MHD FREE-CONVECTION THROUGH

A POROUS

AND MASS TRANSFER MEDIUM

BASANT

KUMAR

WITH

FLOW

HEAT SOURCE

JHA

Department of Mathematics, Banaras Hindu University, Varanasi, lndia and RAVINDRA

PRASAD

Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi, India

(Received 6 July, 1990) Abstract. An analytical study is performed to examine the heat source characteristics on the free-convection

and mass transfer flow past an impulsively started infinite non-conducting vertical plate of a viscous, incompressible electrically-conducting fluid under the action of a uniform magnetic field through porous medium. The effects of various parameters on the velocity field are extensively discussed.

1. Introduction Fluid currents formed in a fluid-saturated porous medium during convective heat transfer have many important applications, such as oil and gas production, cereal grain storage, geothermal energy, and porous insulation. The study of natural convection through porous medium also throws some light on the influence of environment-like temperature and pressure on the germination of seeds. In many situations the influence of heat and mass transfer on the hydromagnetic flow near the vertical plate is encountered, e.g., the cooling of nuclear reactor with electrically-conducting coolants such as liquid sodium and mercury (Rath and Parida, 1982). Raptis et al. (1981) have studied the steady free-convection flow and mass transfer through a porous medium bounded by an infinite vertical plate for the flow near the plate, by using the model of Yamamoto and Iwamura (1976) for the generalized Darcy's law. Gebhart and Pera (1971) have studied the laminar flows which arise in fluids due to the interaction of the force of gravity and density differences caused by the simultaneous diffusion of thermal energy and of chemical species. Raptis and Tzivanidis (1981) have studied the effects of mass transfer, free-convection currents, and heat sources on the Stokes's problem for an infinite vertical plate. The objective of the present investigation is to study the effects of heat source on M H D free-convection and mass transfer flow through a porous medium.

2. Mathematical Analysis This section has been devoted to mathematically analyse the unsteady flow of an incompressible and electrically-conducting viscous fluid along an infinite non-conducting vertical flat plate through porous medium in the presence of heat source. A Astrophysics and Space Science 181: 117-123, 1991. 9 1991 Kluwer Academic Publishers. Printed in Belgium.

118

B. K. JHA AND R. PRASAD

magnetic field of uniform strength is applied in the direction of flow and the induced magnetic field is neglected. The x'-axis is taken on the infinite plate and parallel to the free-stream velocity and y'-axis normal to it. At t' < 0, the plate and the fluid are at same temperature T " in the stationary condition with concentration level C~ at all points. At t' > 0, the plate starts moving impulsively in its own plane with a velocity U, its temperature raised to T~ and concentration level at the plate is raised to C~. Using the Boussinesq's approximation, the governing equations for the flow in the non-dimensional form are: ab/

a2b/

-

ay 2 aO

Pr

Sc

(1)

+GrO+GmO*-(M+K-1)u,

at

a20

-

at

ay 2

a0*

a20 *

-

at

ay 2

SO,

(2)

;

(3)

subjected to the initial and boundary conditions are: t_<0:

u(y,t)=O,

t>0:

~'u(0, t) = 1, {u(oc, t ) = 0 ,

O(y,t)=O,

forally

O*(y,t)=O;

0(0, t) = 1, 0(o%0=0,

0"(0, t) = 1, 0*(0%0=0.

(4)

The non-dimensional quantities introduced in the above equations are: y'U y -

t'U 2 ,

t -

,

U = u'/U,

o = (r'

-

~;)/(~;~-

7"L),

V

O*

=

(C'

-

C'co ]/(C' . . . . - C o o' ) ,

K-

Uak '

•2

v

,

Sc=-- , D (5)

Gm = vgfl (Co)r

Or = v g f l ( T ~ ) - T ' ) / U 3 , M-

aB2v pU 2 '

Pr = vCpp k

S'

m

C ~! ) / U 3,

Qv2 9 kU 2 '

where Sc, Pr, M, K, Gr, Gm, and S are the Schmidt number, Prandtl number, magnetic number, permeability parameter, Grashoff number, modified Grashoff number, and heat source parameter, respectively. In the present investigation, we have considered the heat generation (absorption) of the type Q' = Q ( T L -

T'),

where Q ' / p C p is the volumetric rate of heat generation (absorption).

119

MHD FREE-CONVECTION AND MASS TRANSFER

The solution of Equations (2) and (3) under the boundary condition (4) have been obtained by Raptis and Tzivanidis (1981) and Georgantopoulos etal. (1979). If we apply the usual Laplace transform technique, the solution of Equation (1) subjected to the boundary condition (4) is given by:

CaseI When Pr = Sc va 1

1[ Gr u(y, t) = 2 1 (L-S) Gm] [exp(y x ~ ) erfc(y/2 ,,~ + x/Lt) + +

exp( - y

x/L)

erfc(y/2 x/t -

x/~)]

Gr exp ( - at) x 2(L - S)

+

/(LPr-~)!){erfcIY/2x~t-/(LPr-S) ~ X/ Pr(Pr- 1),/J +exp (y/(L Pr-S)~ {erfc (y/2 x/tt+ /(L_ Pr ; 1) /

- erfc (2N/~ + X//Pr~r-- ~-/]J

S)t ~

,4 (Pr- 1) /

Gm2L

L Sc xexP(scLr 1)[exp(_yx/(S c 1)){erfc(y/2x/t_ /_ Sc_L, x) X/(Sc- 1)J - erf \2 (-y x//Sct - ~/iSc~Lt 1))} + exp (Y N/(LScSc--1))x x {erfc (y/2 x/~+ X/(S c / ScLt__1)/"]- erfc (2 X/7 + X/iScL~ 1i)}1 + m

Or + Om L erfc t2 N/~t + 2(L--S) x erfc

-

Prr + exp(-y x/S) x (6)

120

B . K . JHA AND R. PRASAD

where c~ = (S - L)/(Pr - 1), L = M + K - 1.

CaseH WhenPr= Sc= 1

.(y,t)=

)[

1

Gr (L-S)

Gm][exp(y,,f~)erfc(y/2,,ft+x/Ltt)+ Gm

+ e x p ( - y x/L) erfc(y/2 ~ t - x / ~ ) ] + - L x [exp(y x / S ) erfc (y/2 x/t + ~ )

erfc (y/2 x/t) +

Gr

2(L - S)

x

+ exp( - y x / S ) erfc (y/2 x/t - x/St)] . (7)

3. Discussion and Results In order to study the effects of heat source when plate moves with constant velocity on its own plane, numerical calculations are carried out for different values of Gr, Sc, Gin, Pr, S, M, and K which are listed in Tables I-VIII. The Grashoff number Gr represents the effects of the free-convection currents and have positive, zero, or negative values.

TABLE I

Velocity field distribution for different values o f S, w h e n P r = 0.71, G r = 4.0, G m = 5.0, M = 1.0, K = 2.0, Sc = 0.22, t = 0.2

y/S

0.2

0.5

4.0

0.0 0.4 0.8 1.2 1.6 2.0

1.0000 1.3055 1.0656 0.7046 0.4158 0.2314

1.0000 1.3013 1.0615 0.7024 0.4149 0.2312

1.0000 1.1381 0.8757 0.5864 0.3677 0.2179

T A B L E II

Velocity field distribution for different values o f S, w h e n P r = 0.71, G r = - 2.0, G m = 5.0, M = 1.0, K = 2.0, Sc = 0.22, t = 0.2

y/S

0.2

0.5

4.0

0.0 0.4 0.8 1.2 1.6 2.0

1.0000 0.8512 0.6528 0.4788 0.3308 0.2086

1.0000 0.8533 0.6549 0.4799 0.3312 0.2087

1.0000 0.9349 0.7478 0.5379 0.3548 0.2154

MHD FREE-CONVECTION AND MASS TRANSFER

121

TABLE III Velocity field distribuzion for different values of Gr, when Pr = 0.71, Gm = 5.0, M = 1.0, K = 2.0, Sc = 0.22, S = 0.2, t = 0.2

y/Gr

- 2.0

2.0

4.0

0.0 0.4 0.8 1.2 1.6 2.0

1.0000 0.8512 0.6528 0.4788 0.3308 0.2086

1.0000 1.154t 0.9280 0.6293 0.3874 0.2238

1.0000 1.3055 1.0656 0.7046 0.4158 0.2314

TABLE IV Velocity field distribution for different values of Gm, when Pr = 0.71, Gr = 4.0, M = 1.0, K = 2.0, Sc = 0.22, S = 0.2, t = 0.2

y/Gm

0.0

0.5

5.0

0.0 0.4 0.8 1.2 1.6 2.0

1.0000 0.7739 0.4474 0.1969 0.0655 0.0164

1.0000 0.8270 0.5092 0.2477 0.1006 0.0379

1.0000 1.3055 1.0656 0.7046 0.4158 0.2314

TABLE V Velocity field distribution for different values of Sc, when Pr = 0.71, Gr = 4.0, Gm = 5.0, M = 1.0, K = 2 . 0 , S = 0 . 2 , t = 0 . 2

y/Sc

0.22

0.5

0.6

0.75

1.5

0.0 0.4 0.8 1.2 1.6 2.0

1.0000 1.3055 1.0656 0.7046 0.4158 0.2314

1.0000 1.2331 0.9137 0.4977 0.2092 0.0698

1.0000 1.1987 0.8575 0.4413 0.1694 0.0493

1.0000 1.1297 0.7604 0.3615 0.1246 0.0313

1.0000 0.9237 0.5225 0.2168 0.0688 0.0167

Gr < 0 corresponds

p h y s i c a l l y to a n e x t e r n a l l y h e a t e d p l a t e as t h e f r e e - c o n v e c t i o n

c u r r e n t s are m o v i n g t o w a r d s t h e p l a t e . G r > 0 c o r r e s p o n d s to a n e x t e r n a l l y c o o l e d p l a t e and Gr = 0 corresponds to absence of the free-convection currents. F r o m T a b l e I w e o b s e r v e t h a t , w h e n G r > 0, a n i n c r e a s e in h e a t s o u r c e p a r a m e t e r S l e a d s t o a fall in t h e velocity. T h e o p p o s i t e p h e n o m e n o n is p r e s e n t e d in T a b l e II w h e n G r < 0. F r o m T a b l e s I I I - V it is c l e a r t h a t for fixed v a l u e s o f Pr, S, t, M , a n d K, t h e v e l o c i t y at a n y p o i n t i n c r e a s e s w i t h i n c r e a s e in G m a n d G r for fixed v a l u e o f Sc b u t d e c r e a s e s w i t h i n c r e a s e in Sc for fixed v a l u e s o f G i n

a n d G r . I t is c l e a r f r o m

122

B. K. J H A A N D R. P R A S A D

TABLE VI Velocity field distribution for different values of K, when Pr = 0.50, Gr = 2.0, Gm = 5.0, Sc = 0.60, M = 1.0, S = 0 . 2 , t = 0 . 4 ~K

1.0

2.0

0.0 0.4 0.8 1.2 1.6 2.0

1.0000 1.2376 1.0793 0.7825 0.4923 0.2729

1.0000

1.3540 1.2354 0.9241 0.5947 0.3350

TABLE VII Velocityfielddistribution~rdi~rentvaluesofM, when Pr=0.50, G r = 2 . 0 , Gm =5.0, Sc=0.60, K=l.0, S=0.2, t=0.8

y/M

0.5

0.0 0.4 0.8 1.2 1.6 2.0

1.0000

1.0000

1.5778 1.6489 1.4608 1.1680 0.8633

1.4259 1.4225 1.2199 0.9511 0.6887

1.0

TABLE VIII Velocity field distribution for different values of t, when Pr = 0.50, Gr = 2.0, Gm = 5.0, Sc = 0.60, S = 0.2, M = 1.0, K = 2.0

y/t

0.2

0.0 0.4 0.8 1.2 1.6 2.0

0.4

0.8

1.2

1.6

1.0000

1.0000

1.0697 0.7563 0.4019 0.1653 0.0535

1.3540 1.2354 0.9241 0.5947 0.3350

1.0000

1.0000

1.0000

1.5778 1.6489 1.4608 1.1680 0.8633

1.6920 1.8650 1.7558 1.5114 1.2215

1.7665 2.0075 1.9542 1.7495 1.4811

T a b l e s V I - V I I I , for fixed v a l u e s o f Pr, Sc, G r , G m , a n d S, t h e v e l o c i t y at a n y p o i n t i n c r e a s e s w i t h i n c r e a s e in p e r m e a b i l i t y p a r a m e t e r K w h e n M

a n d t are fixed b u t

d e c r e a s e s w i t h i n c r e a s e in m a g n e t i c n u m b e r M w h e n K a n d t are fixed. It is also s e e n t h a t t h e v e l o c i t y at a n y p o i n t i n c r e a s e s w i t h i n c r e a s e in t for fixed v a l u e s o f M a n d K.

MHD FREE-CONVECTION AND MASS TRANSFER

123

Acknowledgements The authors wish to express their sincere thanks to Prof. K. Lal, Dept. of Mathematics and Dr N. K. Samria, Dept. of Mechanical Engineering, Banaras Hindu University, for their kind guidance in the preparation of this paper.

References Gebhart, B. and Pera, L.: 1971, Int. J. Heat Mass Trans. 14, 2025. Georgantopoulos, G. A., Nanousis, D. N., and Goudas, C. L.: 1979, Astrophys. Space Sci. 66, 13. Raptis, A. A. and Tzivanidis, G.: 1981, Astrophys. Space Sci. 78, 351. Raptis, A. A., Tzivanidis, G., and Kafusias, N.: 1981, Letters Heat Mass Trans. 81,417. Rath, R. S. and Parida, D. N.: 1982, Wear. 78, 305. Yamamoto, K. and Iwamura, N.: 1976, J. Eng. Math. 10, 41.