A N I S O T R O P I C T H E R M O - C O N V E C T I V E E F F E C T S ON T H E

S T A B I L I T Y O F T H E T H E R M O - D I F F U S I V E E Q U I L I B R I U M

T H R O U G H A P O R O U S M E D I U M

M O H A M M A D Y U S U F and A. K. S I N G H

Department of Mathematics, Banaras Hindu University, Varanasi, India

(Received 25 March, 1990)

Abslract. The stability of the onset of thermo-conveetive effects of a fluid layer through porous medium is studied. The existence of marginal states, Rayleigh number, overstability, and the nature of non-oscillatory modes have been examined. We observed that the effect of the permeability has a tendency to increase the stability in the regions.

1. Introduction

The study of thermal convection through porous medium is of great importance in hydrology, oil extraction, and geophysics, etc. When two transport processes take place simultaneously, they are known to interfere with each other and occur cross phenomena. The flow of mass caused by temperature gradient is known as thermal effect or Soret effect and the flux of heat due to a convection gradient is known as thermo-diffusion or Dufour effect. Taslim and Narusawa (I986) analysed the cross-diffusion in a fluid saturated isotropic porous medium where the Dufour effect had been ignored in obtaining the results. Parvathy and Patil ((1989) showed by using Brinkman's model for flow through porous medium, the effect of cross-diffusion on thermohaline convection, the stability region for the Brinkman model is smaller than that for the Darcy model. Patil and Rudraiah (1973, 1984) considered magnetoconvection in a porous medium and showed that heat transport made to increase with an increase in Rayleigh number, the ratio of thermal diffusivity to magnetic diffusivity and porous permeability but decrease with an increase of the Chandrasekhar number (cf. Chandrasekhar, 1961).

Our motivation is by no means to investigate the anisotropic thermo-convective effects on the stability through porous medium. We investigate the existence of marginal states, Rayleigh number, overstability, and the nature of non-oscillatory modes. Numeri- cal results show that for particular range of values of M c and M a, critical Rayleigh numbers lie. Finally, we observed that the permeability has a stabilizing effect in which the stability increases in the regions.

2. Formulation of the Problem

A layer of viscous, incompressible fluid saturating porous medium confined between two horizontal boundaries is considered. For this we take rectangular axes in such a way that the planes at Z 3 = 0 and Z 3 = d represent the lower and the upper boundaries

Astrophysics and Apace Science 175: 125-134, 1991. �9 199 [ Kluwer Academic Publishers. Printed in Belgium.

126 M. YUSUF AND A. K. SINGH

which are maintained at constant temperatures T O and T 1, respectively. If we use Darcy's resistance with permeability Kp under Boussinesq approximation, the governing equations for the disturbances can be written as

v . ~ = o , (1)

- - + q . V ~ = ~t

- 1 v Vp+ vV2~ + ~[1 - ~(T- ro)+S~s(C- Co)] -~_ ~,

Po / /

(2)

~T - - + q. V T = (K T + M).ZKs)V2T + M) .KsV2C, (3) at

(3c - - + q. 7 C = 2KsV2T + K s V z c (4) Dt

and

p = Poll - ~(T - To) + S B , ( C - Co)], (5)

where the constants Po, To, Co are average reference values for the density, temperature, and concentration, respectively, v, K r , Ks, and 2 are the kinematic viscosity, thermo- metric conductivity, solutal diffusivity, and coefficient of thermo-convection while M is the thermodynamic coefficient; ~, acceleration due gravity; e, fis are the coefficients of thermal and mass expansivities; Kp, the permeability of the porous medium;

= (ql, q2, q3), P, T, and C are perturbations in velocity, pressure, temperature, and concentration, respectively; S = + 1, if the density of the solution is greater (or less)4han that of the solvent. Let P~(Z 3 = 0) and P2(Z3 = d) be the pressures which are referred to the orthonormal frame of reference (0, Ki) with K 3 positive upwards. Hence, the boundary conditions are as

q3 = 0 = D2q3 at Z 3 = 0 (6)

and d, where

T = To, C = C o at Z = 0 and T = T 1 , C = C~ at Z = d .

(6)

Let q(Z/t), T(Z/t) , C(Z/t) be small perturbation quantities, then the linearized governing equations become

7 . ~ = O, (7)

Oqc~t- pol Vfi+ v T Z ~ + ~ [ - ~ T + s f l ~ C ] - ~ ~, (8)

c3T - (K r + M/],2Ks)V2T + MJ, KsV2C + flq. K3 , ( 9 )

at

THERMO-CONVECTIVE EFFECTS OF A FLUID LAYER 127

ac - 2KsV2T + KsV2C -}- 7~.K3, (10)

at

where q, T, and C are the perturbed velocity, temperature, and concentration, respec- tively, and ~(Z/t) the perturbed pressure to P. Eliminating fi by taking twice curls of Equation (8) and Equations (7)-(10) follow

&8 gaq3 = vV2(V2q3 ) + g~ ~,~Z2 + az~ ] -

a2c)_ - gSfi~ \aZ21 + aZ2,/ gp V2q3' (1 1)

8T - (KT + M)~2Ks) V2T + M).Ks v2C + fiq3, (12)

at

ac - )'Ks 72T + Ks v 2 C -[- 7q3. (13)

at

We proceed in the similar way of Chandrasekhar (1961) to analyze the disturbances in terms of normal modes and assume that the perturbation quantities are of the form

(q3, T, C) = [W(Z3), T(Z3), F(Z3) ] exp [i(blZ , + b2Z2) + at] . (14)

If we insert Equation (14) in Equations (11)-(13), we have

v(D 2 - b 2 ) ( D 2 - b 2 1 ~ ) W = g c ~ b Z r _ g S f l s b 2 F ' i(p

(15)

K T { ( D 2 - b 2 ) ( l + M22 ~ T ) - ~ T } T = - [ I W - M 2 K ~ ( D 2 - b2)F,

(16)

D _ b2 _ o- ) F E = - y W - 2Ks(D 2 - b2)T, (17)

where D denotes differentiation w.r.t. Z3; 0", the growth rate; bl and b 2 are the wave numbers along Z 1- and Z2-directions and b = (b~ + b2) 1/2 is the resultant wave number.

By setting non-dimensional quantities

= d D , a = db, w - d W , T - K r T v vd

- Ks[" , G = --Gd2 , Pe - Kp vd v d 2 '

(18)

128 M. YUSUF AND A. K. SINGH

in Equations (15)-(17) and if we drop the bars over D, W, T, F, and ain non-dimensional

form we get the equations

(D2 _ a2) (D2 _ a 2 1 ) ~ SB~a2d ~ - - - - a W = g - - T - g F , (19) h" vKw vK~

[(D2 - a 2 ) ( l + M22 P~) - ~TP~JT= - f l W - M2(D2 - a2)F, (20)

( D 2 - a 2 - aSc)F = - v W - 2P~ ( D 2 _ a 2 ) T ' (21) &

where Pr = v/Kr and Sc = v/K s "are the Prandtl and Sehmidt numbers. As we have taken the problem of fluid layers through porous medium with free boundaries,

Equations (19)-(21) are solved under boundary conditions

W = O = D 2 W = T = F at Z 3 = 0 and I . (22)

Now, if we eliminate T and F from Equations (19)-(21), we have

(D2 - a2) (D2 - a 2 1 ~ r ) Pi [ ( D 2 - a e - o ' S ~ ) x

x {(D 2 - a 2) (1 +M22Ra) - air} - (D 2 - a2)aM22Ra]W =

= - (D e - a 2 - ~TSc)RaZW + [(D 2 - a 2 ) (1 + M22Rd) - a P A S R s a e w -

( Mc~RsRa S~R)2a2W, (23) + (D 2 _ a 2) \

where

R - g~ R~ - gyBsd4 and R a = P" vK r vKs Sc

are the Rayleigh numbers for the constant temperature gradient fl and concentration

gradient v and Lewis number.

3. Frequency of Oscillation at the Marginal State

I f we apply the boundary conditions (22) with the help of Equations (19)-(21), we can show that all the even-order derivatives of W vanish at the boundaries (because of free boundaries). Hence, the proper solution of Equation (23) characterizing the lowest mode is W = W o sin feZ 3, where Wo is a constant. I f we substitute this value in

THERMO-CONVECT1VE EFFECTS OF A FLUID LAYER ][29

Equation (23) we get the Rayleigh number R at marginal state (i.e., a = 0)

R = (792 + a2)2 a 2 + ~2 + +

+(I+ Mc+~)R,]/(I+ Ma), (24) where

B~2 M~ = M22Rd and M a -

Again, if we use Equation (23) at the lowest mode and introducing

a 2 0" R R s

X - 7E 2 , i0"1 -- ~ 2 ' R 1 - 7r 4 , R~ ~4

and Pj = PI n2, we obtain the dispersion relation

( 1 ) R 1 =(1 + x ) 1 + x + - - + i ~ 1 [(1 + x + i a l S ~ ) { ( 1 + x ) ( 1 + M 2 2 R a ) + i c r 1 P r } -

P;

- (1 +x)2M22Rd] / [ ( l + x + i a l S c ) + (1 + x ) S : s 2 ] x + [{(1 + x ) •

X (I ~- M),2Rd)+ i~iPr}S+ ( i + x)Mo{~.RdIRts/~( 1 Jc-x -~-i~iSc)-}- Bs A I L

+ ( l + x ) S T s ] .

If we separate the real and imaginary parts of (25), we get

( RlX 1 + s = 1 + x + [ ( l + x ) z ( l + M 2 2 R a ) -

and

(25)

- a~P~S c - (1 + x)2M,~2Ra] - o-~(1 + x) {So(1 + M22Ra) + P r }

{ + ( l + M 2 2 R d ) S + ~ -j , (26)

+ (1 + x) {(1 + x) 2 (1 + M22Rd) - a2prsc -

- (1 + x)2M22Rd} + R; PrS. (27)

130 M. Y U S U F AND A. K. S I N G H

If we eliminate R 1 from Equations (26) and (27), we have

ScI{p , (s c S~B,)+Sc(1 + M22Ra)}b1+ 7z JPrSc]r

+ M22S~Ra+P~) 1 + " +S eT+

+ 1 + PrS- SS c (1 + M22R•) MX~ScR4_R'x, Bs )

w h e r e x + l = b 1. From Equation (28), we can write

A~a 2 + A o = O,

+

(28)

(29)

Fig. 1.

4 4 0 6

Z, 405

T 4404

2115

2117

Rs = 2 5 0 0 , Pt = 0.01

\ \ Mc : 10

\ Md z 15 \

\ N

\ \

/ /

J J

,

I I [ I I

\ Mc : 15 Md = 10

\ \

\ \

j / J

J

2116 I L I I I 2 .75 2.80 2.85 2.90 2.95 3.00 3.05

(3 .~ -

Behaviours o f M c (= M 2 2 R a ) and M a (= B,2/oO in the (R, a) plane.

THERMO-CONVECTIVE EFFECTS OF A FLUID LAYER 131

where

and

A 1 =S~ Pr S~ ~t +S~(1 +MZ2Ra) b a+ Pz d

Ao={(M22ScRd+Pr+ 1)(1 + S~27)+ S Sc2ccB'}b~+

+ (M22Sr 1 + s +S - - + s

{('?) + 1 + + SP~- SS~(1 + M22Rd) MZccScRa]? j R ; x .

Equation (29) gives the required expression for the frequency of oscillation in the marginal state. If a~ is negative, the marginal state and the overstability cannot occur in that case, i.e., when A~ and A o are both positive.

Rs = 2 4 9 5 , !'4 d = 10 , M c = 15

4 3 9 6

4.397

'_\ \

\ \

\ \

\

Pt = 0.01

\ \ /

\ /

T 4 3 7 6 I ~ I I 2 .75 2.80 2 .85 2.90 2.95

c,,, 4107

/ / /

/

3.(]0 3.05

4.106

PL = 0.05

\ \

\ / / "

7

4.105 ] I ; I I 2 .45 2.50 2.55 2.60 2.65 2.70 2.75

Fig. 2. Effect of variation of permeability (Pt) in the (R, a) plane.

132 M. YUSUF AND A. K. SINGH

Overstabi l i ty may occur if A l -~ 0 and Ao X O, i.e.,

and

(Pr + 1 + M22Rd)b~ + p,/j c c~

[{(M22ScRa + Pr + 1)(I + S-2?') + S~S2Bs}b3 +

3P/

+ {(I + S-2Bs) + P,S}R~x]X Sc {S(I + M2~Rs) +

M2~Ra) , + - - } R s x . Bs )

cK

7633

7 6 3 2

7631

7 6 3 0 -

7629 -

R s = 2/ , ,95, NI d = 1 0 , Me = 15

PL = 0.001 ]

/ /

/ \ /

/ \ / \ / \ /

/ /

\ / \ /

/ \ / \ /

\ /

I 1 I I I

762~'~ 9'S 'q 3.00 3 0 5 3.10 3.15 3,20 3.25

Fig. 3. Effect of variation of permeability (Pt) in the (R, a) plane.

(30)

(31)

T H E R M O - C O N V E C T I V E E F F E C T S O F A F L U I D L A Y E R 133

Hence, we conclude that the margnal state exists and overstability occurs only when

the conditions (30) and (31) are satisfied. When P~ -~ o% S = 1, f i> 0, and 7 < 0, the system has a stabilizing effect and at the same time when/3 < 0 and 7 > 0, it has a destabilizing effect which agrees with the Maiellaro and Palese (1989) result.

4. Nature of Non-Oscillatory Mode

Non-oscillatory modes may exist for which a 1 = 0 so that a = a 2 (% is real). Hence, introducing a = % and taking the solution as W = Wo sin rcZ 3 in Equation (23) and on simplification gives

F(a2)= G a 3 + H a ~ + L a 2 + N = O,

where

(32)

G = S c P r ( x 2 + a2),

E ] g = (re 2 + a 2) S o P r n 2 + a 2 + + ( n2 + a2) {Sc(1 + M 2 2 R a ) + Pr} ,

1 L = (n 2 2r- a 2 ) 2 rC 2 + a 2 + {S~(1 + M 2 2 R a ) + Pr} + ( n2 + a 2) +

+ a a ( R s S P r - S c R ) ,

[ { N = ( n 2 + a 2) (n2+a2)2 n 2 + a a + + a 2 (1 +M22Ra)SRs+

Bs

For F(oe) is positive and F(0) is also positive, then the system is stable. Hence, finally we conclude that the effect of permeability of the medium is to increase the stability in the regions.

5. Results and Discussion

Numerical results obtained by Equation (24) are plotted in Figures 1 to 3. After comparing, it is observed that anisotropic thermo-convective effects on the stability of the thermo-diffusive equilibrium through porous medium has a stabilizing effect. Figure 1 shows that for particular range of values of M c and M a, R has critical values and Figures 2 and 3 show that stability region increases with permeability.

134 M. YUSUF AND A. K. SINGH

References

Chandrasekhar, S.: 1961, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford. Maiellaro, M. and Palese, L.: 1989, Int. J. Engng Sci. 27, 329. Parvathy, C. P. and Patil, R. Prabhamani: 1989, J. Math. Phys. Sci. 23, 281. Patil, R. Prabhamani and Rudraiah, N.: 1973, J. Appl. Mech. 40, 879. Rudraiah, N.: 1984, Proc. Indian Acad. Sci. (Math. Sci.) 93, 117. Taslim, M. E. and Narusawa, U.: 1986, ASME, J. Heat Trans. 108, 221.