mhd free convective flow past a porous plate · the stagnation point flow due to a shrinking sheet...

MHD FREE CONVECTIVE FLOW PAST A POROUS PLATE

P. Rama Krishna Reddy* M. C. Raju**

* Research scholar, Department of Mathematics, Shri Venkateshwara University, VenkateshwaraNagar,

Rajabpur, Gajraula, Amroha, Uttar Pradesh 244236, U. P., India. Email: [email protected]

** Research supervisor and corresponding author, Email: [email protected], Cell: 9848998649

Abstract:This manuscript is focused on unsteady magentohydrodynamic (MHD) free convective

flow of a double diffusive fluid past a moving vertical porous plate in the presence of thermal

radiation and first order homogeneous chemical reaction. The temperature of the plate is

assumed span wise cosinusoidally fluctuating with time in the presence of heat generation. The

second order perturbation technique is employed to investigate the non- linear partial differential

equations governing the fluid flow which are non- dimensionalized by introducing the similarity

transformations. The effects of magnetic intensity, radiation, porous permeability, Eckert

number, Schmidt number and heat generation/absorption parameters on velocity, concentration

and temperature. Also, the skin friction coefficients, the rate of heat transfer and rate of mass

transfer at the surface of the plate are computed numerically. The results show that within the

boundary layer, the velocity and temperature are found to decrease with the increasing values of

Prandtl number and radiation parameters however the trend is reverse with respect to porous

permeability and heat generation/absorption parameters.

Key words:

Fluctuatingtemperature,Chemical reaction,Suction,Heat generation, MHD, Double diffusive

fluid and porous medium.

List of symbols

cp Specfic heat at constant pressure (J kg-1 K-1)

C Species concentration (mol m-3)

C Dimensionless concentration

Cf Skin friction

C Concentration away from the wall(mol m-3)

wC Concentration at the wall(mol m-3)

D Chemical molecular diffusivity (m2s-1)

Ec Eckert number

International Journal of Pure and Applied MathematicsVolume 118 No. 5 2018, 507-529ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

507

g Acceleration due to gravity (m/s2)

Gr Grashof number

Gm Solutal Grashof number

H0 Magnetic field(N/Am)

k Thermal conductivity( W m-1K-1)

kc Rosseland mean absorption coefficient (m-1)

K Permeability of the porous medium

K1 Rate of chemical reaction parameter

K* Permeability of porous medium (m2)

K Dimensionless permeability

l Wave length (m)

M Hartmann number

Nu Nusselt number

Pr Prandtl number

0Q

Heat generation or absorption coefficient

rq Radiative heat flux ( W m-2)

Re Raynolds number

R Radiation parameter

So Soret number

Sc Schmidt number

T Fluid temperature (K)

wT Wall temperature (K)

0T Mean temperature (K)

T Free stream dimensional temperature(K)

T Dimensionless time

u Dimensionless velocity

e Magnetic permeability (H m-1 or N A-1)

Dimensionless frequency of oscillation

Dimensionless chemical reaction parameter

Dimensionless temperature

Coefficient of thermal expansion

Coefficient of concentration expansion

Dimensionless heat generation/ absorption parameter

v Kinematic viscosity (m2s)

Density of the fluid ( kgm-3)

Electrical conductivity(S/m)

s Stefan-Boltzmann constant(Wm-2K-4)

International Journal of Pure and Applied Mathematics Special Issue

508

wvu Velocity component along zyx directions (m s-1)

1. Introduction:

The study of heat flow with heat and mass transfer has become industrially more

important as most of the engineering processes occur at a high temperature such as in nuclear

power plants,gas turbines and various propulsion devices for aircraft,missiles,satellites and space

vehicles.In all of these applications,the radiation effects can be quite significant at high

temperature.Further, in astrophysical regimes,the presence of planetary debris,cosmic

dusted,creates a suspended porous medium saturated with plasa fluids.As in other porous media

problems such as aeromechanics and insulation engineering, the conventional approach is to

simulate the pressure drop across the porous regime. Israel-Cookey et al. [1] examined the

influence of viscous dissipation and radiation on unsteady MHD free- convection flow past an

infinite heated vertical plate in a porous medium with time dependent suction. Chamka [2]

examined an unsteady simultaneous convection heat and mass transfer flow along a vertical

permeable plate embedded in a fluid saturated porous medium in the presence of mass blowing

or suction, magnetic field and heat absorption. Pal and Mandal [3] studied the effects of thermal

radiation onMHD Darcy Forchheimer convective flow past a stretching sheet in porous medium.

Turkyilmazoglu and Pop [4] discussed heat and mass transfer effects on unsteadyMHDfree

convection flow past an impulsively started infinite vertical plate with Soret and heat source

effects. Ram et al. [5] examined the effects of porosity on unsteadyMHDfree convective

boundary layer flow along a semi- infinite vertical plate with time dependent suction by taking

into account the effects of dissipation. Parsa et al. [6] presented steady laminar MHD boundary

layer flow past a stretching surface with uniform free stream and internal heat generation or

absorption in an electrically conducting fluid. The transient MHDfree convective flow of a

viscous, incompressible, electrically conducting, gray, absorbing-emitting, but not scattering,

optically thick fluid medium which occupies a semi-infinite porous region adjacent to an infinite

hot vertical plate moving with constant velocity was presented by Ahmed and Kalita [7]. The

influence of thermal radiation and Darcian drag forceon unsteady MHD thermal-convection flow

past a semi-infinite vertical plate immersed in a semi-infinite saturated porous regime with

variable surface temperature in the presence of transversal uniform magnetic field have been

discussed by Ahmed et al.[8]. Singh et al. [9] analysized the unsteady MHD free convection,

incompressible, electrically conducting and fluctuating flow past an impulsively started

isothermal vertical plate by taking into account the effect of viscous dissipation. Raju and Varma

[10] considered an unsteady MHD free convective, thermal diffusive and viscoelastic fluid with

heat and mass transfer in porous medium bounded by an infinite vertical porous plate with heat

absorption and dissipation. Ziyauddin et al. [11] investigated MHDheat transfer flow of a


509

micropolar fluid past a wedge with heat generation or absorption by considering the radiation

effect along with viscous dissipation, Joule heating, and Hall and ion-slip effects. Su [12]

examined unsteady MHDmixed convective flow and heat transfer over an impulsively stretched

permeable vertical surface with partial velocity and thermal slip conditions in the presence of

thermal heat generation and suction/injection. Hazarika and Ahmed [13] analyzed an unsteady

MHD flow of two-dimensional, laminar, incompressible, Newtonian, electrically-conducting and

radiating fluid along a semi-infinite vertical permeable moving plate with periodic heat and mass

transfer by taking into account the effect of viscous dissipation in presence of chemical reaction.

Sahoo et al. [14] consideredMHD unsteady free convection flow past an infinite vertical plate

with constant suction and heat sink. Singh and Kumar [15] studied fluctuating heat and mass

transfer on unsteady MHDfree convection flow of radiating and reacting fluid past a vertical

porous plate in slip- flow regime. Mishara et al. [16] examined heat and mass transfer in the

MHDflow of avisco-elstic fluid in a rotating porous channel with radiative heat and chemical

reaction. Hussaini et al. [17] discussed MHD unsteady radiating memory convective flow with

variable suction. Pal et al. [18] investigated steady magneto hydro dynamic (MHD) boundary

layer flow of a Casson nanofluid over a vertical stretching surface with combined effects of

thermal radiation, Ohmic dissipation, thermophoresis and Brownian motion on heat and mass

transfer. Vajravelu et al. [19] studied non-linear convection effects on the flow past a flatporous

plate. Chand and kumar [20] described the effects of Hall current and slip conditions on heat and

mass transfer in the unsteady flow of the viscoelastic fluid (Walter’s liquid B’ model) past an

infinite porousflat plate subjected to uniform suction through a porous medium. Singh and

Pathak [21] focused the effects of Hall current and rotation on MHD free convection flow in a

vertical rotating channel filled with theporous medium. Srinivsasacharya and Reddy [22]

investigated on mixed convection heat and mass transfer from a vertical plate embedded in a

power-law fluid- saturatedporous mediumwith radiation and chemical reaction effects.

Chaudhary and Kumar [23] described steady two- dimensional, laminar, viscous incompressible

stagnation point flow past a porous medium with heat generation of an electrically conducting

fluid over a stretching surface in the presence of a magnetic fluid. Pal and Mandal [24] examined

mixed convection-radiation interaction on stagnation-point flow of nanofluids over a

stretching/shrinking sheet embedded in a porous medium in the presence of viscous dissipation

and internal heat generation. Kumar and Sood [25] investigated non-linear convection effects on

the stagnation point flow due to a shrinking sheet in the presence of the porous medium.

Soundalgekar [26] analyzed the viscous dissipative heat on the two-dimensional unsteady free

convective flow past an infinite vertical porous platewhen the temperature oscillates in time and

there is constant suction at the plate.

In the chemical engineering processes, the chemical reaction occurs between a foreign

mass and the fluid in which the plate is moving. These processes take place in many industrial

areas that is manufacturing of ceramics or glassware, polymer production, and food

processing.Prakash and Ogulu [27] analylized the problem of unsteady two-dimensional

boundary layer flow of a viscous incompressible, electrically conducting fluid along a semi-


510

infinite vertical plate in the presence of thermal and concentration buoyancy effects. The effects

ofchemical reactionand heat generation on unsteady MHD free convection heat and mass transfer

flow past an infinite hot vertical porous plate are obtained by Singh and Kumar [28]. Ibrahim and

Mankinde [29] investigated radiation effect onchemically reaction MHD boundary layer flow of

heat and mass transfer past a porous vertical flat plate. The influence of heat absorption,

radiation, Joule heating, viscous dissipation, chemical reactionand thermal diffusion on unsteady

MHD Couette flow of a dusty viscoelastic fluid in a vertical irregular porous channel with

convective boundary and varying mass diffusion was discussed by Sivaraj and Kumar[ 30]. Rout

and Pattanayak [31] have examined and reported chemical reaction and radiation effects on

MHD flow past an exponentially accelerated vertical porous plate in the presence of heat

generation with variable temperature embedded in a porous medium. Venkateswarlu and Satya

naraya [32] examined radiation absorption andchemical reactioneffectson MHD free convection

heat transfer flow of a nanofluid bounded by semi-infinite flat plate in a rotating system. Mishra

et al. [33] dealt with heat and mass transfer characteristic of MHD visco-elastic fluid in a rotating

porous channel with radiative heat and chemical reaction. Garg et al. [34] investigated on

chemically reacting, radiating and rotating MHD convective flow of visco-elastic fluid through

porous mediumin vertical channel. Makinde et al. [35] studied the effects of buoyancy forces,

homogeneous chemical reaction, thermal radiation, partial slip, heat source, Thermophoresis and

Brownian motion on hydromagnetic stagnation point flow of nanofluid with heat and mass

transfer over a stretching convective surface. Srinivasacharya and Jagadeeshwar [36]

investigated the boundary layer flow, heat and mass transfer towards the exponentially stretching

sheet in a viscous fluid. Loganathan and Ganesan [37] analyzed the effects of radiation on the

flow past an impulsively started infinite vertical plate in the presence of mass transfer. Nadeem

et al. [38] investigated the boundary layer flow behavior of a Jeffery fluid due to an

exponentially stretching surface and explained the effects of thermal radiation for two cases of

heat transfer analysis known as Prescribed exponential order surface temperature (PEST) and

Prescribed exponential order heat flux (PEHF).

In this paper, we have extended the work of Ramet al. [39] and investigated the effects of

thermal radiation and chemical reaction on free convective heat and mass transfer unsteady flow

with heat generation and constant suction normal to an infinite hot vertical porous plate when the

plate temperature is Cosinusoidallyfluctuating with time.The present study of heat generation or

absorption in moving fluids is important in view of several physical problems involving

exothermic or endothermic chemical reaction. The governing partial equations are transformed

into a set of ordinary differential equations by using a similarity transformation which is further

solved with the help of second order perturbation scheme. To the best of our knowledge,this kind

of work has not yet been investigated.

2. Mathematical Formulation

Consider the flow of a conducting fluid past an infinite hot porous plate lying vertically on zyx plane.The plate is assumed to be infinite in lengthand taken along the fluid in x


511

direction, therefore all physical quantities are independent of x . A magnetic field of uniform

strength B0 is applied normally.Let wvu ,, be the components of the velocity in the

zyx ,, directions respectively.Due to suction at the surface of the plate with constant

velocity V , w is independent of z and assumed as zero.Further, we assume that the

magnetic Reynolds number is very small so that the induced magnetic field is negligible in

comparison to the applied magnetic field. The fluid is considered hereto be gray,

absorbing/emitting radiation but a non-scattering medium. Noexternal electrical field is applied

and effect of polarization of ionized fluid is negligible,therefore, electrical field is assumed to be

zero. There exists a first order chemical reaction between the fluid and species concentration, the

heat generated during chemical reaction cannot be neglected. Followed by Ram et al. [39] and by

considering Boussinesq’s approximation, the flow field is governed by the following set of

equations:

Continuity equation:

0

y

v

(1)

Momentum Equation:

u

ku

B

z

u

y

uCCgTTg

y

uv

t

u*

2

022

*

22)()(

(2)

Energy equation:

)((0

22

22

TTQ

y

q

z

T

y

T

y

Tv

t

Tc r

p

(3)

Species diffusion equation:

))()2222

22

1

*22

z

T

y

TDCC

z

C

y

CD

y

Cv

t

C

(4)

The temperature of the plate is considered to vary spanwisecosinusoidally fluctuating with time

and assumed to be of the form

twl

zTTTtzTw

cos, 00

(5)

The corresponding initial and boundary conditions are as follows:

0u ,

twl

zTTTT

cos00 , wCC at y =0

0u ,

TT ,

CC yas (6) For the case of an optimality thin gray gas, local radiative heat flux in the energy equation


512

44

4

TTk

y

qes

r

(7)

If temperature differences within the flow are sufficiently small,then equation (7) can be linearized by

expanding

4

T

into the Taylors series about

T

,which after neglecting higher order

terms,takes the form

4334

4

TTTT (8)

In the view of Eqs.(7) and (8), Eq.(3) reduces to

)(16 (0

322

22

TTQTTTk

z

T

y

T

y

Tv

t

Tc

sep

(9)

The non-dimensional parameters as follows

l

yy

, tt , v

uu

,l

zz

, l

kk

, twtv

l 2

, l

zz

,

1

2kl ,

TT

TT

C

C

0w

,C

C

(10)

Using the transformation (10) and Eq.(5), the momemtum Eq.(2), energy Eq. (9) and

concentration Eq.(4) reduce to the following dimensionless form

k

uuM

z

u

y

uCGGR

y

uR

t

umree

2

2

2

2

22

(11)

r

e

r

eP

RR

zyPyR

t

2

2

2

2

21

(12)

2

2

2

2

02

2

2

21

zySK

zySyR

tc

C

e

(13)

Where v

VlRe

is the Reynolds number,

3

0

V

TTvgGr

is the thermal Grashof number,

3

0

V

CCvgGm

is the mass / Solutal /modified Grashof number,

v

lBM

2

02

is the magnetic parameter/ Hartmann number,


513

2

3216

kV

TvkR se

is the radiation parameter,

k

cP

p

r

is the Prandtl number,

D

vSc

is the Schmidt number,

vc

lQ

p

2

0

is the heat generation parameter,

CC

TTDS

0

00

is the Soret number.

The corresponding boundary conditions are given by

;01),cos(1,0 yattzu

;0,0,0 yatu

(14)

3. SOLUTION OF THE PROBLEM

To investigate the effects of various parameters like Grashof number, Prandtl number, porous

permeability parameter and Schmidt number etc, on velocity and temperature fields in the

boundary layer generated on the surface, the solution of partial differential equations(11-13)

along with boundary condition(14) is obtained using perturbation technique (Chamkha[2],

Kumar and Singh [15], Ram et al/ [5]) through which it is assumed the components of velocity,

temperature and concentration respectively as follows: )(2

2

2)(

10

tzitzi eueuuu

)(2

2

2)(

10

tzitzi ee (15)

)(2

2

2)(

10

tzitzi ee

Substituting equation (15) into set of equations (11)-(13) and equating the like powers of ε, the

following set of equations are obtained:

Zeroth order equations

00

2

0

1

0

11

0 ReRe mr GGNuuu (16)

0Re 0

1

0

11

0 rr PP (17)

CCcec SSSkRS 00

211

00

1

0

11

0 4 (18)

The corresponding boundary conditions reduce to the following form;

00 u , 10 , 10 ,at y=0


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00 u , 00 , 00 , at y= (19)

First order equations

11

2

1

1

1

11

1 Re)(Re mr GGuwiNuu (20)

0)(Re 1

21

1

11

1 iwPP rr (21)

CCCcec SSiwSSkRS 0

11

11

2

1

21

1

11

1 (22)

The corresponding boundary conditions reduce to

01 u , 11 , 01 at y=0

01 u , 01 , 01 at y= (23)

Second order equations

22

2

2

21

2

11

2 Re)24(Re mr GGuwiNuu (24)

0)24(Re 2

21

2

11

2 iwPP rr (25)

The corresponding boundary conditions reduce to

02 u , 02 , 02 at y=0

02 u , 02 , 02 at y= (26)

Solving the above differential equations along with the corresponding boundary conditions, the

following solutions are obtained: ym

e 1

0

(27)

yme 3

1

(28)

02 (29)

ymymekek 17

110 1

(30)

ymymekek 39

221

(31)

02 (32)

ymymymekekekku 7113

21430

(33)

ymymymekekekku 9315

21651

(34)

02 u (35)

4. Results and discussion:

The set of non-dimensionless governing equations (16) - (18) and subject to the corresponding

appropriate boundary conditions (19 ) are solved analytically by using regular perturbation

technique. The solutions are carried out for different flow parameters on the flow quantities and

the computed results are presented in figures1-11.


515

The effect of Reynolds number on temperature is shown in figure 1, for all values of

Reynolds number the temperature profiles have significant appearance near the plate but as the

values of Reynolds number are increased temperature decreased and reaches ambient

temperature far away from the plate.The effect of Prandtl number on temperature is presented in

Fig.2. It is noticed that the temperature of fluid decreases as Prandtle number increases.The

effect of radiation parameter on temperature is presented in Fig.3. It is noticed that temperature

of the fluid decreases as radiation parameter increases.The effect ofheat absorption /generation

parameter on temperature is presented in Fig.4. From this figure, it observed that temperature

increases for increasing values of absorption parameter but reverse trend in the presence of

generation parameter.The effect of Schmidt number on concentration is presented in the Fig.5.

This figure witnesses that, concentration decreases for increasing values of Schmidt number.The

effect of chemical reaction parameter on concentration is presented in Fig.6, from which it is

noticed that the concentration decreases on increasing the values of chemical reaction parameter.

The effect of Soret number on concentration is presented in Fig.7, which depicts that

concentration increases for increasing values of Soret number. The effect of Solutal Grashof

number on velocity is presented in the Fig.8.It is noticed that the velocity increases on increasing

the values of Solutal Grashof number.The effect of Hartmann number on velocity is presented in

the Fig.9, that shows that the velocity decreases as expected on increasing thevalues of Hartmann

number. The effect of Grashaf number on velocity is shown in the Fig.10, it is noted that the

velocity increases on increasing the values of thermal Grashaf number. This is due the fact that

the presence of buoyancy force that enhances the velocity for both the cases of thermal and

solutal Grashof numbers. The effect of Reynolds number on velocity is presented in the Fig.11

through which it is noticed that the velocity increases on increasing the values of Reynolds

number. The effect of permeability of the porous medium on velocity is presented in the Fig.12,

from this figure it is noticed that velocity increases for increases values of permeability

parameter.


516

Fig.1. Effect ofReynolds number on temperature

Fig.2. Effect of Prandtl number on temperature

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

y

Re=1,3,5,7

Re=7

R=0.5

Pr=7=0.5

w=.5

z=0

e=0.1

t=pi/2

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

y

Re=10R=0.5Pr=7

=0.5

=.5z=0;

=0.1

t=/2

Pr=0.71,7.0


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Fig.3. Effect of Radiation parameter number on temperature

Fig.4. Effect of radiation absrption/generation parameter on temperature

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

y

R=0.2,0.5,1,2

Pr=0.71Re=10

=0.5

=0

t=/2

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

y

=0.1

=0.5

=1.0

=1.5

= -0.1

= -0.5

=-1.0

= -1.5

Pr=0.71Re=10,

=0

t=/2


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Fig.5. Effect of Schmidt number on concentration

Fig.6. Effect of Chemical reaction parameter on concentration

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

y

Sc=0.16,0.22,0.61,0.78,0.96

Re=10R=0.5Pr=7.0

=5z=0

=0.1

t=/2S

0=0.1

Kc=0.2

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

y

Re=10R=0.5Pr=7.0

=5z=0

=0.1

t=/2S

0=0.1

Sc=0.22

Kc=0.2,0.4,0.6,0.8,1.0


519

Fig.7. Effect of Soret number on concentration

Fig.8. Effect of Solutal Grashof number onvelocity

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

y

Re=10R=0.5Pr=7.0

=5z=0

=0.1

t=/2Kc=0.2Sc=0.22

S0=0.1,0.3,0.5,0.7,1.0

0 1 2 3 4 50

1

2

3

4

5

6

y

u

Gm=5,6,7,8

Gr=5Re=5M=2Pr=7

=1z=0K

p=0.1

=0.5Sc=0.22Kc=0.2S

0=2

R=0.5


520

Fig.9. Effect of Hartmann number on velocity

Fig.10. Effect of Grashof number on velocity

0 1 2 3 4 50

1

2

3

4

5

6

7

8

y

u

M=1,2,3,4

Re=5Gr=10Gm=10Pr=7

=1z=0R=0.5K

p=0.1

Ql=0.5

Sc=0.22Kc=0.2S

0=2

0 1 2 3 4 50

2

4

6

8

10

12

14

y

u Gr=5,10,15,20

Gm=15Re=5M=2Pr=7

=1z=0K

p=0.1

Ql=0.5

Sc=0.22Kc=0.2S

0=2

R=0.5


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Fig.11. Effect of Reynolds number on velocity

Fig.12. Effect of Permeability of the porous medium onvelocity

Table 1 :Rate of heat transfer coefficient(Nusselt number)

rP R Re ω χ Nu

0.71 2 5 0.5 0.5 9.0103

7.0 2 5 0.5 0.5 36.2816

0 1 2 3 4 50

2

4

6

8

10

12

14

16

y

u

Re=5,6,7,8

Gr=10

0 1 2 3 4 50

1

2

3

4

5

6

7

y

u

Kp=0.1,0.3,0.5,1.0

Gr=5Gm=8Re=5M=2Pr=7

=1z=0

=0.5Sc=0.22Kc=0.2S

0=2

R=0.5


522

0.71 3 5 0.5 0.5 10.5655

0.71 5 5 0.5 0.5 11.8847

0.71 2 10 0.5 0.5 18.0568

0.71 2 15 0.5 0.5 27.0952

0.71 2 20 0.5 0.5 36.1316

0.71 2 5 0.6 0.5 9.0103

0.71 2 5 0.7 0.5 9.0055

0.71 2 5 0.5 0.6 9.0055

0.71 2 5 0.5 0.7 9.0007

0.71 2 5 0.5 0.8 8.9959

Table 2 :Rate of mass transfer coefficient( Sherwood number)

Sc S0 Re Kc Sh

0.16 1 1 0.2 0.8351

0.22 1 1 0.2 0.8364

0.60 1 1 0.2 1.8567

0.78 1 1 0.2 2.6760

0.16 1 1 0.2 0.8351

0.16 2 1 0.2 1.5901

0.16 3 1 0.2 2.3452

0.16 4 1 0.2 3.1002

0.16 1 3 0.2 1.4516

0.16 1 4 0.2 1.7745

0.16 1 5 0.2 2.2138

0.16 1 6 0.2 2.6645

0.16 1 1 0.3 0.5288

0.16 1 1 0.4 0.4449

0.16 1 1 0.5 0.4119

0.16 1 1 0.6 0.3965

Table 3: Skin friction coefficient

Sc S0 Re Gr Gm M Kc Kp Cf

0.16 1 1 2 2 1 0.2 0.5 5.3505

0.22 1 1 2 2 1 0.2 0.5 4.9752

0.60 1 1 2 2 1 0.2 0.5 3.5759

0.78 1 1 2 2 1 0.2 0.5 2.8403

0.16 2 1 2 2 1 0.2 0.5 6.5608

0.16 3 1 2 2 1 0.2 0.5 7.7711

0.16 4 1 2 2 1 0.2 0.5 8.9814

0.16 5 1 2 2 1 0.2 0.5 10.1916


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0.16 1 0.5 2 2 1 0.2 0.5 2.8327

0.16 1 0.6 2 2 1 0.2 0.5 3.0782

0.16 1 0.7 2 2 1 0.2 0.5 3.5637

0.16 1 0.8 2 2 1 0.2 0.5 4.1045

0.16 1 1 3 2 1 0.2 0.5 5.9934

0.16 1 1 4 2 1 0.2 0.5 6.6363

0.16 1 1 5 2 1 0.2 0.5 7.2792

0.16 1 1 6 2 1 0.2 0.5 7.9220

0.16 1 1 2 3 1 0.2 0.5 7.3829

0.16 1 1 2 4 1 0.2 0.5 9.4153

0.16 1 1 2 5 1 0.2 0.5 11.4477

0.16 1 1 2 6 1 0.2 0.5 13.4801

0.16 1 1 2 2 2 0.2 0.5 4.5206

0.16 1 1 2 2 3 0.2 0.5 3.6090

0.16 1 1 2 2 4 0.2 0.5 2.4587

0.16 1 1 2 2 5 0.2 0.5 1.8016

0.16 1 1 2 2 1 0.3 0.5 4.8098

0.16 1 1 2 2 1 0.4 0.5 4.5280

0.16 1 1 2 2 1 0.5 0.5 4.3562

0.16 1 1 2 2 1 0.6 0.5 4.2382

0.16 1 1 2 2 1 0.2 0.6 5.5390

0.16 1 1 2 2 1 0.2 0.7 5.6997

0.16 1 1 2 2 1 0.2 0.8 5.8384

0.16 1 1 2 2 1 0.2 0.9 5.9593

Table 1 shows the effects of important physical parameters on tangential Nusselt number. It is

indicated that Nusselt increases with increasing values of Prandtl number, Radiation parameter,

Reynolds number, dimensionless frequency oscillation and Nusselt number decreases with

increase the value of χ. Table 2 shows the effects of pertinent physical parameters on Sherwood

number. Sherwood number increases with the increasing values of Schmidt number, Soret

number, Reynolds number but reverse effect in the presence of chemical reaction. Table 3 shows

the effects of physical parameters on tangential skin friction coefficient. It is indicated that skin

friction coefficient decreases for increasing values of Schmidt number, Hartmann number,

permeability of the porous medium and reverse trend is noticed in the presence of Soret number,

Reynolds number, thermal Grashof number, and Solutal Grashof number.

5. Conclusions:

The problem of unsteady MHD free convective heat generation and absorption and mass transfer

flow of thermal radiation and chemical reaction fluid a past a vertical porous plate has been


524

investigated .The present study may also be more useful in the study of the effects of magnetic

fields on blood circulation,cardiovascular events, crude oil transportation,etc. The conclusions of

the study are as follows:

Temperature decreases for increasing values of Reynolds number,Prandle number,Radiation

parameter whereas it decreases for increasing values of radiation absorption parameter but mixed

effect is noticed in the presence of heat generation/absorption parameter.Concentration decreases

for increasing values Schmidt number and Chemical reaction parameter but reverse effect is seen

in the presence of Soret number. Velocity decreases for increasing values of Hartmann

number,Reynolds number and Permeability of the porous medium but reverse effect is

noticed in the presence of thermal Grashof number, solutal Grashof number. Skin friction

coefficient increases for increasing values of Soret number,Raynolds number,Grashof

number,Solutal Grashof number,Permeability of porous medium ofcourse reverse trend is

noticed in the case of Schmidt number,Hartmann number,Rosseland mean absorption coefficient.

Nusselt number increases for increasing values of Prandtl number,Radiation parameter,Raynolds

number and opposite impact is noticed in the case of Dimensionless frequency of

oscillation,Dimensionless heat generation/absorption parameter. Similarly Sherwood number

increases for increasing values of Schmidt number,Soret number,Raynolds number and opposite

reaction is noticed in the case of Rosseland mean absorption coefficient.

Appendix

Where 2

4Re22

1

rrre PPPRm

2

)(4Re 222

3

iwPPPRm

rrre

2

4Re22

7

cccec SkSRSm

2

4Re 222

9

ccccec iwSSkSRSm

2

4Re2

13

NRm e

2

4Re2

15

wiNRm e

Pr

Re2 R

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