mhd boundary layer flow of a visco-elastic fluid past a ... · chandra reddy et al. [14, 15]...

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2717-2733

© Research India Publications

http://www.ripublication.com

MHD Boundary Layer Flow of a VISCO-Elastic

Fluid Past a Porous Plate with Varying Suction and

Heat Source/Sink in the Presence of Thermal

Radiation and Diffusion

K. Vijayakumar1 and E. Keshava Reddy1

Department of Mathematics, JNTUA College of Engineering Anantapur,

Anantapuramu -510003, A.P, India.

Abstract

This manuscript consists of the properties of natural convective flow of a viscous

incompressible electrically conducting fluid past a vertical porous plate bounded

by a porous medium in the presence of thermal radiation and variable

permeability. A magnetic field of uniform strength is applied perpendicular to the

plate and the presence of heat source is also considered. The novelty of the study

is to investigate the effects of thermal radiation and Eckert number. The coupled

dimensionless non-linear partial differential equations are solved by finite

difference method. The numerical computations have been studied through

graphs. The presence of thermal radiation decreases the temperature and an

opposite nature is shown in the case of Eckert number. The influence of neat

source leads to enhance the temperature.

Keywords: MHD, thermal radiation, variable suction, variable permeability,

vertical porous plate, heat and mass transfer.

1. INTRODUCTION:

The studies related to MHD visco-elastic fluids with radiation effect past a porous

media in the presence of heat source/sink plays significant role in many scientific,

industrial and engineering applications. These flows were basically utilized in the

fields of petroleum engineering concerned with the oil, gas and water through

reservoir to the hydrologist in the analysis of the migration of underground water. To

recover the water for drinking and irrigation purposes the principles of this flow are

followed. In the recent days many researchers recognized the significance of these

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2718 K. Vijayakumar and E. Keshava Reddy

flows and contributed in studying the application of visco-elastic fluid flow of several

types past porous medium in the presence of thermal radiation and heat source/sink.

Chen [1] studied magneto-hydrodynamic mixed convection of a power-law fluid past

a stretching surface in the presence of thermal radiation and internal heat

generation/absorption. Hayat et al. [2] analyzed Soret and Dufour effects for three-

dimensional flow in a viscoelastic fluid over a stretching surface. Ganesan et al. [3]

analyzed and reported radiation and mass transfer effects on flow of an

incompressible viscous fluid past a moving vertical cylinder. Lavanya et al. [4]

discussed radiation and mass transfer effects on unsteady MHD natural convective

flow past a vertical porous plate embedded in a porous medium in a slip flow regime

with heat source/sink and Soret effect. Hayat et al. [5] explained slip and Joule

heating effects in mixed convection peristaltic transport of nanofluid with Soret and

Dufour effects. Abo-Eldahab et al. [6] considered blowing/suction effect on

hydromagnetic heat transfer by mixed convection from an inclined continuously

stretching surface with internal heat generation/absorption. Chamka [7] examined

double-diffusive convection in a porous enclosure with cooperating temperature and

concentration gradients and heat generation or absorption effects. Ahmed et al. [8]

studied boundary layer flow and heat transfer due to permeable stretching tube in the

presence of heat source/sink utilizing nanofluids. Hayat et al. [9] explained effect of

Joule heating and thermal radiation in flow of third-grade fluid over radiative surface.

Motsumi et al. [10] discussed effects of thermal radiation and viscous dissipation on

boundary layer flow of nanofluids over a permeable moving flat plate. Mishra et al.

[11] considered and pointed out the mass and heat transfer effect on MHD flow of a

visco-elastic fluid through porous medium with oscillatory suction and heat source.

Uma et al. [12] discussed Unsteady MHD free convective visco-elastic fluid flow

bounded by an infinite inclined porous plate in the presence of heat source, viscous

dissipation and Ohmic heating. Motivated by the above investigations, in this work a

fully developed free convective flow of a viscous incompressible electrically

conducting fluid past a vertical porous plate bounded by a porous medium in the

presence of thermal radiation, heat source/sink, variable suction and variable

permeability is analyzed. We have extended the work of Chandra Reddy et al. [13]

with the novelty of considering Eckert number, heat generation/absorption and

thermal radiation.

We also have gone through the literature related to the present work and followed.

Chandra Reddy et al. [14, 15] analyzed magnetohydrodynamic convective double

diffusive laminar boundary layer flow past an accelerated vertical plate as well as

Soret and Dufour effects on MHD free convection flow of Rivlin-Ericksen fluid past a

semi infinite vertical plate. Further Chandra Reddy et al. [16] studied the properties of

free convective magneto-nanofluid flow past a moving vertical plate in the presence

of radiation and thermal diffusion. Mahdy et al. [17] studied thermosolutal marangoni

boundary layer magnetohydrodynamic flow with the Soret and Dufour effects past a

vertical flat plate. Kairi and Murthy [18] discussed the effect of melting and thermo-

diffusion on natural convection heat mass transfer in a non-Newtonian fluid saturated

non-Darcy porous medium. Swati & Murthy [19] analyzed Magnetohydrodynamic

MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2719

(MHD) mixed convection slip flow and heat transfer over a vertical porous plate.

Mukhopadhyay et al. [20] investigated forced convective flow and heat transfer over a

porous plate in a Darcy-Forchheimer porous medium in presence of radiation. Ahmed

and Kalita [21] discussed magnetohydrodynamic transient flow through a porous

medium bounded by a hot vertical plate in presence of radiation. Kumar and Verma

[22] considered thermal radiation and mass transfer effects on MHD flow past a

vertical oscillating plate with variable temperature effects variable mass diffusion.

Ravi kumar et al. [23] examined combined effects of heat absorption and MHD on

convective Rivlin-Ericksen flow past a semi-infinite vertical porous plate with

variable temperature and suction. Sharma and Ajaib [24] investigated bounds for

complex growth rate in thermosolutal convection in Rivlin–Ericksen viscoelastic fluid

in a porous medium. Raju et al. [25] considered and studied MHD thermal diffusion

natural convection flow between heated inclined plates in porous medium. Raju et al.

[26] attended and discussed Soret effects due to Natural convection between heated

inclined plates with magnetic field. Makinde et al. [27] considered and established

free convection flow with thermal radiation and mass transfer past a moving vertical

porous plate. Samad et al. [28] analyzed thermal radiation interaction with unsteady

MHD flow past a vertical porous plate immersed in a porous medium. Bhattacharyya

et al. [29] analyzed effects of thermal radiation on micropolar fluid flow and heat

transfer over a porous shrinking sheet. Hayat et al. [30] discussed heat and mass

transfer for Soret and Dufour effects on mixed convection boundary layer flow over a

stretching vertical surface in a porous medium filled with a viscoelastic fluid. Khan

and Aziz [31] examined the properties of natural convection flow of a nanofluid over

a vertical plate with uniform surface heat flux.Rohni et al. [32] considered unsteady

mixed convection boundary-layer flow with suction and temperature slip effects near

the stagnation point on a vertical permeable surface embedded in a porous medium.

Nomenclature:

Cl

D

Gr

Species concentration

Molecular diffusivity

Grashof number of heat transfer

C Non-dimensional Species

concentration

σ Electrical conductivity ω Non-dimensional frequency of

oscillation

K1 Permeability of the medium Gc Grashof number for mass transfer

k Thermal diffusivity G Acceleration due to gravity

Rc

Ec

Elastic parameter

Eckert number

Kp

F

Porosity parameter

Thermal radiation

Nu Nusselt number M Magnetic parameter

S Heat source parameter B0 Magnetic field of uniform strength

Sh Sherwood number Pr Prandtl number

T Non-dimensional temperature Sc Schmidt number

T Non-dimensional time T1 Temperature of the field


u Non-dimensional velocity tl Time

vo Constant suction velocity ul Velocity component along x-axis

y Non-dimensional distance along y-axis V Suction velocity

Ԑ A small positive constant yl Distance along y-axis

β Volumetric coefficient of expansion for

heat transfer

ρ

τ

Density of the fluid

Skin friction

β1 Volumetric coefficient of expansion

with species concentration

ω1

So

Frequency of oscillation

Soret number

υ Kinematic coefficient of viscosity W Condition on porous plate

2. FORMULATION OF THE PROBLEM:

The unsteady free convection heat and mass transfer flow of a well-known non-

Newtonian fluid, namely Walters B visco-elastic fluid past an infinite vertical porous

plate, embedded in a porous medium in the presence of thermal radiation, oscillatory

suction as well as variable permeability is considered. In addition to this the existence

of heat generation / absorption is also considered. A uniform magnetic field of

strength B0 is applied perpendicular to the plate. Let x1 axis be taken along with the

plate in the direction of the flow and y1 axis is normal to it. Let us consider the

magnetic Reynolds number is much less than unity so that the induced magnetic field

is neglected in comparison with the applied transverse magnetic field. The basic flow

in the medium is, therefore, entirely due to the buoyancy force caused by the

temperature difference between the wall and the medium. It is assumed that initially,

at tl ≤ 0, the plate as fluids are at the same temperature and concentration. When tl >

0, the temperature of the plate is instantaneously raised to 1

wT and the concentration of

the species is set to 1

wC . Under the above assumption with usual Boussinesq’s

approximation, the governing equations and boundary conditions are given by

1

13

11

13

0

11

1

12

0111

1

12

1

1

1

1

2

2

)(

)()(

y

uvytuk

tKu

uBCCgTTgyu

yuv

tu

(1)

2

2*1 1 2 1 *

1 1

1 1 * *1( ) rqT T T uv S T T

t y y yy

(2)

22 1

12

11

12

1

1

1

1

yTD

yCD

yCv

tC

(3)


with the boundary conditions

,)(,0111 tn

ww eTTTTu 0)(111 yateCCCC tn

ww (4)

yasCCTTu 11 ,,0

Let the permeability of the porous medium and the suction velocity be of the form

)1()(11111 tn

p eKtK (5)

)1()(11

0

1 tnevtv (6)

where v0 ˃ 0 and ԑ « 1 are positive constants.

Introducing the non-dimensional quantities

(7)

The equations (1), (2), (3) reduce to the following non-dimensional form: 2

2

2

3

2

1(1 )

4

(1 ) 4

nt

nt

u u ue GrT GcC M ut y y

u Rc uKp e t y

(8)

(9)

2 2

2 2

1 1(1 )

4

ntC C C Te Sot y Sc y y

(10)

11 2 1 11 1

0 0

2

0 0

2 1 2 210 2 0 0 0

2 2 2 2

0 0

11

1

2 3 3

0 0 0

*

4, , , , , ,

4

, , Pr , , , ,

( ) ( ) ( )4, , , ,

( )

4(

w w

p

w w w

w

r

v y v t T T C Cw uy t w u T Cv v T T C C

v K B k vSS Kp Sc M Rcv D v

g C C g T T D T Tnn Gc Gr Sov v v C C

q T Ty

2

0

2 1 10

4) , , .

p p w

UII F EC U C T T

22

2

1 1(1 )

4 Pr

ntT T T ue E ST FTt y y y


with the boundary conditions

01,1,0 yateCeTu ntnt

yasCTu 0,0,0 (11)

3. SOLUTION OF THE PROBLEM:

Equations (8)-(10) are coupled non-linear partial differential equations and are to be

solved by using the initial and boundary conditions (11). However exact solution is

not possible for this set of equations and hence we solve these equations by finite-

difference method. For this, a rectangular region of the flow field is chosen, and the

region is divided into a grid of lines parallel to X and Y-axes, where the X-axis is

taken along the plate and the Y-axis is normal to the plate as shown in Fig.1.

Fig. 1 Finite difference space grid

The equivalent finite difference schemes of equations for (8)-(10) are as follows:

, 1 , 1, , 1, , 1,

, , 2

1, 1 , 1 1, 1 1, , 1, 2,2

,

21(1 )

4

2 2

4

1

(1 )

i j i j i j i j i j i j i jnti j i j

i j i j i j i j i j i ji j

i jntp

u u u u u u ue GrT GcC

t y yu u u u u uRc M u

t y

uK e

(12)


2

, 1 , , 1 , 1, , 1, 1, ,

2

, ,

21 1(1 )

4 Pr

i j i j i j i j i j i j i j i j i jnt

i j i j

T T T T T T T u ue E

t y yyST FT

(13)

, 1 , , 1 , 1, , 1,

2

1, , 1,

2

21 1(1 )

4

2

i j i j i j i j i j i j i jnt

i j i j i j

C C C C C C Ce

t y Sc y

T T TSo

y

(14)

Here, the index i refer to y and j to time. The mesh system is divided by taking

∆y = 0.1. From equation (11), we have the following equivalent initial condition

( ,0) 0, ( ,0) 0, ( ,0) 0u i T i C i for all i (15)

The boundary conditions from (11) are expressed in finite-difference form as follows

1, 1,

max max max

(0, ) 1, (0, ) 1, 2.

( , ) 0, ( , ) 0, ( , ) 0

i j i ju j T j C C y for all ju i j T i j C i j for all j

(16)

(Here imax was taken as 20)

First the velocity at the end of time step viz, u(i,j+1)(i=1,20) is calculated from (12) in

terms of velocity, temperature and concentration at points on the earlier time-step.

Then T(i, j +1) is computed from (13) and C(i, j +1) is computed from(14). The

procedure is repeated until t = 0.5 (i.e. j = 500). During computation ∆t was chosen as

0.001.

4. RESULTS AND DISCUSSION:

The influence of different physical parameters like Grashof number modified Grashof

number, magnetic parameter, thermal radiation, Prandtl number, Eckert number, Soret

number and Schmidt number on velocity, temperature and concentration is discussed

by using graphical representations. The general nature of the velocity profile is

parabolic with picks near the plate. Figure 2 show that the velocity enhances for rising

values of magnetic parameter. The effect of Prandtl number on velocity is displayed

in figure 3. The Prandtl number is a dimensionless number approximating the ratio of

momentum diffusivity (kinematic viscosity) and thermal diffusivity. The fluid

velocity decreases for increasing values of Prandtl number. This is due to the effect of

transverse magnetic field, which has the nature of reducing the velocity. These results

are similar to that of Mishra et al. [11]. Figures 4, 5 depict the velocity variations


under the effect of Grashof number and modified Grashof number respectively. The

velocity of the flow grows when the values of these parameters increases. Figure 6

represents the impact of porous medium on velocity. It is evident that the velocity

enhances for increasing values of porosity parameter. The changes in velocity under

the existence of heat source / sink are depicted in figure 7. It is noticed that the

velocity enhances in the presence of heat source where as it falls down in the case of

heat sink. The variation in velocity under the influence of Schmidt number is shown

in figure 8. It is evident that velocity comes down when the values of Schmidt number

are increased. The existence of thermal diffusion results in improving the flow

velocity which is clear from figure 9.These results coincide with that of Chandra

Reddy et al. [13]. The temperature decreases in the presence of thermal radiation

which is shown in figure 10. The effect of Prandtl number on temperature is presented

in figure 11. The temperature decreases for increasing values of Prandtl number.

Figure 12 reveals the same nature as that of velocity under the influence of heat

source/sink. Figure 13 depicts the influence of Eckert number on temperature. It is

observed that the temperature rises with increasing values of Eckert number. The

effect of Schmidt number on concentration is shown in figure 14. The concentration

reduces with an increase in Schmidt number. Increasing values of Soret number leads

to enhance the concentration which is clear from figure 15.

Fig.2: Effect of magnetic parameter on velocity

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

y

u

M=1,1.2,1.4,1.6

Sc=0.22; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; Gr=5;F=0.2; S=0.1;S0=2; Pr=0.71;

=0.01; n=1;


Fig.3: Effect of Prandtl number on velocity

Fig.4: Effect of Grashof number on velocity

Fig.5: Effect of modified Grashof number on velocity

0 5 10 15 20 250

1

2

3

4

y

u

S=0.1;F=5;n=1;Sc=0.22;S0=0.1;M=1;Kp=5;Gr=5;Gc=5;Rc=0.001;Ec=0.1;t=0.1;

Pr=0.71,1,7.1

M=1; S=0.1;F=5; n=2; Sc=0.22; So=0.1;Kp=5; Gr=5;Gc=5; Rc=0.001;Ec=0.1; t=0.1;

=0.01

0 2 4 6 8 10 12 14 16 18 20 22

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y

u

Sc=0.22; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;S0=2; Pr=0.71;

=0.01; n=1;

Gr=5,10,15,20

0 2 4 6 8 10 12 14 16 18 20 22

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y

u

Sc=0.22; Gr =5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;S0=2; Pr=0.71;

=0.01; n=1;

Gc=5,10,15,20


Fig.6: Effect of porosity parameter on velocity

Fig.7: Effect of heat source/sink on velocity

Fig.8: Effect of Schmidt number on velocity

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

y

u

K=0.2,0.4,0.6,0.8

Sc=0.22; Gc=5;Gr=5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;So=2; Pr=0.71;

=0.01; n=1;

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

y

u

_____ S = 0.2, 0.4, 0.6........... S = -0.2, -0.4, -0.6

Sc=0.22; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; Gr=0.1;So=2; Pr=0.71;

=0.01; n=1;

0 2 4 6 8 10 12

0

0.05

0.1

0.15

0.2

0.25

0.3

y

u

Gr=5; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;S0=2; Pr=0.71;

=0.01; n=1;

Sc=0.60,0.78,0.96


Fig.9: Effect of Soret number on velocity

Fig.10: Effect of radiation parameter on temperature

Fig.11: Effect of Prandtl number on temperature

0 2 4 6 8 10 12 14 16 18 20 220

0.5

1

1.5

2

y

u

Gr=5; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;Sc=0.22; Pr=0.71;

=0.01; n=1;

So=0.5,1,1.5,2

2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

F=1,3,5,7

y

T

Gr=5; Gc=5;

Kp=0.5; Rc=0.1;

Ec=0.01; M=1;

Pr=0.71; S=0.1;

Sc=0.22; So=1;

=0.01; n=1;

2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

y

T

Pr=0.71,1,3,5,7.1

Gr=5; Gc=5;

Kp=0.5; Rc=0.1;

Ec=0.01; M=1;

F=0.2; S=0.1;

Sc=0.22; So=1;

=0.01; n=1;


Fig.12: Effect of heat source/sink on temperature

Fig.13: Effect of Eckert number on temperature

Fig.14: Effect of Schmidt number on concentration

0 1 2 3 4 50

0.5

1

1.5

2

2.5

y

_____ S = 0.2, 0.4, 0.6, 0.8........... S = -0.2, -0.4, -0.6, -0.8

Sc=0.22; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; Gr=0.1;So=2; Pr=0.71;

=0.01; n=1;

5 10 15 20 250

0.2

0.4

0.6

0.8

1

y

T

Gr=5; Gc=5;

Kp=0.5; Rc=0.1;

Pr=0.71; M=1;

F=0.2; S=0.1;

Sc=0.22; So=1;

=0.01; n=1;

Ec=0.1,0.3,0.5,0.7

0 2 4 6 8 10 120

0.5

1

1.5

2

y

C

Pr=0.71; So=2;F=5; n=2;S=0.1; M=2;Kp=20; Gr=5;Gc=5; Rc=0.001;Ec=0.1; t=2;

=0.001

Sc=0.60,0.78,0.96


Fig.15: Effect of Soret number on concentration

The present work is extended to observe the changes in skin friction, Nusselt number

and Sherwood number under the influence of thermal radiation, Eckert number,

Schmidt number and magnetic parameter. This is done with the help of tabular values.

Table 1 shows that the skin friction coefficient reduces for increasing values of Eckert

number and Schmidt number. A reverse trend is shown in the case of radiation

parameter and magnetic parameter. The rate of heat transfer increases under the

influence of thermal radiation whereas it decreases in the case of Eckert number.

Increasing values of Schmidt number leads to enhance the Sherwood number.

Table1. Effect of various physical parameters on skin friction, Nusselt number and

Sherwood number

Ec F Sc M Nu Sh

0.2 1 0.22 5 2.0359 0.6742 0.2116

0.4 1 0.22 5 1.9848 0.6645 0.2116

0.6 1 0.22 5 1.5212 0.5932 0.2116

0.2 1 0.22 5 2.0361 0.6742 0.2116

0.2 3 0.22 5 2.0856 0.7356 0.2116

0.2 5 0.22 5 2.1589 0.7931 0.2116

0.2 1 0.60 5 1.9611 0.6742 0.5755

0.2 1 0.78 5 1.9292 0.6742 0.7472

0.2 1 0.96 5 1.9002 0.6742 0.9194

0.2 1 0.22 1 6.9669 0.6742 0.2116

0.2 1 0.22 1.2 7.9647 0.6742 0.2116

0.2 1 0.22 1.4 9.3238 0.6742 0.2116

0 5 10 15 20 25

0

0.2

0.4

0.6

0.8

1

1.2

y

C

So=1,2,3,4

Gr=5; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;Sc=0.22; So=1;

=0.01; n=1;


4. CONCLUSION:

In the present study the effect of thermal radiation due to natural convection on MHD

flow of a visco-elastic fluid past a porous plate with variable suction and heat

source/sink is analyzed. The governing equations for the velocity field, temperature

and concentration by finite difference method. The main findings of this study are as

follows.

Velocity of the fluid reduces for increasing values of Prandtl number and

magnetic parameter.

Temperature of the fluid grows for rising values of Eckert number, but a

reverse effect is noticed in the case of Prandtl number and radiation

absorption parameter.

The concentration reduces with an increase in Schmidt number.

The existence of heat source leads to enhance the temperature and a

reverse trend is observed in the presence of heat sink.

Skin friction decreases with an increase of Eckert number and Schmidt

number but a reverse effect is noticed in the case of radiation absorption

parameter and magnetic parameter.

Nusselt number increases as radiation absorption parameter increases but

in the case of Eckert number it decreases.

Sherwood number increases with an increase in Schmidt number.

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