EFFECT OF STRATIFIED VISCOUS FLUID ON MHD FREE CONVECTION FLOW WITH HEAT AND MASS TRANSFER IN THE PRESENCE OF RADIATION AND HEAT SOURCE

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EFFECT OF STRATIFIED VISCOUS

FLUID ON MHD FREE CONVECTION

FLOW WITH HEAT AND MASS

TRANSFER IN THE PRESENCE OF

RADIATION AND HEAT SOURCE

N. K. VARSHNEY

and

BHAGWAN SINGH (R.S.)

Deptt. of Mathematics, S. V. College, Aligarh (U. P. )

Abstract:

A study of the effect of stratified viscous fluid on MHD free convection flow past a vertical porous

plate with heat and mass transfer in the presence of radiation and heat source taking viscous and Darcy

resistance terms into account and the constant permeability of the medium numerically and neglecting induced

magnetic field in comparison to applied magnetic field is presented. The velocity, temperature and concentration

distributions are derived and discussed numerically with the helps of graphs and tables.

Keywords : Heat and mass transfer, MHD flow, Porous medium, Vertical plate, Stratified viscous fluid,

Radiation, Heat source.

ABSTRACT:

A study of the effect of stratified viscous fluid on MHD free convection flow past a vertical porous

plate with heat and mass transfer in the presence of radiation and heat source taking viscous and Darcy

resistance terms into account and the constant permeability of the medium numerically and neglecting induced

magnetic field in comparison to applied magnetic field is presented. The velocity, temperature and concentration

distributions are derived and discussed numerically with the helps of graphs and tables. It is observed that

velocity increases with the increase in Gr (Grashof number), K (Permeability parameter) and b (Stratification

parameter), but it decreases with the increase in M(Magnetic parameter).

INTRODUCTION:

The convection problem in a porous medium has important applications in geothermal reservoirs and

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rector, chemical process industries and many engineering applications in which the fluid is the working medium.

The wide range of technological and industrial applications has stimulated considerable amount of interest in the

study of heat and mass transfer in convection flows. Free convective flow past a vertical plate has been studied

extensively by Ostrach (1953). Siegel (1958) investigated the transient free convection from a vertical flat plate.

Cheng and Lau (1977) and Cheng and Teckchandani (1977) obtained numerical solutions for the convective

flow in a porous medium bounded by two isothermal parallel plates in the presence of the withdrawal of the

fluid. In all the above mentioned studies, the effect of porosity, permeability and the thermal resistance of the

medium is ignored or treated as constant. However, porosity measurements by Benenati and Broselow (1962)

show that porosity is not constant but varies from the surface of the plate to its interior to which as a result

permeability also varies. In case of unsteady free convective flow, Soundalgekar (1972) studied the effects of

viscous dissipation on the flow past an infinite vertical porous plate. The combined effect of buoyancy forces

from thermal and mass diffusion on forced convection was studied by Chep. et al. (1980). The free convection

on a horizontal plate in a saturated porous medium with prescribed heat transfer coefficient was studied by

Ramanaiah and Malarvizhi (1991). Bejan and Khair (1985) have investigated the vertical free convective

boundary layer flow embedded in a porous medium resulting from the combined heat and mass transfer. Lin and

Wu (1995) analyzed the problem of simultaneous heat and mass transfer with the entire range of buoyancy ratio

for most practical and chemical species in dilute and aqueous solutions. Rushi Kumar and Nagarajan (2007)

studied the mass transfer effects of MHD free convection flow of incompressible viscous dissipative fluid past

an infinite vertical plate. Mass transfer effects on free convection flow of an incompressible viscous dissipative

fluid have been studied by Manohar and Nagarajan (2001). Sivaiah, et. al (2009) studied heat and mass transfer

effects on MHD free convective flow past a vertical porous plate. Recently, Sharma et al (2011) analyzed

radiation and heat source effect on MHD free convection flow with heat and mass transfer past a vertical porous

plate.

In the present section we have considered the problem of Sharma et al (2011) by the introducing

stratified viscous fluid under the same conditions taken by Sharma et al (2011).

MATHEMATICAL ANALYSIS:

We study the two-dimensional free convection and mass transfer flow of stratified viscous fluid past an

infinite vertical porous plate with radiation and heat source under the following assumptions:

 The plate temperature is constant

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 Boussinesq’s approximation is valid.

 The suction velocity normal to the plate is constant and can be written as,

1

0

v

 

U

A system of rectangular co-ordinates O (x1,y1,z1) is taken, such that y1 = 0 on the plate and z1 axis is along its leading edge. All the fluid properties considered constant except that the influence of the density

variation with temperature is considered. The influence of the density variation in other terms of the momentum

and the energy equation and the variation of the expansion coefficient with temperature is considered negligible.

The variations of density, viscosity, elasticity, Stefan-Boltzman constant, thermal conductivity and heat source

are supposed to be of the form

 _{= oe}–b 1

y1

, = oe–b

1

y1

, = oe–b

1

y1

,

  

* _o*e–b

1

y1



kT = koe–b

1

y1



S_{= Soe}–b

1

y1

where o, o, o, o* and ko are the coefficients of density, viscosity, elasticity, Stefan-Boltzman constant,

thermal conductivity and heat source respectively at y = 0, b > 0 represents the stratification factor. Under these conditions, the problem is governed by the following system of Equations:

Equation of continuity:

1 1

v

0 y



_



………(1)

Equation of Momentum:

1 1

1 1 1 1 1 1

1 1

u

v

g (T

T )

g (C

C )

t

y

 









_



_

  



  









1 2 1 0

1 1 1

u

B

u

y

K















_



_{ }

 



_



_



_

_

_

…… (2)

Equation of Energy:

1 1 1 1

1 r 1 1

T

1 1 1 1

p p p

T

1 T

1 q

S

v

k

(T

T )

C

y

C

t

y









_



_



_



_

_













_



_

.….(3)

(4)

2

1 1 2 1

1

1 1 ₁

C

v

D

t

y

_y







_



 

_



_



_

_

_

…..(4)

where

u , v

1 1 are the velocity components.

T , C

1 1 are the temperature and concentration components,



is the kinematic viscosity, ρ is the density, σ is the electric conductivity, B0 is the magnetic induction, KT is the

thermal conductivity and D is the concentration diffusivity, S1_{is the coefficient of heat source, qr is the radiative}

heat flux, Cp is the specific heat at constant pressure.

The boundary conditions for the velocity, temperature and concentration fields are:

1 1 1 1 1

w w

u



0, T



T , C



C

at

y

1



0

………(5)

1 1 1

u



0, T



T , C

_



C

_ at

y

1

 

By using Rosseland approximation for the radiation, we take

4

* 1

r * 1

4 T

q

3 y

  



 

………(6)

where



*is the Stefan - Boltzman constant and



*is the mean absorption coefficient.

We assume that the temperature difference with in the flow is such that

4

1

T

may be expressed as a

linear function of temperature

This is accomplished by expending

4

1

T

in a Taylor series about

T

_1 and neglecting higher- order

terms, thus

4 3 4

1 1 1 1

T



4T T

_



3T

_ ……… (7)

Let us introduce the non-dimensional variables

1 0

u

U



, 1 2 0 o

t U

t





, 1 0 o

y U

y





, 1 1 1 1 w

T

 



 



, 1 1 1 1 w

C

 







1 2 0 2 o

K U

K





, o p r o

C

P

k





,

S

_c o

D





,

2

o 0 o

(5)

1 1 1

w

0 1 1

w

(C

C )

N

(T

T )

 











, 1 1 o w r 3 0

g (T

T )

G

U



 





, 3

*

* 1

o

k

N

4 T

_







2 o o 2 o o

S

k U





, 1 o o

b

U





where

P

_r is the Prandtl number,

G

_ris the Grashof number, N₀ is the buoyancy ratio,

S

_c is the Schmidt number, M is the magnetic parameter, K is the permeability parameter, N is the radiation parameter, S is the

heat source parameter, b is the stratification parameter. Other physical variables have their usual meaning.

Introducing the non-dimensional quantities describes above, the governing equations reduce to

2

r 0 2

u

1 (1 b)

G (

N C)

M

u

t

y

K



_{ }



_

_{ }

_



_



_













…….. (8)

2

r r 2

4N

P

(P

b)

(1

)

S

t

y

3 y



_



_{ }

 

_{ }



……… (9)

2 2 c

C

1 C

t

y

S

y



_



_



……… (10)

and the corresponding boundary conditions are

u

  

0,

1, C 1



at

y



0

……… (11)

u

  

0,

0, C



0

at

y

 

METHOD OF SOLUTION:

We assume the solution of eq. (8), (9), (10) as

nt 0

u(y, t)



u (y)e

 ,

nt 0

(y, t)

(y)e





 

,

nt 0

C(y, t)



C (y)e



……… (12)

Using eq.(12) in eq. (8), (9), (10) and we get

'' '

0 0 0 r 0 r 0 0

1 u

(1 b)u

M

n

u

G

G N C

K









 



_

_





_

_

   





(6)





'' '

0 r 0 r 0

4N

(1

)

(P

b)

S nP

0

3 

 

  



 

……… (14)

C

''₀



S C

_c '₀



S nC

_c ₀



0

……… (15) Now the corresponding boundary conditions are

0 0 0

u

  

0,

1, C



1

at

y



0

……… (16)

0 0 0

u

  

0,

0, C



0

at

y

 

Equations (13) to (15) are ordinary linear differential equations, now

u

₀,



₀and

C

₀ with boundary conditions (16) are

3 1 2

m y m y m y

0 1 2 1 2

u



(A



A )e



A e



A e

 ……… (17)

1 m y 0

e



 

……… (18) 2 m y 0

C



e



……… (19)

where





2

r r r

1

4N

(P

b)

(P

b)

4(1

) S nP

3 m

4N

2(1

)

3  









2

c c c

2

S

4S n

m

2 





2 3

1 (1 b)

(1 b)

4 M

n

K

m

2 



 





_





_







r 1 2 1 1

G

A

1 m

(1 b)m

M

n

(7)

r 0 2

2

2 2

G N

A

1 m

(1 b)m

M

n

K





_{ }

_



_

_











_

_







Hence, The equations for u,



and C will be as follows

 

m y3 m y1 m y2 nt

1 2 1 3

u y, t





_

(A



A )e



A e



A e



_



e

 ……… (20)

 

_{y, t}

_e

m y1

_e

nt





……… (21)

 

m y2 nt

C y, t



e



e

 ……… (22)

Skin Friction:

The skin friction coefficient at y = 0 is given by





nt

3 1 2 1 1 2 2

y 0

u

m (A

A )

m A

m A e

y

 







 

_

_

 









……… (23)

RESULT AND DISCUSSION:

Fluid velocity distribution of fluid flow is tabulated in Table -1 and plotted in Fig. -1 having

six graphs at

P

_r= 0.71,

S

_c= 0.4,

n

= 0.1,

t

= 0.1, N₀ = 1.5, N = 0.1, S = 0.001for following

different value of

G

_r, M, K and b

Gr M K b

For Graph-1 2 0.02 100 0 For Graph-2 2 0.02 100 0.05 For Graph-3 4 0.02 100 0.05

For Graph-4 2 0.04 100 0.05 For Graph-5 2 0.02 1000 0.05

For Graph-6 2 0.02 100 0.10

It is observed from Fig.-1 that all velocity graphs are increasing sharply up to y = 1.2 after that velocity

in each graph begins to decrease and tends to zero with the increasing in y. It is also observed from Fig. -1 that

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The skin friction distribution is tabulated in Table -2 and plotted in Fig. -2 having six graphs. It is

observed from Fig. -2 that skin friction increases with the increase in Gr, K and b, but it decreases with the

increase in M.

The temperature distribution is tabulated in Table -3 and plotted in Fig. -3 having three graphs. It is

observed from Fig. -3 that temperature increases with the increase b.

The concentration does not change with the change in above parameters taken for velocity.

PARTICULAR CASE

When b is equal to zero, this problem reduces to the problem of Sharma et al (2011).

CONCLUSION

1. The velocity increases with the increase in bStratification parameter).

2. The skin friction increases with the increase in b.

3. The temperature also increases with the increase in b.

ACKNOWLEDGEMENT

The authors are very thankful to the Principal and the Head, Department of Mathematics, S.V. College,

Aligarh for providing necessary facilities.

The authors are also very thankful to U.G.C., New Delhi, for providing full financial assistance under

J.R.F. scheme.

Table-1: Value of velocity u for Fig-1 at Pr = 0.71, Sc = 0.4, n = 0.1, t =0.1, N₀ = 1.5, N = 0.1, S = 0.001 different values of Gr, M, K

and b.

y

Graph 1

Graph 2

Graph 3

Graph 4

Graph 5

Graph 6

0 0 0 0 0 0 0

1 16.15431 17.73285

35.46571

14.84031

19.79119 19.93860

2 19.17832 21.44272

42.88544

17.75351

23.95914 24.66625

3 17.77949 20.13157

40.26313

16.52665

22.52001 23.54857

4 15.20598 17.35202

34.70403

14.15180

19.43208 20.52753

5 12.60005 14.43610

28.87221

11.71582

16.18298 17.19355

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Table-2: Value of skin friction for Fig-2 at Pr = 0.71, Sc = 0.4, n = 0.1, N₀ = 1.5, N = 0.1, S = 0.001 different values of Gr, M, K and b.

t

Graph 1

Graph 2

Graph 3

Graph 4

Graph 5

Graph 6

0 28.66380 30.70210 61.40420

26.03192

34.22860

33.52975

0.2 28.09622 30.09416 60.18832

25.51645

33.55083

32.86581

0.4 27.53988 29.49825 58.99651

25.01119

32.88648

32.21503

0.6 26.99455 28.91415 57.82830

24.51594

32.23528

31.57713

0.8 26.46002 28.34161 56.68322

24.03049

31.59698

30.95186

1 25.93608 27.78041 55.56082

23.55466

30.97132

30.33897



Table-3: Value of temperature  for Fig-3 at Pr = 0.71, n = 0.1, t =0.1, N = 0.1, S = 0.001 different values of b.

y

Graph 1

Graph 2

Graph 3

0 0.99005 0.99005 0.99005

1 0.60050 0.63763 0.68867

2 0.36422 0.41066 0.47903

3 0.22091 0.26448 0.33320

4 0.13399 0.17033 0.23177

5 0.08127 0.10970 0.16122

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VELOCITY DISTRIBUTION

0 5 10 15 20 25 30 35 40 45

0 1 2 3 4 5

y 

u



Graph 1 Graph 2

Graph 3 Graph 4

Graph 5 Graph 6

Fig.-1



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SKIN FRICTION DISTRIBUTION

0 5 10 15 20 25 30 35 40 45 50 55 60 65

0 0.2 0.4 0.6 0.8 1

t 





Graph 1 Graph 2 Graph 3 Graph 4 Graph 5 Graph 6

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TEMPERATURE DISTRIBUTION

0.0 0.2 0.4 0.6 0.8 1.0

0 1 2 3 4 5

y







Graph 1

Graph 2

Graph 3

Fig.-3

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[3] Chenge, P. and Lau, K. H. (1977) : “In Proc., 2nd_{Nation’s Symposium Development”, Geothermal Resources}_,_{pp. 1591-1598.}

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