profile for different values of Rd when Pr =071 and 120591 = 05

radiation transfer Finally the effect of dimensionless time 120591on temperature profiles is sketched in Figure 12 It is also clearthat temperature is an increasing function of 120591

8 Conclusion

This study presents a theoretical investigation of thermalradiation in unsteady MHD free convection flow of Jeffreyfluid with ramped wall temperature The dimensionless gov-erning equations are solved by using the Laplace transformmethod Graphical results for velocity and temperature areobtained to understand the physical behavior for severalembedded parameters The results show that increasing 120582Rd and 120591 decreases fluid velocity for both ramped walltemperature and isothermal plates Meanwhile the behaviorof velocity and temperature profiles for air is greater thanwater However increasing Gr boosts the fluid motion due tothe enhancement in buoyancy force As expected the effectof Ha shows the opposite behavior It is because the Lorentz

IsothermalRamped

0

02

04

06

08

5 100120578

120591 = 03 05 08 14

120579(120578120591)

Figure 12 Temperature profile for different values of 120591 when Pr =071 and Rd = 2

force tends to decelerate the fluid flow Further the influencesof Rd and 120591 increase the temperature profile Finally weobserved that the boundary layer thickness for ramped walltemperature is always less than isothermal plate

Conflict of Interests

The authors of this paper do not have any conflict of interestsregarding its publication

References

[1] S Nadeem B Tahir F Labropulu and N S Akbar ldquoUnsteadyoscillatory stagnation point flow of a Jeffrey fluidrdquo Journal ofAerospace Engineering vol 27 no 3 pp 636ndash643 2014

[2] T Hayat and M Mustafa ldquoInfluence of thermal radiation onthe unsteady mixed convection flow of a Jeffrey fluid over astretching sheetrdquo Zeitschrift fur Naturforschung vol 65 pp 711ndash719 2010

[3] T Hayat S A Shehzad M Qasim and S Obaidat ldquoRadiativeflow of Jeffery fluid in a porous medium with power law heatflux and heat sourcerdquo Nuclear Engineering and Design vol 243pp 15ndash19 2012

[4] S A Shehzad A Alsaedi and T Hayat ldquoInfluence of ther-mophoresis and joule heating on the radiative flow of Jeffreyfluid with mixed convectionrdquo Brazilian Journal of ChemicalEngineering vol 30 no 4 pp 897ndash908 2013

[5] S A Shehzad Z Abdullah A Alsaedi F M Abbasi and THayat ldquoThermally radiative three-dimensional flow of Jeffreynanofluid with internal heat generation and magnetic fieldrdquoJournal of Magnetism and Magnetic Materials vol 397 pp 108ndash114 2016

[6] T Hussain S A Shehzad T Hayat A Alsaedi F Al-Solamyand M Ramzan ldquoRadiative hydromagnetic flow of Jeffreynanofluid by an exponentially stretching sheetrdquo PLoS ONE vol9 no 8 Article ID e103719 9 pages 2014

[7] K Das N Acharya and P K Kundu ldquoRadiative flow of MHDJeffrey fluid past a stretching sheet with surface slip andmeltingheat transferrdquoAlexandria Engineering Journal vol 54 no 4 pp815ndash821 2015

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[8] T Hayat and N Ali ldquoPeristaltic motion of a Jeffrey fluid underthe effect of a magnetic field in a tuberdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 7 pp1343ndash1352 2008

[9] A M Abd-Alla S M Abo-Dahab and M M AlbalawildquoMagnetic field and gravity effects on peristaltic transport of aJeffrey fluid in an asymmetric channelrdquo Abstract and AppliedAnalysis vol 2014 Article ID 896121 11 pages 2014

[10] A M Abd-Alla and S M Abo-Dahab ldquoMagnetic field androtation effects on peristaltic transport of a Jeffrey fluid inan asymmetric channelrdquo Journal of Magnetism and MagneticMaterials vol 374 pp 680ndash689 2015

[11] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of a Jeffrey fluidin an asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[12] K Kavita K Ramakrishna Prasad and B Aruna KumarildquoInfluence of heat transfer on MHD oscillatory flow of jeffreyfluid in a channelrdquo Advances in Applied Science Research vol 3no 4 pp 2312ndash2325 2012

[13] A S Idowu K M Joseph and S Daniel ldquoEffect of heat andmass transfer on unsteadyMHD oscillatory flow of Jeffrey fluidin a horizontal channel with chemical reactionrdquo IOSR Journalof Mathematics vol 8 no 5 pp 74ndash87 2013

[14] A Ali and S Asghar ldquoAnalytic solution for oscillatory flow in achannel for Jeffrey fluidrdquo Journal of Aerospace Engineering vol27 no 3 pp 644ndash651 2014

[15] M Khan ldquoPartial slip effects on the oscillatory flows of afractional Jeffrey fluid in a porous mediumrdquo Journal of PorousMedia vol 10 no 5 pp 473ndash487 2007

[16] T Hayat M Khan K Fakhar and N Amin ldquoOscillatoryrotating flows of a fractional Jeffrey fluid filling a porous spacerdquoJournal of Porous Media vol 13 no 1 pp 29ndash38 2010

[17] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur Naturforschungvol 65 no 6-7 pp 540ndash548 2010

[18] M A AHamad SM AbdEl-Gaied andWA Khan ldquoThermaljump effects on boundary layer flow of a jeffrey fluid near thestagnation point on a stretchingshrinking sheet with variablethermal conductivityrdquo Journal of Fluids vol 2013 Article ID749271 8 pages 2013

[19] T Hayat S Asad M Mustafa and A Alsaedi ldquoMHDstagnation-point flow of Jeffrey fluid over a convectively heatedstretching sheetrdquo Computers amp Fluids vol 108 pp 179ndash1852015

[20] M Qasim ldquoHeat and mass transfer in a Jeffrey fluid over astretching sheet with heat sourcesinkrdquo Alexandria EngineeringJournal vol 52 no 4 pp 571ndash575 2013

[21] N Dalir ldquoNumerical study of entropy generation for forcedconvection flow and heat transfer of a Jeffrey fluid over astretching sheetrdquo Alexandria Engineering Journal vol 53 no 4pp 769ndash778 2014

[22] N Dalir M Dehsara and S S Nourazar ldquoEntropy analysisfor magnetohydrodynamic flow and heat transfer of a Jeffreynanofluid over a stretching sheetrdquo Energy vol 79 pp 351ndash3622015

[23] T Hayat M Awais and S Obaidat ldquoThree-dimensional flow ofa Jeffery fluid over a linearly stretching sheetrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 2 pp699ndash707 2012

[24] T Hayat M Awais and A Alsaedi ldquoNewtonian heating andmagnetohydrodynamic effects in flow of a Jeffery fluid over aradially stretching surfacerdquo International Journal of the PhysicalSciences vol 7 no 21 pp 2838ndash2844 2012

[25] T Hayat S Asad M Qasim and A A Hendi ldquoBoundary layerflow of a Jeffrey fluid with convective boundary conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 69no 8 pp 1350ndash1362 2012

[26] S A Shehzad T Hayat M S Alhuthali and S Asghar ldquoMHDthree-dimensional flow of Jeffrey fluid with newtonian heatingrdquoJournal of Central South University vol 21 no 4 pp 1428ndash14332014

[27] M Farooq N Gull A Alsaedi and T Hayat ldquoMHD flow of aJeffrey fluid with Newtonian heatingrdquo Journal of Mechanics vol31 no 3 pp 319ndash329 2015

[28] I Khan ldquoA note on exact solutions for the unsteady freeconvection flow of a jeffrey fluidrdquo Zeitschrift fur NaturforschungA vol 70 no 6 pp 272ndash284 2015

[29] Samiulhaq I Khan F Ali and S Shafie ldquoFree convection flowof a second-grade fluid with ramped wall temperaturerdquo HeatTransfer Research vol 45 no 7 pp 579ndash588 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of