Heat and Mass Transfer 38 (2002) 551±563 Ó Springer-Verlag 2001 DOI 10.1007/s002310100245

Rayleigh±Benard convection subject to time dependent wall temperature/gravity in a fluid-saturated anisotropic porous medium M. S. Malashetty, D. Basavaraja

Abstract The effect of time-periodic temperature/gravity modulation at the onset of convection in a Boussinesq ¯uid-saturated anisotropic porous medium is investigated by making a linear stability analysis. Brinkman ¯ow model with effective viscosity larger than the viscosity of the ¯uid is considered to give a more general theoretical result. The perturbation method is applied for computing the critical Rayleigh and wave numbers for small amplitude temperature/gravity modulation. The shift in the critical Rayleigh number is calculated as a function of frequency of the modulation, viscosity ratio, anisotropy parameter and porous parameter. We have shown that it is possible to advance or delay the onset of convection by time-periodic modulation of the wall temperature and to advance convection by gravity modulation. It is also shown that the small anisotropy parameter has a strong in¯uence on the stability of the system. The effect of viscosity ratio, anisotropy parameter, the porous parameter and the Prandtl number is discussed. List of symbols d distance between the plates g gravitational acceleration k anisotropy vector, (kx ; kx ; kz ) kx ; kz permeability of the porous medium in horizontal and vertical directions k1 anisotropy parameter, kx =kz k2 porous parameter, kz =d2 Km effective thermal conductivity of porous media l; m wave numbers in x; y directions M viscosity ratio, le =lf Pr Prandtl number, m=j p pressure pH basic state pressure q velocity vector R Rayleigh number, bgDTdkz =mj T temperature TH basic state temperature distribution TR reference temperature

t time x; y; z space co-ordinates Greek symbols a horizontal wave number ac critical wave number b co-ef®cient of thermal expansion c …qcp †m =…qcp †f , ratio of speci®c heats DT temperature difference between the plates e amplitude of modulation j thermal diffusivity, K=…qcp †f le effective viscosity lf viscosity of the ¯uid m kinematic viscosity, lf =qR q density qR reference density qH basic state density / phase angle X frequency of the modulation x non-dimensional frequency, Xd2 =j 2 o2 o2 r1 ox2 ‡ oy2 r2

2

o r21 ‡ oz 2

1 Introduction The convective instability of a horizontal porous layer subject to uniform temperature gradient, in the vertical direction has been investigated extensively by several authors, using a Darcy model [1±4]. However, Darcy model is applicable only under special circumstances, and a generalized model for the accurate prediction of convection in a porous media must include Forchheimer's inertia term and Brinkman's viscous term. Rudraiah et al. [5], Georgiadis and Catton [6], Kladias and Prasad [7±9], and Prasad et al. [10] have used the Darcy±Brinkman and Darcy±Brinkman±Forchheimer models for studying Benard convection in porous media. The above-mentioned studies on porous medium assumes that, the medium is isotropic. However porous mediums in general are not isotropic. In some natural Received on 28 July 2000 / Published online: 29 November 2001 con®gurations the porous layer, though homogeneous, may be anisotropic. This is frequently the case with sediM. S. Malashetty (&), D. Basavaraja mentary rocks for which the permeability tensor, for inDepartment of Mathematics stance, is not spherical because the vertical permeability is Gulbarga University less than the horizontal one. Such a con®guration can also Gulbarga-585106 India be observed in snow layers or with glass, ®ber materials The Research of the one of the authors (BD) is supported by the used for insulating purposes. In an aquifer the grain-size Research Scholarship grant of Gulbarga University, Gulbarga. distribution as well as the packing of constituent materials

551

552

depend on the depth. Likewise in geothermal region the crack pattern may be quite irregular. A lot of work has to be done on the effect of anisotropy on the onset of convection and on the heat transfer they induce. It appears that only few studies are available on the Rayleigh±Benard convection in an anisotropic porous medium. Castinel and Combarnous [11] ®rst studied Rayleigh±Benard instability in an anisotropic porous medium and their work was extended by Epherre [12] to porous layers with anisotropy in the thermal diffusivity also. Mckibbin [13] studied the fraction of the total ¯ow that recirculates with an anisotropic layer at the onset of convection. Nielsen and Storesletten [14] considered the case of a rectangular channel non-uniformly heated to establish a linear temperature distribution in the vertical direction. The effect of anisotropy on the onset of motion and on the steady ¯ow patterns at moderately supercritical Rayleigh numbers were discussed. Tyvand and Storesletten [15] analyzed the onset of nonlinear development of thermal convection in an anisotropic porous medium. Recently Mamou et al. [16] have studied the effect of anisotropy on the onset of convection in a horizontal porous layer with constant heat ¯ux. A linear stability analysis is performed and the resulting governing equations are solved analytically and numerically. There have been many investigations concerning how a time-dependent boundary temperature affects the onset of Rayleigh±Benard convection. Most of the ®ndings relevant to these problems have been reviewed by Davis [17]. In case of small amplitude temperature modulations, a linear stability analysis was performed by Venezian [18]. Rosenblat and Herbert [19], Rosenblat and Tanaka [20] and Roppo et al. [21] have studied the effect of temperature modulation on the onset of thermal convection in a ¯uid layer. The studies on the effect of thermal modulation on the onset of convection in a ¯uid-saturated porous medium have received marginal attention. The effect of time dependent wall temperature on the onset of convection in a porous medium has been studied by Caltagirone [22], Rudraiah and Malashetty [23], Malashetty and Wadi [24]. Further the combined effect of anisotropy of the porous medium and time dependent wall temperature/ gravity on the onset of convection in a porous medium is not studied so far. Since the amplitude and frequency of the modulation are externally controlled parameters the onset of convection can be delayed or advanced by the proper tuning of these parameters. The thermal modulation can be used as a mechanism to delay convection in the case of material processing applications to attain higher ef®ciencies and to advance it in achieving major enhancement of mass, heat and momentum transfer. Recently, there has been a great deal of interest in the study of the effect of complex body forces on convection in a ¯uid and ¯uid saturated porous layer. Grasho and Sani [25] studied the effect of gravity modulation on the onset of convection in a horizontal ¯uid layer with small amplitude approximation. Clever et al. [26] studied the problem of two-dimensional oscillatory convection in a gravitationally modulated ¯uid layer. Farooq and Homsy [27] studied linear and non-linear convection in a vertical slot under gravity modulation. Recently Malashetty and

Padmavathi [28, 29] studied the effect of small amplitude gravity modulation on the onset of convection in a ¯uid and ¯uid saturated porous layer. However the study of the combined effect of anisotropy and gravity modulation is not available. The purpose of present paper therefore, is to study the combined effect of anisotropy of the porous medium and the time dependent wall temperature and gravity modulation on the onset of convection in a horizontal anisotropic porous medium considering the Brinkman ¯ow model with the effective viscosity larger than the ¯uid viscosity, as this work is not available in the literature.

2 Mathematical formulation We consider a ¯uid-saturated anisotropic porous medium bounded between two horizontal surfaces at z ˆ 0 and z ˆ d. A vertical gravitational force acts on the ¯uid. The time-periodic wall temperatures are externally imposed, and they are T…0; t† ˆ TR ‡

DT ‰1 ‡ e cos XtŠ 2

…1†

and

DT ‰1 e cos…Xt ‡ /Š ; …2† 2 where  represents small amplitude of the oscillation of the wall temperature, X is the frequency and / is the phase angle. We consider three types of temperature modulation namely (a) symmetric (in phase, / ˆ 0), (b) asymmetric (out of phase, / ˆ p) and (c) only lower wall temperature is oscillating (/ ˆ i1). We assume that the amplitude is very small and can be used to control convection. We use the amplitude of thermal modulation as a parameter to ®nd the solution of the basic equation. The porous medium is assumed to posses horizontal isotropy. With the Boussinesq approximation made in the study of thermal convection problems in a ¯uid-saturated sparsely packed porous medium, the governing equations are taken as (30) T…d; t† ˆ TR

r
qˆ0 ; …3†   oq lf q ‡ le r2 q ; …4† ‡ q
rq ˆ rp ‡ qf g qf k ot oT …5† …qcp †m ‡ …qcp †f q
rT ˆ Km r2 T ; ot …6† qf ˆ qR ‰1 b…T TR †Š : These are the continuity, the momentum, the energy equations, and the equation of state for the ¯ow through anisotropic porous medium, which has horizontal isotropy in permeability. 2.1 Basic state The basic state of the ¯uid is quiescent and is given by qH ˆ 0; T ˆ TH …z; t†; p ˆ pH …z; t†; qf ˆ qH …z; t† : …7† The temperature TH , pressure pH and density qH satisfy the following equations:



oTH o2 TH ˆj 2 ; c ot oz opH ˆ qH g oz and qH ˆ qR ‰1

b…TH

…8† …9†

    o o 1 o2 2 4 ‡ 1 r1 ‡ ‡ Mk2 r ot ot k1 oz2     o 1 2 1 oTH r21 W ˆ 0 :  Pr k2 r W ‡ Rk2 Pr oz ot …17†

TR †Š ;

…10†

We express the boundary conditions (15) in terms of only vertical component of velocity, which in non-dimensional where c ˆ …qcp †m =…qcp †f and j ˆ K=…qcp †f : Under the boundary conditions (1) and (2) the solution of form are given by Eq. (8) is 2 4

nh TH ˆ TS ‡ e Re a…k†ekz=d ‡ a… k†e

i kz=d e

iXt

o

; …11†

where

DT…d 2z† TS ˆ TR ‡ ; 2d

k ˆ …1

 1=2 cXd2 ; i† 2j

oW oW ˆ 4 ˆ 0 at z ˆ 0; 1 : …18† oz2 oz The dimensionless temperature gradient appearing in Eq. (17) can be obtained from the Eq. (11) as, Wˆ

oTH ˆ oz where

1 ‡ ef ;

…19†

DT …e i/ e k † kz kz ixt and Re stands for the real part: f ˆ Re‰A…k†e ‡ A… k†e Še 2 …ek e k † with   x1=2 k …e i/ e k † 2.2 A…k† ˆ : and k ˆ …1 i† 2 …ek e k † 2 Linear stability analysis For the small disturbances, we assume a solution for q, T, p and q in the form The horizontal dependence of W is factorable, and we look for solutions with a single wave number a, such that 0 0 0 0 q ˆ q ; T ˆ TH ‡ T ; p ˆ pH ‡ p ; qf ˆ qH ‡ q ; r21 W ˆ a2 W. The dependence exp‰ …lx ‡ my†Š of W on …12† the horizontal co-ordinates is implied throughout. where q0 , T 0 , p0 and q0 represents the perturbed quantities which are assumed to be small. Substituting Eq. (12) into 3 Eqs. (3)±(6) and using the basic state equations and ne- Perturbation procedure glecting nonlinear terms in perturbed quantities, we obtain We seek eigenfunctions W and the eigenvalues R of Eq. (17) for the basic temperature pro®le (19) that departs the following equations for the perturbed quantities: from the linear pro®le …oTH =oz ˆ 1† by quantities of oq0 1 v 0 le 2 0 0 0 order e. Thus the eigenvalues of the present problem difˆ q ‡ rq ; rp ‡ bT g^ n …13† ot qR qR k fers from the classical Rayleigh±Benard problem by   quantities of order e. We seek solution of Eq. (17) in the oT 0 oTH ‡ w0 ˆ jr2 T 0 ; …14† form c ot oz 2 ^ is the unit vector in the positive z direction and …W; R† ˆ …W0 ; R0 † ‡ e…W1 ; R1 † ‡ e …W2 ; R2 † ‡


where n …20† the value of c is set equal to 1. The boundary conditions for the perturbed velocity and temperature are given by Malkus and Veronois [31] ®rst used this type of expansion o2 w in connection with convection problem to study the effects 0 w ˆ 2 ˆ 0; T ˆ 0 at z ˆ 0; d : …15† of ®nite-amplitude convection. Here W and R are the 0 0 oz eigenfunction and eigenvalue respectively of the unmod0 0 We eliminate p and T from Eqs. (13) and (14) and render ulated system and Wi ; Ri …i  1† are the correction to W0 the resulting equations dimensionless by setting and R0 in the presence of modulation.   1 tm k Substituting Eq. (20) into Eq. (17) and equating the z w0 ; …x ; y ; z † ˆ …x; y; z†; t ˆ ; W ˆ corresponding terms, we obtain the following system of vd d kz equations: T0 kx kz ; k1 ˆ ; k2 ˆ 2 ; T ˆ LW0 ˆ 0 ; …21† DT kz d 1 2 2 bgDTdkz m Xd2 …22† LW1 ˆ k2 Pr …R1 r1 W0 R0 f r1 W0 † ; Rˆ ; Pr ˆ ; x ˆ ; …16† mj j j 1 2 2 2 LW2 ˆ k2 Pr …R1 r1 W1 ‡ R2 r1 W0 R0 f r1 W1 to obtain the following equation for the vertical compoR1 f r21 W0 † ; …23† nent of velocity W on dropping all asterisks:

a…k† ˆ

553

where



    o o 1 o2 2 4 ‡ 1 r1 ‡ ‡ L Mk2 r ot ot k1 oz2   o  Pr 1 k2 r2 R0 k2 Pr 1 r21 : ot

554

Each W0 , W1 , W2 is required to satisfy the boundary conditions of Eq. (18). Equation (21) is the one used in the study of convection in an anisotropic porous medium subject to uniform temperature gradient. The marginally stable solutions of …n† Eq. (21) is W0 ˆ W0 ; n ˆ 1; 2; 3; . . . ; where …n† W0 ˆ sin…npz†, with corresponding eigenvalues …n† R0 ˆ R0 given by …n† R0


1 2
1 2 2 ˆ 2 a a ‡ n2 p2 ‡ a ‡ n2 p2 n2 p2 a k1 
3 ‡ Mk2 a2 ‡ n2 p2 :

…24†

For a ®xed value of a the least eigenvalue occurs for n ˆ 1: R0 assumes a minimum value for a ˆ ac where ac satis®es the equation


3

2 2Mk2 a2c ‡ 1 ‡ 3k2 Mp2 a2c
Mk2 p6 ‡ …p4 =k1 † ˆ 0 :

…25†

Clearly the critical wave number ac depends on anisotropy parameter k1 and the Brinkman number (porous parameter) k2 . It is interesting to note that the expression for the critical wave number ac given by (25) is similar to the one given by Rudraiah and Siddheshwar [32] in the study of double diffusive convection with cross-diffusion in a ¯uid saturated porous medium when k1 ˆ 1: The dependence of the unmodulated critical Rayleigh number R0 and the critical wave number ac on the anisotropy parameter k1 for ®xed value of k2 is shown in Fig. 1. It is interesting to note that small anisotropy parameter k1 has a signi®cant effect on both critical wave number and critical Rayleigh number. Equation (22) is inhomogeneous and its solution poses a problem, because of the presence of resonance terms. In the study of nonlinear convection problems, e is usually the amplitude of perturbations, whereas in the present problem it is the amplitude of thermal modulation. In this power integral technique, the solubility condition requires that the time-independent part of the right-hand side of Eq. (22) should be orthogonal to W0 . The term independent of time in the right-hand side of Eq. (22) is k2 Pr 1 R1 r21 W0 so that R1 ˆ 0; it follows that all the odd coef®cients R1 ; R3 ; . . . ; in Eq. (20) are zero. We expand the right-hand side of Eq. (22) in a Fourier series of the form

ekz sin…mpz† ˆ

1 X nˆ1

where

gnm …k† ˆ

gnm sin…npz† ;

  4mnp2 1 ‡ … 1†m‡n‡1 ek k  2   k ‡ …n ‡ m†2 p2 k2 ‡ …n m†2 p2

…26†

Fig. 1. Variation of the critical wave number ac and the critical Rayleigh number R0c with anisotropy parameter k1

and we obtain the following expression for W1 by inverting the operator L term by term 1 2

W1 ˆ R0 k2 Pr a Re

X

Bn …k† e L…x; n†

ixt

 sin…npz† ;

…27† where

Bn …k† ˆ A…k†gn1 …k† ‡ A… k†gn1 … k†; L…x;n†


ˆ x2 a2 ‡ n2 p2  

2 1 2 2 2 2 2 2 1 M ‡ Pr ‡ a ‡ n p ‡ ix k2 a ‡ n p k1   2 2
2 np
1 2 2 2 2 2 2 2 ‡ Mk2 a ‡ n p a ‡ Pr k2 a ‡ n p k1  
1 2
2
1 2 2 2 2 2 2 3 a a ‡p ‡ : a ‡ p p ‡ Mk2 a ‡ p ‡ k2 Pr k1 …28†

The Eq. (23) for W2 then reduces to

LW2 ˆ

k2 Pr 1 R2 a2 W0

k2 Pr 1 R2 a2 fW1 :

…29†

We shall not require the solution of this equation, but merely use it to determine R2 , the ®rst non-zero correction to R: Again the solubility condition requires that the steady part of the right-hand side should be orthogonal to sin…pz†, and this gives

Z1 R2 ˆ 2R0

fW1 sin…pz†dz ;

…30†

0

where the over bar represents a time average. Now, rewriting Eq. (22) we obtain

f sin…pz† ˆ

1

Pr

1 2 aR

0 k2

R…a; e† ˆ R0 …a† ‡ e2 R2 …a† ‡




LW1 ;

2

a ˆ a0 ‡ e a2 ‡




so that

1 R0 fW1 sin…pz† ˆ W1 LW1 ˆ k2 Pr 1 2 Pr 1 a2 R0 k2 X  X Bn …k† 2   a Re sin…npz† Bn …k† sin…npz† L…x; n† and ®nally Eq. (30) gives

R2 ˆ k2 Pr

1

1 R20 2 X jB2n …k†j2  a 2 ‰L…x; n† ‡ L …x; n†Š ; 4 jL…x; n†j nˆ1

…31† where  denotes a complex conjugate. we simplify the expression for R2 given by (31) and obtain

R2 ˆ k2 Pr

1

R20 2 X 2 a Bn Cn =Dn ; 2

…32†

where

16p4 n2 x2  B2n ˆ  x2 ‡ …n ‡ 1†4 p4 x2 ‡ …n and

R2c  O…x 4 † and hence the effect of modulation disappears for large x. The value of R obtained by this procedure is the eigenvalue corresponding to the eigenfunction W, which though oscillating remains bounded in time. Since R is a function of the horizontal wave number a and amplitude of perturbation e, we expand

1†4 p4



…33† …34†

The critical value is computed upto O…e2 † by evaluating R0 and R2 at a ˆ a0 . It is only when one wishes to evaluate R4 and a2 must be taken into account. In view of this we write Eq. (33) in the form

Rc ˆ R0c ‡ e2 R2c ;

…35†

where R0c and R2c can be obtained from (24) and (32), respectively. If R2c is positive, supercritical instabilities exists and Rc has a minimum at e ˆ 0. If R2c is negative, subcritical instabilities are possible. In this case we have from equation (35) e2 < R0c =R2c . From this we can determine the maximum range of e, by assigning values to the various physical parameters involved in the problem. Thus the range of the amplitude of modulation, which causes subcritical instabilities in different physical situations, can be explained. To the order of e2 , R2 is obtained for the cases where the oscillating temperature ®eld is (a) symmetric (/ ˆ 0), and (b) asymmetric (/ ˆ p), and (c) when only the lower wall temperature is oscillating while the upper wall is held at constant temperature …/ ˆ i1†. In the next section we address another important problem of gravity modulation using the analysis of the previous sections.

4 Cn ˆ x2 …a2 ‡ n2 p2 † Pr 1 k2 …a2 ‡ n2 p2 † Time dependent gravity   n2 p2 2 2 2 2 2 Under the in¯uence of a periodically varying vertical ‡ Mk2 …a ‡ n p †  a ‡ gravity ®eld k1   2 g ˆ g0 …1 ‡ e cos Xt† ; …36† p ‡ k2 Pr 1 …a2 ‡ p2 † a2 ‡ p2 ‡ Mk2 …a2 ‡ p2 †2 ; k1 where g0 is the mean gravity, e is the small amplitude of " gravity modulation, X is the frequency and t is the time, the governing equations for the Boussinesq ¯uid-saturated Dn ˆ x2 …a2 ‡ n2 p2 † Pr 1 k2 …a2 ‡ n2 p2 † anisotropic porous medium are essentially the same   Eqs. (3)±(6) but with g given by Eq. (36). 2 2 np 2 2 2 2 2 The basic state of the ¯uid is quiescent and is described ‡ Mk2 …a ‡ n p †  a ‡ k1 by  #2 2 qH ˆ 0; T ˆ TH …z†; p ˆ pH …z†; qf ˆ qH …z† : p ‡ k2 Pr 1 …a2 ‡ p2 † a2 ‡ ‡ Mk2 …a2 ‡ p2 †2 …37† k1 " #2 This differs from the one in the thermally modulated case. n2 p2 2 2 2 2 2 1 2 ‡x : Temperature TH in the basic state satis®es k2 …a ‡ n p † …M ‡ Pr † ‡ a ‡ k1 r2 TH ˆ 0 ; …38† In Eq. (32) the summation extends over even values of n wherein the pressure pH balances the buoyancy force. Following the analysis of Sect. 2, the non-dimensional for case (a), odd values of n for case (b) and for all integer values of n for case (c). The in®nite series (32) converges form of the linearized perturbation equation, on using Eq. (36), are given by rapidly, as n 6 . Further, it follows from Eq. (32) that

555





556

   o o 1 o2 2 ‡ 1 r1 ‡ ‡ ot ot k1 oz2

o ot

Mk2 r

4



 W ˆ RPr 1 k2 …1 ‡ e cos xt†r21 T 0 ;  1 2 Pr k2 r T 0 ˆ W :

…39† …40†

Equations (39) and (40) can be combined to obtain      o o 1 o2 2 4 ‡ 1 r1 ‡ ‡ Mk2 r ot ot k1 oz2   o 1 2 Pr k2 r T0  ot

ˆ Rk2 Pr 1 …1 ‡ e cos xt†r21 T 0 ; …41† where the parameters R; M; Pr, k1 and k2 are as de®ned earlier. Following the analysis of Sect. 3, we obtain R0 same as the one given by Eq. (24) and R2 in the form,

X 1 k2 Pr 1 R20 a2 Cn =Dn : …42† 2 The expressions for Cn and Dn are essentially the same as in Sect. 3. The variation of R0c with k1 and R2c with x for different values of M, k1 , k2 and Pr is shown in Figs. 1±17 and the results are discussed in Sect. 5. R2 ˆ

5 Results and discussion In the present paper we make an analytical study of the effect of time-periodic temperature/gravity modulation on the onset of convection in an anisotropic porous medium. An approximate linear stability analysis proposed by Venezian [18] is used to ®nd the critical Rayleigh number as a function of frequency of the modulation, Prandtl number, anisotropy parameter, viscosity ratio and porous parameter.

The solution obtained in this paper is based on the assumption that the amplitude of the temperature/gravity modulation is small. Furthermore, the validity of the results depends on the value of the modulating frequency x. When x  1, the period of modulation is large and the temperature/gravity modulation affects the entire volume of the ¯uid. On the other hand, the effect of modulation disappears for large frequencies. This is due to the fact that the buoyancy force takes a mean value leading to the equilibrium state of the unmodulated case. In view of this we choose only moderate values of x in our present study. We now discuss the results obtained in this paper. Figure 1 shows the effect of anisotropy of the porous medium for the ®xed value of the porous parameter k2 and the viscosity ratio M on the critical wave number ac and the critical Rayleigh number R0c of the unmodulated system. We observe from this ®gure that, as the anisotropy parameter k1 becomes smaller, both the critical wave number ac and the critical Rayleigh number R0c increases linearly. Thus the effect of small k1 is more pronounced. However for large values of anisotropy parameter k1 , the effect is almost negligible. The variation of the shift in the Rayleigh number R2c with frequency x for different values of the viscosity ratio M is shown in Fig. 2. We observe from this ®gure that R2c is negative for the whole range of the frequency, indicating that the symmetric modulation advances the convection. The effect of large viscosity ratio is to further advance the convection in this case. However for small M, R2c become positive for the values of x  45. Figure 3 shows the variation of R2c with x, for different values of anisotropy parameter k1 and ®xed values of viscosity ratio M, porous parameter k2 and Prandtl number Pr, in respect of symmetric modulation of the wall temperature. We ®nd that R2c is positive for the moderate values of the frequency x …x  45†, indicating that the effect of modulation is to stabilize the system. However, for small value of the frequency …x < 45†, R2c will be negative, indicating that the effect is destabilizing one. The

Fig. 2. Variation of R2c with x for different values of the viscosity ratio M

557

Fig. 3. Variation of R2c with x for different values of the anisotropy parameter k1

effect of the increasing value of the anisotropy parameter k1 is to inhibit the effect of modulation. The effect of porous parameter k2 on the stability of the system in the presence modulation is shown in Fig. 4, in respect of the symmetric modulation of the wall temperature. We observe from this ®gure that an increase in the value of k2 decreases the value of R2c . Thus in case of symmetric modulation, the effect of large value of the porous parameter k2 is to advance the onset of convection. However for small values of k2 , R2c will become positive over a moderate value of x …x > 45†, which implies that the effect is to stabilize the system. Physically k2 small means the medium is densely packed and naturally inhibit convection. The effect of the viscosity ratio on the stability of the system for the case of asymmetric modulation is shown in Fig. 6. The effect of increasing the viscosity ratio is to make the system more stable. This is in contrast to its effect in the case of symmetric modulation.

The effect of anisotropic parameter k1 on R2c for the case of asymmetric modulation is shown in Fig. 7. We observe that the effect is stabilizing over the whole range of the frequency. We can also observe that an increase in the value of k1 reduces the effect of modulation. The effect of porous parameter k2 on R2c for the case of asymmetric modulation is shown in Fig. 8. We ®nd from this ®gure that, the effect is stabilizing one. An increase in the value of k2 increases R2c , hence making the system more stable. In this case, the effect of modulation is to delay the onset of convection compared to the plane problem for the whole range of the porous parameter k2 . The results of only lower wall temperature modulation are similar to the asymmetric modulations (Figs. 10±12). As R2c is found to be always positive for the out-of-phase modulation and also for lower wall temperature modulation, the subcritical motions are ruled out in these two cases. This is due to the fact that in these two cases temperature ®eld has essentially a linear gradient varying in

Fig. 4. Variation of R2c with x for different values of the porous parameter k2

558

Fig. 5. Variation of R2c with x for different values of the Prandtl number Pr

Fig. 6. Variation of R2c with x for different values of the viscosity ratio M

Fig. 7. Variation of R2c with x for different values of the anisotropy parameter k1

559

Fig. 8. Variation of R2c with x for different values of the porous parameter k2

Fig. 9. Variation of R2c with x for different values of the Prandtl number Pr

Fig. 10. Variation of R2c with x for different values of the viscosity ratio M

560

Fig. 11. Variation of R2c with x for different values of the anisotropy parameter k1

Fig. 12. Variation of R2c with x for different values of the porous parameter k2

Fig. 13. Variation of R2c with x for different values of the Prandtl number Pr

time so that the instantaneous Rayleigh number is supercritical over half a cycle and subcritical during the other half cycle with respect to the critical Rayleigh number of unmodulated system. The effect of Prandtl number on the onset of convection in the presence of modulation for symmetric, asymmetric and only lower wall temperature modulation cases are shown in Figs. 5, 9, and 13 respectively. We observe that the effect of increasing Prandtl number is to minimize the effect of modulation and also the effect of modulation disappears for large values of Prandtl number. In all the three cases, the effect of modulation disappears for large frequencies. We now discuss the results of gravity modulation. The effect of viscosity ratio M on the stability of the system in the presence of gravity modulation is shown in Fig. 14. We observe that an increase in the value of M increases the value of R2c in the negative direction, indicating that the effect is destabilizing one.

Figure 15 shows the variation of R2c with frequency for different values of anisotropy parameter k1 . It is found that the effect of increasing k1 is to reduce the effect of gravity modulation. Figure 16 depicts the variation of R2c with x, for different values of porous parameter k2 . We ®nd that an increase in the value of k2 , increases the value of R2c in the negative direction, indicating that the effect of large k2 advances the convection. The effect of the Prandtl number on the onset of convection in the presence of gravity modulation is shown in Fig. 17. We ®nd that the effect of increasing Pr is to decrease the absolute value of R2c in the presence of gravity modulation, indicating that the larger Prandtl number reduces the effect of modulation. The results of this study are expected to be useful in controlling convection by thermal modulation, in an anisotropic porous medium. The gravity modulation leads to advancement of convection. It is observed that for large frequencies, the effect of temperature and gravity

Fig. 14. Variation of R2c with x for different values of the viscosity ratio M

Fig. 15. Variation of R2c with x for different values of the anisotropy parameter k1

561

562

Fig. 16. Variation of R2c with x for different values of the porous parameter k2

Fig. 17. Variation of R2c with x for different values of the Prandtl number Pr

modulation disappears. The low frequency temperature modulation in the case of symmetric modulation makes the system unstable and in case of asymmetric and when only lower temperature modulation it makes the system more stable. Thus, the low-frequency thermal/gravity modulation has a signi®cant effect on the stability of the system.

6. Georgiadis JG; Catton I (1986) Prandtl number effect on Benard convection in a porous media. ASME J Heat Transfer 108: 284±290 7. Kladias N; Prasad V (1987) Numerical study for inertia and viscous diffusion effects on Benard convection in a porous media. Proc 5th Int Conf Num Meth Thermal Problems 1: 797±810 8. Kladias N; Prasad V (1988) Natural convection in a horizontal porous layer: effects of Darcy and Prandtl References numbers. Proc National Heat Transfer Conf, ASME HTD 96: 1. Horton CW; Rogers FT (1945) Convective currents in a 593±604 porous medium. J Appl Phys 16: 367±370 9. Kladias N; Prasad V (1990) Flow transitions in a buoyancy2. Lapwood ER (1948) Convection of a ¯uid in a porous induced non-Darcy convection in a porous medium heated medium. Proc Camb Phil Soc 44: 508±521 from below. ASME J Heat Transfer 112: 675±684 3. Palm E; Weber JE; Kvernvold O (1972) On steady convection 10. Prasad V; Lauriat G; Kladias N (1988) Re-examination of in a porous medium. J Fluid Mech 64: 153±161 Darcy-Brinkman solutions for free convection in porous 4. Straus JM (1974) Large amplitude convection in porous media. Proc National Heat Transfer Conf, ASME HDT 96: media. J Fluid Mech 64: 51±63 569±579 5. Rudraiah N; Veerappa B; Balachandra Rao S (1980) Effects of non-uniform thermal gradient and adiabatic boundaries on 11. Castinel G; Combarnous M (1974) Critere d'apparition de la convection naturelle dans une couche poreuse anisotrope convection in a porous medium. ASME J Heat Transfer 102: horizontale. C R Acad Sci B 278: 701±704 254±260

12. Epherre JF (1975) Critere d'apparition de la convection naturelle dans une couche poreuse anisotrope. Rev Gen Thermique 168: 949±950 13. Mckibbin R (1986) Thermal convection in a porous layer: effects of anisotropy and surface boundary conditions. Transport Por Media 1: 271±292 14. Nielsen J; Storesletten L (1990) An analytical study on natural convection in isotropic and anisotropic porous channels. Trans ASME J Heat Transfer 112: 396±401 15. Tyvand PA; Storesletten L (1991) Onset of convection in an anisotropic porous medium with oblique principal axes. J Fluid Mech 226: 371±382 16. Mamou M; Mahidjiba A; Vasseur P; Robillard L (1998) Onset of convection in an anisotropic porous medium heated from below by a constant heat ¯ux. Int Comm Heat Mass Transfer 25(6): 799±808 17. Davis SH (1976) The stability of time periodic ¯ows. Ann Rev Fluid Mech 8: 57±74 18. Venezian G (1969) Effect of modulation on the onset of thermal convection. J Fluid Mech 35: 243±254 19. Rosenblat S; Herbert DM (1970) Low frequency modulation of thermal instability. J Fluid Mech 43: 385±389 20. Rosenblat S; Tanaka GA (1971) Modulation of thermal convection instability. Phys Fluids 14: 1319±1322 21. Roppo MH; Davis SH; Rosenblat S (1984) Benard convection with time-periodic heating. Phys Fluids 27: 796±803 22. Caltagirone JP (1976) Stabilite d'une couche poreuse horizontale soumise a des conditions aux limite periodiques. Int J Heat Mass Transfer 19: 815±820

23. Rudraiah N; Malashetty MS (1990) Effect of modulation on the onset of convection in a sparsely packed porous layer. ASME J Heat Transfer 122: 685±689 24. Malashetty MS; Wadi VS (1999) Rayleigh±Benard convection subjected to time dependent wall temperature in a ¯uid saturated porous layer. Fluid Dynamics Research 24: 293±308 25. Gresho P; Sani RL (1970) The effect of gravity modulation on the stability of a heated ¯uid layer. J Fluid Mech 40: 783±806 26. Clever R; Schubert G; Busse F (1993) Two dimensional oscillatory convection in a gravitationally modulated ¯uid layer. J Fluid Mech 253: 663±680 27. Farooq A; Homsy GM (1996) Linear and non-linear dynamics of a differentially heated slot-under gravity modulation. J Fluid Mech 313: 1±38 28. Malashetty MS; Padmavathi V (1997) Effect of gravity modulation on the onset of convection in a porous layer. J Porous Media 1(3): 219±226 29. Malashetty MS; Padmavathi V (1997) Effect of gravity modulation on the onset of convection in a ¯uid and porous layer. Int J Eng Sci 35: 829±839 30. Rudraiah N (1984) Non-linear convection in a porous medium with convective acceleration and viscous force. Arabian J Sci Eng 9(2): 153±167 31. Malkus WVR; Veronis G (1958) Finite amplitude cellular convection. J Fluid Mech 4(3): 225±232 32. Rudraiah N; Siddheshwar PG (1998) A weak non-linear stability analysis of double diffusive convection with cross diffusion in a ¯uid saturated porous medium. Heat and Mass Transfer 33: 287±293

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