Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model

International Journal of Non-Linear Mechanics 43 (2008) 600 – 621 www.elsevier.com/locate/nlm

Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model M.S. Malashetty ∗ , Rajashekhar Heera Department of Mathematics, Gulbarga University, Jnana Ganga Campus, Gulbarga 585 106, India Received 31 October 2007; received in revised form 17 February 2008; accepted 17 February 2008

Abstract Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out. 䉷 2008 Elsevier Ltd. All rights reserved. Keywords: Double diffusive convection; Local thermal non-equilibrium; Porous layer; Rotation

1. Introduction The problem of double diffusive convection in porous media has attracted considerable interest during the last few decades because of its wide range of applications, from the solidification of binary mixtures to the migration of solutes in water-saturated soils. The other examples include geophysical systems, electrochemistry and the migration of moisture through air contained in fibrous insulation. Early studies on the phenomena of double diffusive convection in porous media are mainly concerned with problem of convective instability in a horizontal layer heated and salted from below. A comprehensive review of the literature concerning double diffusive convection in a fluid-saturated porous medium may be found in the book by Nield and Bejan [1]. The study of double diffusive convection in porous medium is first under taken by Nield [2] on the basis of linear stability theory for various thermal and solutal boundary conditions. Rudraiah et al. [3] have used non-linear perturbation theory to study the onset of double diffusive convection in a horizontal porous layer. The linear stability analysis of the thermosolutal convection is carried out by Poulikakos [4] using the Darcy–Brinkman model. The double diffusive convection in porous media in the presence of cross-diffusion effects is analyzed by Rudraiah and Malashetty [5]. The problem of double diffusive convection in a fluid-saturated porous layer was later on investigated by many authors [6–14]. The study of double diffusive convection in a rotating porous media is motivated both theoretically and by its practical applications in engineering. Some of the important areas of applications in engineering include the food and chemical process, solidification and centrifugal casting of metals, rotating machinery, petroleum industry, biomechanics and geophysical problems. There are only few

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E-mail address: [email protected] (M.S. Malashetty). 0020-7462/$ - see front matter 䉷 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2008.02.006

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

Nomenclature a c d D Da g h H k k K l, m Le Nu Pr p q RaT RaS S Sh T Ta t Va x, y, z

horizontal wavenumber specific heat depth of the porous layer solute diffusivity Darcy number, dK2 gravitational acceleration inter-phase heat transfer coefficient non-dimensional inter-phase heat transfer coefficient, hd 2 /kf unit vector in the vertical direction thermal conductivity permeability of the porous layer wavenumbers in x-, and y-directions Lewis number, f /D Nusselt number, Prandtl number, /f pressure velocity vector, (u, v, w) thermal Rayleigh number, T gT dK/f solute Rayleigh number, S gS dK/f solute concentration Sherwood number temperature Taylor number (2K/)2 time Pr Vadasz number,  Da space coordinates

Greek symbols  S T

2    



the ratio of diffusivities, (0 c)s kf /(0 c)f ks = f /s solute expansion coefficient thermal expansion coefficient the porosity modified conductivity ratio, kf /(1 − )ks 2 + a 2 porosity thermal diffusivity, k/(0 c) dynamic viscosity kinematic viscosity, /0 fluid density frequency stream function

Subscripts/superscripts b c f F l Osc s St u 0 *



basic state critical fluid phase finite amplitude lower wall oscillatory solid phase stationary upper wall reference non-dimensional perturbed quantity

601

602

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studies available on double diffusive convection in the presence of rotation. Chakrabarti and Gupta [15] have analyzed the non-linear thermohaline convection in a rotating porous medium. The effect of rotation on linear and non-linear double diffusive convection in a sparsely packed porous medium was studied by Rudraiah et al. [16]. The effect of anisotropy of the porous medium on the onset of double diffusive convection in a rotating porous medium has been studied by Patil et al. [17]. The Lyapunov direct method is applied to study the non-linear conditional stability problem of a rotating doubly diffusive convection in a sparsely packed porous layer by Guo and Kaloni [18]. The non-linear stability of the conduction–diffusion solution of a fluid mixture heated and salted from below and saturating a porous medium in the presence of rotation is studied by Lombordo and Mulone [19] using Lyapunov direct method. In modeling a fluid-saturated porous medium all the above investigations on double diffusive convection have assumed a state of local thermal equilibrium (LTE) between the fluid and the solid phases at any point in the medium. This is a common practice for most of the studies where the temperature gradient at any location between the two phases is assumed to be negligible. For many practical applications, involving high-speed flows or large temperature differences between the fluid and solid phases, the assumption of LTE is inadequate and it is important to take account of the thermal non-equilibrium effects. Due to applications of porous media theory in drying, freezing of foods and other mundane materials and applications in everyday technology such as microwave heating, rapid heat transfer from computer chips via use of porous metal foams and their use in heat pipes, it is believed that local thermal non-equilibrium (LTNE) theory will play a major role in future developments. Recently, attention has been given to the LTNE model in the study of convection heat transfer in porous media. Much of this work has been reviewed in recent books by Ingham and Pop [20,21]. Kuznetsov [22] studied a perturbation solution for a thermal non-equilibrium fluid flow through a three-dimensional sensible storage packed bed. The review of Kuznetsov [23] gives detailed information about the most but very latest works on thermal non-equilibrium effects on internal forced convection flows. An excellent review of research on LTNE phenomena in porous medium convection, primarily free and forced convection boundary layers and free convection within cavities, is given by Rees and Pop [24]. Free convection in a square porous cavity using a thermal non-equilibrium model is studied by Baytas and Pop [25] while Baytas [26] investigated the thermal non-equilibrium natural convection in a square enclosure filled with a heat generating solid phase and with the Brinkman–Forchheimer extended Darcy law for the momentum equation. A review of thermal non-equilibrium free convection in a cavity filled with non-Darcy porous medium is also given by Baytas [27]. The problem of two-dimensional steady mixed convection in a vertical porous layer using thermal non-equilibrium model is investigated numerically by Saeid [28]. The effect of thermal non-equilibrium on the onset of convection in a porous layer using the Lapwood–Brinkman model and also including anisotropy in permeability and thermal diffusivity in a densely packed porous layer have been investigated by Malashetty et al. [29,30]. Straughan [31] has considered a problem of thermal convection in a fluid-saturated porous layer using a global non-linear stability analysis with a thermal non-equilibrium model. He has established the equivalence of the linear instability and non-linear stability boundaries for the thermal convection in a rotating porous layer with the Darcy law using the non-equilibrium model. More recently, Malashetty et al. [32] have studied the effect of rotation and LTNE on thermal convection in a porous layer. While the rotation effect on double diffusive convection in porous media is studied for the case of LTE model the same is not studied for the case of LTNE model. In this paper, we perform the linear and non-linear stability analysis of the double diffusive convection in a fluid-saturated porous layer subjected to rotation with the assumption that the fluid and solid phases are not in LTE. Our objective is to study how the onset criterion is affected by the combined effect of rotation and LTNE in steady, oscillatory and finite amplitude modes and also to measure the heat and mass transport. A general theory has been developed over five decades that extends the seminal contributions of Fick and Darcy, the theory of mixtures, to systematically study the interactions between continua. Governing equations are written for each component and the interactions between the components are taken into account. Darcy’s equation and Fick’s equations are shown to follow as approximations of these general equations. The readers may refer the excellent book on the theory of mixture by Rajagopal and Tao [33]. 2. Mathematical formulation We consider an infinite horizontal fluid-saturated porous layer confined between the planes z = 0 and d, with the vertically downward gravity force g acting on it. A uniform adverse temperature gradient T = (Tl − Tu ) and a stabilizing concentration gradient S = (Sl − Su ) where Tl > Tu and Sl > Su are maintained between the lower and upper surfaces. A Cartesian frame of reference is chosen with the origin in the lower boundary and the z-axis vertically upwards. The porous layer is subjected to the rotation with an angular velocity  = (0, 0, ) about the z-axis. The Darcy model with time derivative and Coriolis terms is used to model the momentum equation 1 jq  2 1  + q +  × q = − ∇p + f g,  jt K  0 0

(2.1)

where  and K denote porosity and permeability, respectively,  and  are density and kinematic viscosity, respectively. In modeling energy equation for a fluid-saturated porous system, two kinds of theoretical approaches have been used. In the first model, the fluid and solid structures are assumed to be in LTE. This assumption is satisfactory for small-pore media such

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as geothermal reservoirs and fibrous insulations and small temperature differences between the phases. In the second kind of approach, the fluid and solid structures are assumed to be in thermal non-equilibrium. For many applications involving highspeed flows or large temperature difference between the fluid and solid phases, it is important to take account of the thermal non-equilibrium effects. If the temperatures difference between phases is a very important safety parameter (e.g., fixed bed nuclear propulsion systems and nuclear reactor modeling) the thermal non-equilibrium model in the porous media is an indispensable model. The LTNE, which account for the transfer of heat between the fluid and solid phases is considered. A two-field model that represents the fluid and solid phase temperature fields separately, is employed for the energy equation [1] (0 c)f

jTf + (0 c)f (q · ∇)Tf = kf ∇ 2 Tf + h(Ts − Tf ), jt

(2.2)

jTs = (1 − )ks ∇ 2 Ts − h(Ts − Tf ), jt

(2.3)

(1 − )(0 c)s

where c is the specific heat, k, the thermal conductivity and h being the inter-phase heat transfer coefficient. In two-field model the energy equations are coupled by means of the terms, which accounts for the heat lost to or gained from the other phase. The inter-phase heat transfer coefficient h depends on the nature of the porous matrix and the saturating fluid and the value of this coefficient has been the subject of intense experimental interest. Large values of h correspond to a rapid transfer of heat between the phases (LTE) and small values of h gives rise to relatively strong thermal non-equilibrium effects. In Eqs. (2.2)–(2.3) Tf and Ts are intrinsic average of the temperature fields and this allows one to set Tf = Ts = Tw whenever the boundary of the porous medium is maintained at the temperature Tw . The equation of continuity, solute concentration and state are ∇ · q = 0,

(2.4)

jS 1 + (q · ∇)S = D∇ 2 S, jt 

(2.5)

 = 0 [1 − T (Tf − Tl ) + S (S − Sl )],

(2.6)

where T , S and D are the thermal and solute expansion coefficients and solute diffusivity, respectively. The basic state is assumed to be quiescent and is given by qb = (0, 0, 0),

Tf = Tfb (z),

Ts = Tsb (z),

S = Sb t (z),

h = 0.

(2.7)

The basic state temperatures and concentration satisfy the equations d2 Tfb = 0, dz2

d2 Tsb = 0, dz2

d 2 Sb = 0, dz2

(2.8)

with boundary conditions Tfb = Tsb = Tl

and

Sb = Sl at z = 0,

(2.9)

Tfb = Tsb = Tu

and

Sb = Su at z = d,

(2.10)

so that the conduction state solutions are given by Tfb = Tsb = − Sb = −

T z + Tl , d

(2.11)

S z + Sl . d

(2.12)

We now superimpose the infinitesimal perturbations on the basic state and study the stability of the system. Let the basic state be perturbed by an infinitesimal thermal perturbation, so that q = q ,

Tf = Tfb + Tf ,

Ts = Tsb + Ts ,

S = Sb + S  ,

p = pb + p  ,

 = b +   ,

(2.13)

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where the prime indicates that the quantities are infinitesimal perturbations. Substituting Eq. (2.13) into Eqs. (2.1)–(2.6) and using the basic state solutions, we obtain the equations governing the perturbations in the form 1 jq 2 1  + q +  × q = − ∇p  + (S S  − T Tf )g,  jt K  0   jT  dTfb (0 c)f f + (0 c)f (q · ∇)Tf + (0 c)f w  = kf ∇ 2 Tf + h(Ts − Tf ), jt dz jTs = (1 − )ks ∇ 2 Ts − h(Ts − Tf ), jt   dSb jS  1 1 + (q · ∇)S  + w  = D∇ 2 S  . jt   dz

(1 − )(0 c)s

(2.14)

(2.15) (2.16) (2.17)

We consider only two-dimensional perturbations by ignoring the variations in y-direction. By operating curl twice on Eq. (2.14) we eliminate p from it, and then render the resulting equation and Eqs. (2.15)–(2.17) dimensionless using the following transformations (x, y, z) = (x ∗ , y ∗ , z∗ )d, Tf = (T )Tf∗ ,

t=

Ts = (T )Ts∗ ,

(0 c)f d 2 ∗ t , kf

(u , v  , w ) =

kf (u∗ , v ∗ , w∗ ), (0 c)f d

(2.18)

S  = (S)S ∗ ,

to obtain non-dimensional equations in the form (on dropping the asterisks for simplicity),     2   1 j 1 j 1 j j2 2 + 1 ∇ + T a 2 w = RaT + 1 ∇12 Tf − RaS + 1 ∇12 S, Va jt Va jt Va jt jz  j − ∇ 2 Tf + (q · ∇)Tf = w + H (Ts − Tf ), jt   j 2  − ∇ Ts = H (Tf − Ts ), jt   1 2 j − ∇ S + (q · ∇)S + w = 0, jt Le

(2.19)



(2.20) (2.21) (2.22)

where Va =  Pr /Da, the Vadasz number, Ta = (2K/)2 , the Taylor number, RaT = T gT dK/f , the Darcy–Rayleigh number, RaS = S gS dK/f , the solute Rayleigh number, H = hd 2 /kf , the inter-phase heat transfer coefficient,  = f /s , the diffusivity ratio, =kf /(1−)ks , the porosity modified conductivity ratio, Le=f /D, the Lewis number, with f =kf /(0 c)f being the thermal diffusivity. Vadasz [34] in his work, defined the Nield number as Ni = (1 − )ks /(hd 2 ). It should be noted that the dimensionless group H used in the present paper may be defined as H = 1/Nif where Nif is the fluid related Nield number given by Nif = kf /(hd2 ). It is worth mentioning that the Rayleigh number RaT defined above is based on the properties of the fluid while    g (T )Kd RaTLTE = RaT = f T (2.23) 1+ [kf + (1 − )ks ] is the Rayleigh number based on the mean properties of the porous medium and it is this latter definition, which is used in the thermal equilibrium model. Since the fluid and solid phases are not in thermal equilibrium, the use of appropriate thermal boundary conditions may pose a difficulty. However, the assumption that the solid and fluid phases share the same temperatures at that of the boundary temperatures helps in overcoming this difficulty. Accordingly, Eqs. (2.19)–(2.22) are solved for stress free isothermal isosolutal boundaries. Hence the boundary conditions for the perturbation variables are given by w = Tf = Ts = S = 0

at z = 0, 1.

(2.24)

3. Linear stability analysis In this section we predict the thresholds of both marginal and oscillatory convections using linear theory. The eigenvalue problem defined by Eqs. (2.19)–(2.22) subject to the boundary conditions (2.25) is solved using the time-dependent periodic disturbances

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in a horizontal plane, upon assuming that amplitudes are small enough and can be expressed as ⎛ ⎞ ⎛ ⎞   sin(ax) ⎜T ⎟ ⎜  cos(ax) ⎟ ⎜ f⎟ ⎜ ⎟ ⎜ ⎟ = e t ⎜ ⎟ sin(z), ⎝ Ts ⎠ ⎝  cos(ax) ⎠

605

(3.1)

 cos(ax)

S

where a is a horizontal wavenumbers and is the growth rate. Infinitesimal perturbations of the rest state may either damp or grow depending on the value of the parameter . Substituting Eq. (3.1) into the linearized form Eqs. (2.19)–(2.22) of we obtain a matrix equation ⎛

2

⎞

2 + 1 + 2 Ta −aRaT +1 0 aRaS +1 ⎜ ⎟ ⎛⎞ ⎛0⎞ Va Va Va ⎜ ⎟ 2 ⎜ ⎟ ⎜⎟ ⎜0⎟ 1 −( + + H ) H 0 ⎜ ⎟⎝ ⎠ = ⎝ ⎠, (3.2) ⎜ ⎟  0 ⎜ ⎟ 0 H −( + 2 + H ) 0 ⎝ 0

⎠ 

2 1 0 0 − + Le where 2 = 2 + a 2 is the total wavenumber. For the above matrix Eq. (3.2) to have the non-trivial solution, we require   

2 [ 2 + H (1 + )] + [ + 2 (1 + ) + H ( + )] 2 Ta a 2 RaS 2 RaT = 2 +1 + +

. Va 1 + /Va + 2 /Le a  + 2 + H

(3.3)

The growth rate is in general a complex quantity such that = r + i i . The system with r < 0 is always stable, while for

r > 0 it will become unstable. For neutral stability state r = 0. Therefore, we now set = i i in Eq. (3.3) and clear the complex quantities from the denominator, to obtain RaT = 1 + i i 2 ,

(3.4)

1 = A0 (A1 + A2 + A3 ),

(3.5)

2 = A0 (A4 + A5 + A6 ),

(3.6)

where

with A0 = a −2 [Va( 4 + Le2 2i ){( 2 + H )2 + 2 2i }]−1 , 2

4

A1 = ( + Le

2

2i )[ 2Va( 2

+ H ){

2

(3.7)

+ H ( + 1)} + 2i [−H {−Va2

+ H ( + )}{Va − 2 H } − 2 + 4 ] − 2 4i ], 2

(3.8)

A2 = a 2 LeRaSVa[2 Le 4i + { 4 (2 + Le) + 2 H (2 + 2 Le) + H 2 Le( + )} 2i + 4 ( 2 + H ){ 2 + H ( + 1)}], A3 =

(3.9)

2Va2 Ta [{(Va 2 + 2i )( 4 + Le2 2i )( 4 + 2 2i ) + H 2 ( 4 + Le2 2i ) Va2 + 2i × {Va 2 (1 + ) + ( + ) 2i } + H ( 4 + Le2 2i ){Va 4 (1 + 2 ) + (Va2 + 2 2 ) 2i }],

(3.10)

A4 = 2 ( 4 + Le2 2i )[ H 2 {Va( + ) + (1 + ) 2 } + (Va + 2 )( 4 + 2 2i ) + H {2Va 2 + {1 + 2 } 4 + 2 2i }],

(3.11)

A5 = − a 2 LeRaSVa[2 { 2 (Le − 1) + H Le} 2i + 6 (Le − 1) + 4 H {Le + 2 (Le − 1)} + 2 H 2 { (Le − 1) + Le − }], A6 =

(3.12)

2Va2 Ta [H 2 {Va ( + ) − (1 + ) 2 }( 4 + Le2 2i ) + H ( 4 + Le2 2i ) Va2 + 2i × {2Va 2 − (1 + 2 ) 4 − 2 2i } + (Va − 2 )( 4 + Le2 2i )( 4 + 2 2i )].

(3.13)

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Since RaT is a physical quantity, it must be real. Hence, from Eq. (3.4) it follows that either i = 0 (steady onset) or 2 = 0 ( i = 0, oscillatory onset). 3.1. Stationary convection The direct bifurcation (steady onset) corresponds to i = 0 and the steady convection occurs at RaSt T =

[ 2 + H ( + 1)]( 4 + a 2 LeRaS + 2 Ta) a 2 ( 2 + H )

.

(3.14)

It is worth mentioning that the stationary Rayleigh number is independent of the diffusivity ratio of the fluid and solid phases and also the Vadasz number. For a single component system, i.e. when RaS = 0, Eq. (3.14) reduces to RaSt T =

[ 2 + H ( + 1)]( 4 + 2 Ta) a 2 ( 2 + H )

.

This expression coincides with the one given by Malashetty et al. [32]. When H → ∞, Eq. (3.14) gives    2 1+ ( + a 2 )2 + 2 Ta St + LeRaS , RaT = a2 Using definition (2.23) the above equation takes the form   (2 + a 2 )2 + 2 Ta St St = + LeRaS , RaTLTE = RaT +1 a2

(3.15)

(3.16)

(3.17)

which is the classical result for the problem of double diffusive convection in a rotating porous layer with LTE model. For RaS = 0 Eq. (3.17) gives 1 {(2 + a 2 )2 + 2 Ta}, a2 which coincides with the result of Vadasz [35]. Further when Ta = 0, Eq. (3.18) gives RaSt TLTE =

(2 + a 2 )2 , a2 the classical result of Horton and Rogers [36] and Lapwood [37] for single component convection in a porous layer. Further if we set RaS = 0 and Ta = 0 in Eq. (3.14) we obtain   (2 + a 2 )2 (2 + a 2 ) + H (1 + ) St . RaT = a2 (2 + a 2 + H ) RaSt TLTE =

(3.18)

(3.19)

(3.20)

This is identical with the result of Banu and Rees [38] for the single component Darcy–Benard convection in a porous layer with the LTNE model. √ The stationary Rayleigh number RaSt T given by Eq. (3.14) attains the critical value for the wavenumber ac = x, which satisfies the equation x 4 + 2x 3 (2 + H ) + x 2 [−4 Ta − H {LeRaS + 2 (1 + Ta − 2 )} + H 2 (1 + )] − 24 x(1 + Ta)(H + 2 + H ) − 4 (1 + Ta)(2 + H )(H + 2 + H ) = 0.

(3.21)

Now we discuss the asymptotic analysis for both small and large values of H. Very small H physically represents that there is almost no transfer of heat between the fluid and solid phases. The solid phase ceases to affect the thermal field of the fluid, which is free to act independently. On the other hand for large H, the solid and the fluid phases have nearly identical temperatures and may be treated as a single phase. The respective mathematical problems are identical except for a rescaling of RaSt T . The expression for the Rayleigh number and the corresponding wavenumber for small as well as large values of the inter-phase heat transfer coefficient H are obtained and the same are discussed below. Case 1: For very small values of H When H is very small the critical value of the Rayleigh number RaSt T is slightly above the critical value for the LTE case. Accordingly we expand RaSt given by Eq. (3.14) in a power series in H as T   H 2 1 H St 2 2 2 2 2 − + ··· . (3.22) RaT = 2 [( + a ){a +  (1 + Ta)} + a LeRaS ] 1 + 2 a ( + a 2 ) (2 + a 2 )2

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607

St 2 To minimize RaSt T up to O(H ) we set jRaT /ja = 0 and obtain an expression of the form

(a 2 + 2 )3 {a 4 − 4 (1 + Ta)} + H [(a 2 + 2 )3 {−2 (1 + Ta)} − a 4 LeRaS (a 2 + 2 )] + H 2 [a 6 + 3a 2 4 (1 + Ta) + 6 (1 + Ta) + a 4 {2LeRaS + 2 (3 + 2Ta)}] = 0.

(3.23)

We also expand a in power series of H as a = a 0 + a1 H + a 2 H 2 + · · · ,

(3.24)

where a0 is critical wavenumber for the LTE case and is given by the expression a0 = (1 + Ta)1/4 .

(3.25)

Substituting Eq. (3.24) into Eq. (3.23), then equating the coefficients of the same powers of H we obtain the values of a1 and a2 , given by a1 =

1 , 

a2 =

2 , 

where 1 = (1 + Ta)(2a02 4 + 6 ) + a04 {LeRaS + 2 (1 + Ta)}, 2 = a1 {−45a08 a1 − 84a06 a1 2 + 15a04 a1 4 (Ta − 2) + 6a0 6 (1 + Ta) + 3a1 8 (1 + Ta) + 6a02 a1 6 (2 + 3Ta) + 6a05 {LeRaS 2 (1 + Ta)} + 4a03 {LeRaS 2 + 34 (1 + Ta)}} − [a06 + 3a02 4 (1 + Ta) + 6 (1 + Ta) + a04 {2LeRaS + 2 (3 + 2Ta)}],  = 2a0 (a02 + 2 )2 {5a04 + 2a02 2 − 34 (1 + Ta)}. With these values of a0 , a1 and a2 Eq. (3.24) gives the critical wavenumber and consequently using this in Eq. (3.22) one can obtain the critical Rayleigh number for small H. Case 2: For very large values of H For large values of H, expression for the stationary Rayleigh number takes the form   1 1+ 2 2 2 2 2 = [( + a ){a +  (1 + Ta)} + a LeRa ] RaSt S T a2   (2 + a 2 ) −1 (2 + a 2 )2 −2 H + ··· . × 1− H + 2 (1 + ) (1 + )

(3.26)

We minimize this with respect to a in a similar way as we did in the small H case and obtain an algebraic equation in a as a 4 2 (1 + ) − 4 2 (1 + Ta)(1 + 2 ) + H −1 [−2a 6 + 6 (1 + Ta) − a 4 {LeRaS 2 (3 + Ta)}] × H −2 [3a 8 − 8 (1 + Ta) + a 4 2 {2LeRaS + 32 (2 + Ta)} + 2a 6 {LeRaS 2 (4 + Ta)}] + · · · = 0.

(3.27)

Similarly we expand a in the form a = a0 +

a1 a2 + 2 + ···, H H

(3.28)

where a0 is given by Eq. (3.25) and a1 , a2 are to be found. Substituting Eq. (3.28) into Eq. (3.27) and equating the coefficients of like powers of H −1 we find a1 and a2 in the form a1 =

1 , 

a2 =

2 , 

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where 1 = 2a06 − 6 (1 + Ta) + a04 {LeRaS + 2 (3 + Ta)}, 2 = 8 (1 + Ta) − 3a08 − a04 {2LeRaS 2 + 34 (2 + Ta)} − 2a06 {LeRaS + 2 (4 + Ta)} × 12a05 a1 + 4a03 a1 {LeRaS + 2 (3 + Ta)} − 6a02 a12 2 (1 + ),  = 4a03 2 (1 + ). Again with these values of a1 and a2 , we compute the critical wavenumber ac using Eq. (3.28) and finally using this value of ac , one can obtain the critical Rayleigh number RaSt T for stationary convection from Eq. (3.26) for large H. 3.2. Oscillatory convection For oscillatory onset 2 = 0 ( i  = 0) and this gives a dispersion relation of the form (on dropping the subscript i) b0 ( 2 )3 + b1 ( 2 )2 + b2 ( 2 ) + b3 = 0,

(3.29)

where b0 = 2 Le2 2 ( 2 + H + Va), b1 = 8 (2 + Le2 ) + 6 {(H + Va)(2 + Le2 ) + 2Le2 H } + 4 Le2 {Va2 2 + 2HVa + H 2 (1 + )} + 2 Le2Va{HVa2 + Va2 (Va − 2 Ta) + H 2 ( + )} + a 2 2 LeR SVa{ 2 − Le(H + 2 )} − Le2 2Va2 Ta2 (H − Va), b2 = 12 + 10 (H + Va + 2 H ) + 8 {Va2 (Le2 + a 2 )H (H + 2Va) + H 2 2 } + 6Va{Va(H + Va − 2 Ta)(Le2 + a 2 ) + H (2Le2Va + H ) + 2 H 2 } + 4Va2 {−2 Ta(H − Va)(Le2 + 2 ) + H Le2 (H + 2Va − 22 Ta) + H 2 Le2 2 } + 2 H Le2 Va2 {22VaTa + H {−2 Ta(1 + ) + Va( + )}} + a 2 LeRaSVa[−H Le × Va2 2 − H 2 2 {Le −  + (Le − 1)} − H 4 {Le + 2 (Le − 1)} − 2 (Le − 1) × (Va2 2 + 4 )] + H 2 Le2 2 TaVa3 ( + ), b3 = Va2 2 [ 10 + 8 (H + Va − 2 Ta + 2H ) + 6 {2 Ta(−H + Va) + H (H + 2Va − 22 Ta) + H 2 2 } − 4 H {−22VaTa + H {2 Ta(1 + ) − Va( + )}} + 2 H 2 2VaTa ( + ) + a 2 Le Pr RaS {H 2 { + − Le(1 + )} − H 2 {Le + 2 (Le − 1)} 2 − 4 (Le − 1)}. Now Eq. (3.4) with 2 = 0 gives RaOsc T = A0 (A1 + A2 + A3 )

(3.30)

with the values of A0 , A1 , A2 , and A3 given by Eqs. (3.7)–(3.10). For the oscillatory convection to occur 2 must be positive. Since Eq. (3.29) is a cubic in 2 , it can give rise to more than one positive root, for fixed values of the governing parameters. This has important implications for the linear stability of double diffusive porous layer. We find the oscillatory neutral solutions from Eq. (3.30). It proceeds as follows: First determine the number of positive solutions of Eq. (3.29). If there are none, then no oscillatory instability is possible. If there are more than one, then the minimum (over a 2 ) of Eq. (3.30) with 2 given by Eq. (3.29) gives the oscillatory neutral Rayleigh number Raosc T,c corresponding to the critical wavenumber ac and the critical frequency of the oscillation

2c . The analytical expression for oscillatory Rayleigh number given by Eq. (3.30) is evaluated at a = ac and 2 = 2c for various values of the physical parameters in order to know their effects on the onset of oscillatory convection. 4. Finite amplitude steady convection In this section we consider the non-linear analysis using a truncated representation of Fourier series considering two terms. Although the linear stability analysis is sufficient for obtaining the stability condition of the motionless solution and the corresponding

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609

eigenfunctions describing qualitatively the convective flow, it cannot provide information about the values of the convection amplitudes, nor regarding the rate of heat transfer. To obtain this additional information, we perform the non-linear analysis, which is useful to understand the physical mechanism with minimum amount of mathematical analysis and is a step forward toward understanding full non-linear problem. A minimal double Fourier series which describes the finite amplitude steady-state convection is given by  = A sin(ax) sin(z),

(4.1)

Tf = B1 cos(ax) sin(z) + B2 sin(2z),

(4.2)

Ts = B3 cos(ax) sin(z) + B4 sin(2z),

(4.3)

S = B5 cos(ax) sin(z) + B6 sin(2z),

(4.4)

where the steady-state amplitudes A, Bi ’s are constants and are to be determined from the dynamics of the system. Substituting Eqs. (4.1)–(4.4) into the steady part of coupled non-linear system of partial differential equations (2.19)–(2.22), and equating the coefficients of like terms we obtain the following non-linear system equations: [(2 + a 2 ) + 2 Ta]A + aRaT B1 − aRaS B5 = 0,

(4.5)

aA + [(2 + a 2 ) + H ]B1 − H B 3 + aAB 2 = 0,

(4.6)

2[42 + H ]B2 − 2H B 4 − aAB 1 = 0,

(4.7)

H B 1 − [(2 + a 2 ) + H ]B3 = 0,

(4.8)

H B 2 − [42 + H ]B4 = 0,

(4.9)

aA +

1 2 ( + a 2 )B5 + aAB 6 = 0, Le

(4.10)

82 B6 − aAB 5 = 0. Le

(4.11)

The steady state solutions are useful because they predict that a finite amplitude solution to the system is possible for subcritical values of the Rayleigh number and that the minimum values of RaT for which a steady solution is possible lies below the critical values for instability to either a marginal state or an overstable infinitesimal perturbation. Elimination of all amplitudes, except A, yields   2 ( + a 2 ){2 + a 2 + H (1 + )} A (2 + a 2 ) + 2 Ta − a 2 RaT (2 + a 2 + H )  −1  2  2 −1  2) a 2 (42 + H ) A2 + a ( A + 2 = 0. (4.12) + a 2 RaS + a 2 Le Le 8 4 + H (1 + ) 8 The solution A = 0 corresponds to pure conduction, which we know to be a possible solution though it is unstable when RaT is sufficiently large. The remaining solutions are given by A2 1 [−x2 + = 8 2x1



x22 − 4x1 x3 ],

(4.13)

where x1 = x2 = x3 =

a 4 Le(42 + H )(2 Ta + 2 ) , 42 + H ( + 1) a 2 2 Le[ 2 + H ( + 1)](2 Ta + 2 )

2 + H

4 (H ( + 1) + 2 )(2 Ta + 2 ) Le( 2 + H )

+

+

a 2 2 (42 + H )(2 Ta + 2 ) a 4 RaS (42 + H ) + − a 4 LeRaT , Le[42 + H ( + 1)] [42 + H ( + 1)]

a 2 2 RaS (H ( + 1) + 2 ) ( 2 + H )

−

a 2 2 RaT . Le

610

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

When we let the radical in the above equation to vanish, we obtain the expression for finite amplitude Rayleigh number RaFT , which characterizes the onset of finite amplitude steady motions. The finite amplitude Rayleigh number can be obtained in the form  1 RaFT = [−y2 + y22 − 4y1 y3 ], (4.14) 2y1 where y1 = a 4 Le2 ( 2 + H )[42 + H ( + 1)], y2 = 2a 2 [−a 2 LeRaS ( 2 + H )(42 + H ) − 2 (2 Ta + 2 ){H 2 [ (Le − 1) + Le] × [Le( + 1) + ] + 42 2 (Le2 − 1) + H {Le2 + (Le2 − 1)}( 2 + 42 )}], y3 =

1 Le2 ( 2

+ H )[42 + H ( + 1)]

[a 2 LeRaS (42 + H )( 2 + H ) − 2 (2 Ta + 2 )

× {H 2 [Le + (Le − 1) ][Le( + 1) + ] + 42 2 (Le2 − 1) + H ( 2 + 42 )[Le2 + (Le2 − 1) ]}]2 . 5. Heat and mass transports Once we know the amplitude we can find the heat and mass transfer. In the study of convection in porous medium, the quantification of heat and mass transports is important. This is because onset of convection, as Rayleigh number is increased, is more readily detected by its effect on the heat transport. If H¯ and J¯ are the rate of heat and mass transports per unit area, respectively, for the fluid phase, then   jTftotal H¯ = −f , (5.1) jz z=0   jStotal ¯ J = −D , (5.2) jz z=0 where the angular bracket corresponds to a horizontal average and z Tftotal = Tl − T + Tf (x, z), d z Stotal = Sl − S + S(x, z). d

(5.3) (5.4)

Substituting Eqs. (4.2) and (4.4) into Eqs. (5.3) and (5.4) and using the resultant equations in Eqs. (5.1) and (5.2), we get f T (1 − 2B2 ), H¯ = d

(5.5)

DS (1 − 2B6 ). J¯ = d

(5.6)

The Nusselt number for the fluid phase and the Sherwood number are defined by Nu =

H¯ = 1 − 2B2 , f T /d

(5.7)

Sh =

J¯ = 1 − 2B6 . DS/d

(5.8)

Writing B2 and B6 in terms of A, using Eqs. (4.5)–(4.11) and substituting into Eqs. (5.7) and (5.8), we obtain   ( 2 + H )(42 + H ) 2 2 Nu = 1 + 2a (A /8) 2 2 ,

[ + H ( + 1)][42 + H (1 + )] + a 2 ( 2 + H )(42 + H )(A2 8)   2 Le . Sh = 1 + 2a 2 (A2 /8) 2

+ a 2 Le2 (A2 /8)

(5.9)

(5.10)

The second term on the right-hand side of Eqs. (5.9) and (5.10) represent the convective contribution to heat and mass transports, respectively. It is obvious that Nu = Sh = 1 for all RaT RaSt T,c , indicating that the convection heat and mass transfers branches off

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611

from the conductive heat transfer line at the critical value of the Rayleigh number. It is important to note that our finite amplitude analysis is valid for thermal Rayleigh number around the convection threshold. Therefore, the Nusselt number and the Sherwood number in the present study are limited by an upper bound value of 3. Better results can only be obtained by including more number of terms in the Fourier series representation, which allows the variation of wave number as the value of thermal Rayleigh number varies. Now we intend to obtain the quantification of heat and mass transports for the limiting cases H → 0 and ∞. When H → 0, Eq. (5.9) yields   1 Nu = 1 + 2a 2 (A2 /8) , (5.11) a 2 + 2 (A2 /8) while Eq. (5.10) remains unchanged, with (A2 /8) given by Eq. (4.13) where x1 = a 4 Le(2 Ta + 2 ), x2 =

a2 2 [ (1 + Le2 )(2 Ta + 2 ) + a 2 Le(RaS − RaT Le)], Le

x3 =

2 4 [ + 2 Ta 2 + a 2 (LeRaS − RaT )]. Le

Further, when H → ∞, Eqs. (5.9) reads   1 2 2 , Nu = 1 + 2a (A /8) a 2 + 2 (1 + 1/ )2 (A2 /8)

(5.12)

and Eq. (5.10) remains invariant, with (A2 /8) given by Eq. (4.13) where   (2 Ta + 2 )a 4 Le, x1 = +1 x2 =

a2 [ 2 (2 Ta + 2 ){ 2 + Le2 ( + 1)2 } − a 2 Le {− RaS + RaT Le( + 1)}], Le ( + 1)

x3 =

2 2 [ ( + 1)(2 Ta + 2 ) + a 2 {LeRaS ( + 1) − RaT }]. Le

6. Results and discussion An analytical study of linear and non-linear double diffusive convection in a horizontal fluid-saturated rotating porous layer is carried out by considering a thermal non-equilibrium model. The onset criterion for both marginal and oscillatory convections is derived using the linear theory. The expression for finite amplitude Rayleigh number is derived analytically using minimal representation of the Fourier series. The effect of rotation, solute diffusion and thermal non-equilibrium on the stability of the system is investigated. It is found that in most of the situations the instability sets in via finite amplitude motions, prior to the marginal or oscillatory convection. However, for large values of solute Rayleigh number and moderate values of Lewis number the finite amplitude motions become weaker and the onset of double diffusive convection ceases to be in the form of finite amplitude convection and the instability sets in via oscillatory mode. Further the measure of heat and mass transports across the layer is determined by evaluating the Nusselt number and the Sherwood number for the case of steady convection. The neutral stability curves in RaT –a plane for various parameter values are shown in Figs. 1–7. From these figures it is clear that the neutral curves are connected in a topological sense for the fixed values of H = 100, = 0.5, Ta = 20, RaS = 100, Le = 2, Va = 15 and  = 0.25. This connectedness allows the linear stability criteria to be expressed in terms of the critical Rayleigh number, below which the system is stable and unstable above. The points where the overstable solutions branch off from the stationary convection can be easily identified from these figures. We also observe that for smaller values of the wavenumber each curve is a margin of the oscillatory instability and at some fixed wavenumber depending on the other parameters the overstability disappears and the curve forms the margin of stationary convection. The effect of inter-phase heat transfer coefficient H on the neutral stability curves is shown in Fig. 1 for the fixed values of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, Vadasz number, Taylor number and diffusivity ratio. It is clear from this figure that for the set of values chosen for the parameters, the onset of convection is through the oscillatory state. Further, we observe that the minimum of Rayleigh number for oscillatory mode increases with H, indicating that the effect of inter-phase heat transfer coefficient is to stabilize the system.

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

1000 Stationary 500 Oscillatory

100

800

50

RaT

10

600

H=1

400 Rs = 100, = 0.5, Le = 2,  = 0.25, Va = 15, Ta = 20

200 0

2

4

6

8

10

12

14

16

18

a Fig. 1. Neutral stability curves for different values of inter-phase heat transfer coefficient H.

1000 H = 100, = 0.5, Le = 2, Va = 0.25,  = 0.25, Ta = 20

150 100

800 50

RaT

20

600

RaS = 10

400 Stationary Oscillatory

200 0

2

4

6

8

10

12

14

16

18

a Fig. 2. Neutral stability curves for different values of solute Rayleigh number Ra s .

1000 50 20 800 10 5 RaT

612

600

Ta-0

400

200 0

2

4

6

8

10

12

14

16

a Fig. 3. Neutral stability curves for different values of Taylor number Ta.

18

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

1000

Stationary Oscillatory

= 0.1 0.5 1

800

RaT

5 10

600

400 H = 100, Ras = 100, Le = 2,  = 0.25, Va = 15, Ta = 20 200

0

2

4

6

8

10

a

12

14

16

18

Fig. 4. Neutral stability curves for different values of porosity modified conductivity ratio .

1000 Stationary Oscillatory

10 5 3

800

2

RaT

Le = 1 600

400 H = 100, RaS = 100, = 0.5,  = 0.25, Va = 15, Ta = 20 200 0

2

4

6

8

10

12

14

16

18

a Fig. 5. Neutral stability curves for different values Lewis number Le.

1000

RaT

800

Va = 20,15,13,10, 5

600

Stationary Oscillatory

400

H = 100, RaS = 100, = 0.5, Le = 2,  = 0.25, Ta = 20 200 0

2

4

6

8

10

12

14

16

a Fig. 6. Neutral stability curves for different values of Vadasz number Va.

18

613

614

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

1000 H = 100, RaS = 100, = 0.5, Le = 2, Va = 15, Ta = 20

RaT

800

600 5 1 0.5 0.3 0.2

400

 = 0.01.0.1

Stationary Oscillatory

200 0

2

4

6

8

10

12

14

16

18

a Fig. 7. Neutral stability curves for different values of ratio of diffusivities .

1500

1200

Stationary Oscillatory Finite amplitude

900

10

RaS = 10, 50, 100

600

RaS = 0.5, Le = 2,  = 0.25, Va = 15, Ta = 15

900

0.5 1

300

RaS = 100, 50, 10

0

0 -2

-1

0

1

2

3

4

-2

5

-1

0

log10H

Le = 5 2.5

3

4

5

RaTc

2.5 Le = 1

300

RaS = 100, = 0.5, Le = 2, Va = 15, Ta = 15

1200

5

600

2

1500

 = 0.25, Va = 15, Ta = 15

900

1

log10H

1500 Ra = 100, = 0.5, S

RaTc

1

= 0.01

600

300

1200

0.5

50

RaTc

1200 RaTc

1500

= 0.5, Le = 2,  = 0.25, Va = 15, Ta = 15 RaS = 100

900

Le = 1 2.5

600

5

300

0

 = 0.1, 0.2, 0.3

0 -2

-1

0

1

2

log10H

3

4

5

-2

-1

0

1

2

3

4

5

log10H

Fig. 8. Variation of critical Rayleigh number Ra T,c with H for different values of (a) Ra S , (b) , (c) Le and (d) .

In Fig. 2 the effect of solute Rayleigh number RaS on the marginal stability curves for the fixed values of other governing parameters is depicted. We observe that the convection sets in first as oscillatory mode even for small values of RaS in the presence of rotation in contrast to the case of non-rotating doubly diffusive system. It is also clear from this figure that the minimum of Rayleigh number for stationary and oscillatory modes increases with the solute Rayleigh number indicating that the solute Rayleigh number delay the onset of convection. In Fig. 3 we show the effect of Taylor number, Ta on the onset of convection when all other parameters are fixed. The onset of instability manifests as oscillatory convection for the values of the parameters chosen for this figure. It is apparent from this figure that the minimum of Rayleigh number for both stationary and oscillatory states increases with the Taylor number, indicating that the effect of rotation is to enhance the stability of the system in both stationary and oscillatory modes.

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

14

14

Le = 2.5, = 0.5, RaS = 100  = 0.25, Va = 15, Ta = 15

12

12

615

= 0.01

RaS = 100, Le = 2.5,

 = 0.25, Va = 15, Ta = 15

0.5

50

8

RaS = 10, 50, 100

6

1

10

10

Stationary Oscillatory Finite amplitude

ac

ac

10

0.5

8

1

= 1, 0.5, 0.01

6

100 50

4

4 -2

16

-1

0

1 2 log10H

3

4

5

-2

14

RaS = 100, = 0.5,

12

2.5

12

2 3 log10H

4

5

6

ac

ac

1

10

1

10

0

RaS = 100, = 0.5, Le = 2.5, Va = 15, Ta = 15

Le = 5

 = 0.25, Va = 15, Ta = 15

14

-1

1

8

8

Le = 2.5,5

6

2.5

4 -2

-1

0

1 2 log10H

 = 0.1, 0.2, 0.3

6

5

4 3

4

5

-2

-1

0

1 2 log10H

3

4

5

Fig. 9. Variation of critical wavenumber ac with H for different values of (a)Ra S , (b) , (c) Le and (d) .

Fig. 4 indicates the effect of porosity modified conductivity ratio on the marginal stability curves. We observe that with the increase in the value of the minimum of oscillatory Rayleigh number decreases. The effect of therefore, is to advance the onset of oscillatory convection. The effect of Lewis number on the neutral curves is unveiled in Fig. 5. It is found that the minimum of oscillatory Rayleigh number decreases with Le, indicating that the effect of Lewis number is to destabilize the system. Further, with the increasing Le, the two branches of the oscillatory neutral curves comes closer indicating that the Le shrinks the region of oscillatory convection. The similar effect is observed with solute Rayleigh number. It is also interesting to note that the effect of Lewis number is insignificant beyond Le = 10. The variation of marginal curves of oscillatory mode for different values of the Vadasz number Va, when all other parameters are kept fixed, is depicted in Fig. 6. It is clear that the critical value of oscillatory Rayleigh number increases with the increase in Vadasz number, indicating that, the Vadasz number delay the onset of oscillatory convection. In Fig. 7 we display the effect of ratio of diffusivities, , on the neutral stability curves. In this case the effect similar to that of Va is observed. That is, the diffusivity ratio makes the system more stable. The behavior of the critical values of Rayleigh number for stationary, oscillatory and finite amplitude convection and also the wavenumber and frequency of the oscillatory mode are shown in Figs. 8–10, as the functions of inter-phase heat transfer coefficient H. In general, it is observed that for very small and large values of H the stability criterion is found to be independent of H . However, the effect of H on the stability of the system is significant only for intermediate values of H . The physical reason for this is that when H → 0 there is almost no transfer of heat between the fluid and solid phases and the properties of solid phase have no significant influence on the onset criterion. When H → ∞ the fluid and solid phases have almost equal temperatures and therefore may be treated as single phase (i.e. LTE model). Between these two extremes H gives rise to a strong non-equilibrium effect. The variation of critical Rayleigh number with inter-phase heat transfer coefficient H for different parameter values is shown in Figs. 8(a)–(d). These figures indicate that the critical Rayleigh number increases from the LTE value when H is small to a LTNE value when H is large. Thus, the inter-phase heat transfer coefficient makes the system more stable for its intermediate values. Fig. 8(a) indicates the effect of RaS on the critical Rayleigh number. For the chosen values of the parameters, the convection sets in first as oscillatory mode even for small values of the solute Rayleigh number. For RaS = 10, 50 the stationary convection occurs prior to the finite amplitudes motions, however, the oscillatory motions occur prior to these two modes. The variation of the critical Rayleigh number with H for different values of porosity modified conductivity ratio is depicted in Fig. 8(b) when all other parameters are fixed. We observe from this figure that for small H, RaT,c is independent of and its value is close to that of the LTE case, since for very small values of H, there is no significant transfer of heat between the phases and the onset criterion is not affected by the properties of the solid phase. On the other hand, for large values of H, though the stability

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

1200

c2

1000

1200

= 0.5, Le = 2,  = 0.25, RaS = 100

 = 0.25, Va = 15, Ta = 15

1000

80 60

800

RaS = 0.5, Le = 2,  = 0.25,

800

50

600

600

400

400

200

200

= 0.01

 = 0.25, Va = 15, Ta = 15

0.1

c2

616

0.5 1

-1

0

1 2 log10H

3

4

-2

5

1400

1200

1200

1100

-1

0

1 2 log10H

RaS = 100, = 0.5, Le = 2.5, Va = 15, Ta = 15

3

4

5

 = 0.1 0.2

1000

1000

Le = 5 4

800

c2

c2

-2

3 2.5

600

RaS = 100, = 0.5,

900

0.3

800

0.4

700

 = 0.25, Va = 15, Ta = 15

400

600 -2

-1

0

1 2 log10H

3

4

5

-2

-1

0

1 2 log10H

3

4

5

Fig. 10. Variation of critical frequency 2c with H for different values of (a)Ra S , (b) , (c) Le and (d) .

criterion is independent of H, the condition for the onset of convection is based on the mean properties of the medium, and therefore the critical Rayleigh number is a function of . This figure also indicates that for moderate and large values of H, critical Rayleigh number for each of stationary, oscillatory and finite amplitude convection decreases with the increasing values of . Therefore, the effect of porosity modified conductivity ratio is to reduce the stabilizing effect of inter-phase heat transfer coefficient. It is important to note that for sufficiently large values of , the critical Rayleigh number becomes independent of H. Fig. 8(c) displays the variation of critical Rayleigh number with H for different values of Lewis number. It is found that for small Le the oscillatory motions are not possible and the finite amplitude convection occurs prior to the stationary convection when H is in the range of very small to moderate values. Further for large values of H, convection sets in first as oscillatory mode. For Le > 5, the finite amplitude motions occur prior to the oscillatory motions when H is small and the oscillatory motions occur prior to the finite amplitude motions when H is large. This figure also reveals that with the increasing values of Le, the critical Rayleigh number for stationary mode increases while that for oscillatory and finite amplitude convections decreases. Therefore, the Lewis number enhances the double diffusive convection in stationary mode while it advances the oscillatory and finite amplitude convection. In Fig. 8(d) the variation of the critical Rayleigh number RaT,c with H for different values of diffusivity ratio  is indicated. We find that the stationary and finite amplitude motions are independent of the diffusivity ratio. This figure also indicates that for moderate and large values of H the oscillatory critical Rayleigh number increases with increasing  while its effect is insignificant for small values of H. As  increases, the contribution of heat conduction from the solid phase becomes negligible, and therefore the critical Rayleigh number for oscillatory mode increases toward a constant value. The diffusivity ratio therefore reinforces the stabilizing effect of inter-phase heat transfer coefficient in case of the overstable mode. The variation of critical wavenumber with inter-phase heat transfer coefficient H is shown in Figs. 9(a)–(d) for different parameter values. We observe from these figures that the critical wavenumber increases with H from LTE value when H is small to its maximum LTNE value and then decreases back to the LTE value with further increase in H. We found that the critical wavenumber approaches to that of LTE case when H → 0 and ∞. This is quite obvious as the corresponding physical problems are equivalent. As H → 0, the solid phase ceases to affect the thermal field of the fluid, which is free to act independently, while as H → ∞ the solid and fluid phases have identical temperatures and may be treated as a single phase. At intermediate values of H we observe that the critical wavenumber for stationary mode attains a maximum value and returns back to the LTE value. The effect of solute Rayleigh number on critical wavenumber is displayed in Fig. 9(a). This figure indicates that the critical wavenumber increases with the increasing RaS in stationary and oscillatory modes, whereas the effect of RaS on critical wavenumber for finite amplitude mode is not significant. Fig. 9(b) indicates the variation of critical wavenumber with H for different values of . As stated earlier for very small and very large values of H the solid phase has little effect the onset criterion, and therefore the critical wave number ac becomes almost independent

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

617

1500 RaS = 100, Le = 2.5, = 0.5,  = 0.25, Ta = 15

1200

900 RaTc

Stationary Oscillatory Finite amplitude

600 Va = 5 10,15

300

0 -2

-1

0

1

2

3

4

5

2

3

4

5

2

3

4

5

log10H 15 RaS = 100, Le = 2.5, = 0.5,  = 0.25, Ta = 15

ac

12

9

6

10,15 Va = 5

3 -2

-1

0

1 log10H

1500 RaS = 100, Le = 2.5, = 0.5,  = 0.25, Ta = 15

1200 900

c2

Va = 20

600

15 10

300

5

0 -2

-1

0

1 log10H

Fig. 11. Variation of critical values of (a) Rayleigh number (b) wavenumber and (c) frequency with H for different values Vadasz number.

of . We also observe from this figure that for intermediate values of H the critical wavenumber for each of the stationary, oscillatory and finite amplitude modes decreases with . Further, it is interesting to note that the maximum value shift toward the smaller values of H as the value of is increased. In Fig. 9(c) we display the effect of Lewis number on the critical wavenumber. This figure indicates that the critical wavenumber for both stationary and oscillatory modes increases with increase in Le while that for the finite amplitude motions decreases with Le. Further the effect of Le on finite amplitude critical wavenumber becomes less significant for large values of Le. The variation of

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1500 15

RaS = 100, Le = 2.5, = 0.5,  = 0.25, Va = 15

10

1200 15

Stationary Oscillatory Finite amplitude

RaTc

900

5

10

600

Ta = 5

300 Ta = 15, 10, 5

0 -2

-1

0

1

2

3

4

5

log10H 14

(b)

RaS = 100, Le = 2.5,

Ta = 15

= 0.5,  = 0.25, Va = 15

12

10 5

ac

10

15 10

8 5

6 4

Ta =15, 10,5

2 -2

-1

0

1

2

3

4

5

log10H 1500 RaS = 100, Le = 2.5, = 0.5,  = 0.25, Va = 15

Ta = 20

1200

c2

15

900 10

600

5

300 -2

-1

0

1

2

3

4

5

log10H Fig. 12. Variation of critical values of (a) Rayleigh number (b) wavenumber and (c) frequency with H for different values Taylor number Ta.

critical wavenumber with H for different values of diffusivity ratio is shown in Fig. 9(d). From this figure it is found that for large values of H the critical wavenumber for oscillatory mode falls from its LTE value with the decreasing values of . However, the critical wavenumber for stationary and finite amplitude motions is independent of diffusivity ratio. The variation of critical frequency 2c for the oscillatory mode with H for different parameter values is shown through Figs. 10(a)–(d). It is clear from these figures that the critical frequency increases from a constant value, when H is very small

M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

3

619

3 1000 100

2.5 RaS = 100, 50, 10, 0

2 1.5

H = 100, = 0.5, Le = 0.8, Ta = 10

Nu/Sh

Nu/Sh

2.5

10 H=1

2 H = 1000, 100, 10, 1 RaS = 100, Le = 0.8, = 0.5, Ta = 10

1.5

Nu Sh

1

Nu Sh

1 1

2

3

4

5

1

2

St RaT RT,c

3

4

5

St RaT RT,c

3

3 = 10

2.5

1

2.5

0.1

Nu/Sh

Nu/Sh

Le = 1, 0.9, 0.8, 0.7

2 RaS = 100, H = 100, = 0.5, Ta = 10

1.5

2

0.001 RaS = 100, H = 100, Le = 0.8, Ta = 10

1.5

Nu Sh

1

= 10, 1, 0.1, 0.001

Nu Sh

1 1

2

3 St Ra T RT,c

4

5

1

2

3

4

5

St Ra T RT,c

Fig. 13. Variation of Nusselt/Sherwood number with Rayleigh number RaT for different values of (a) RaS , (b) H, (c) Le and (d) .

to its maximum value and then with the further increase in H, it decreases back to another constant value or remains constant, when H is large. The effect of RaS on the critical frequency is displayed in Fig. 10(a). It is found that the critical frequency increases with increase in the value of RaS . A similar effect on 2c has been observed in Fig. 10(c) with the Lewis number. In these two cases, the frequency increases from an LTE to an LTNE case with increase in the value of H. In Fig. 10(b) the effect of porosity modified conductivity ratio on the critical frequency is displayed. Since for small H the solid phase ceases to affect the onset criteria we observe from these figures that 2c remains independent of for very small H. Also since in the very large H limit the stability criterion depends on the mean properties of medium, 2c depends on . It is observed from this figure that the critical frequency of oscillations decreases with increase in the value of . The effect of diffusivity ratio  on the critical frequency is revealed in Fig. 10(d). It clear that for small H, the diffusivity ratio has no influence on the frequency while for large H, it has a damping effect. The variation of critical Rayleigh number, wavenumber and frequency of oscillations with H for different values of Vadasz number is unveiled in Figs. 11(a)–(c). We observe from Fig. 11(a) that the oscillatory critical Rayleigh number decreases with increasing Va. The effect of Vadasz number therefore, is to advance the onset of double diffusive convection, in the oscillatory mode. In Fig. 11(b) we display the effect of Vadasz number on the oscillatory critical wavenumber. This figure indicates that the critical wavenumber acOsc for oscillatory mode increases with Va. It is clear from Fig. 11(c) that the critical frequency increases with increasing Va. Fig. 12(a)–(c) displays the effect of the Taylor number on the critical Rayleigh number, wave number and the frequency of the oscillations. Fig. 12(a) indicates the effect of Taylor number on the critical Rayleigh number. It is important to note that the onset of convection is through the oscillatory mode even for small Taylor number. The critical Rayleigh number for stationary, oscillatory and finite amplitude mode is found to increase with the Taylor number. Therefore, the rotation enhances the stability of the system in stationary, oscillatory and finite amplitude modes. Fig. 12(b) show the variation of critical wavenumber with H for different Taylor number. This figure indicates that the critical wavenumber increases with the increasing Ta in stationary, oscillatory and finite amplitude mode. Fig. 12(c) depicts the effect of Taylor number on the critical frequency. It is found that the critical frequency increases with increase in the value of Ta. The quantity of heat and mass transfer across the layer is given by the Nusselt number Nu and Sherwood number Sh, respectively, which represent the ratio of heat or mass transported across the layer to the heat and mass transported by conduction alone. In Figs. 13(a)–(d) and Fig. 14 we exhibit the variation of the Nusselt number of the fluid phase and the Sherwood number with thermal Rayleigh number for different values of RaS , H, Le , and Ta. From each of these figures it is clear that as thermal Rayleigh number increases from one to three times of its critical value, the heat and mass transfer increase sharply and as the thermal Rayleigh number is increased further, they remain almost constant. It is also found that in most of the cases the Sherwood number is above the Nusselt number. We also note that the effect of each of RaS , H, , Le and Ta is to increase the values of Nu and Sh. Therefore, the

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M.S. Malashetty, R. Heera / International Journal of Non-Linear Mechanics 43 (2008) 600 – 621

3

Nu/Sh

2.5 Ta = 20, 15, 10, 5. 2

RaS = 100, H = 100, Le = 0.8, = 0.5

1.5

Nu Sh 1

1

2

3 RaT

4

55

St RaT,c

Fig. 14. Variation of Nusselt/Sherwood number with Rayleigh number RaT for different values of Ta.

effect of each of these parameters is to enhance the heat and mass transport across the layer. Although the presence of a stabilizing gradient of solute will inhibit the onset of convection, due to the strong finite amplitude motions, which exist for large Rayleigh numbers, tend to mix the solute and redistribute it so that the interior layers are more neutrally stratified. As a consequence of that the inhibiting effect of solute gradient is greatly reduced and hence fluid will convect more and more heat and mass when there is an increase in the value RaS . 7. Conclusion The linear and non-linear double diffusive convection in a horizontal fluid-saturated rotating porous layer is investigated analytically when the fluid and solid phases are not in LTE. In case of linear theory the thresholds of both stationary and oscillatory convection are derived as the functions of solute Rayleigh number, inter-phase heat transfer coefficient, Lewis number, porosity modified conductivity ratio, Vadasz number, diffusivity ratio and Taylor number. The non-linear theory predicts the occurrence of finite amplitude motions. We found that there is a competition between the processes of thermal and solute diffusion that causes the convective instability to set in as oscillatory and finite amplitude mode rather than stationary. It is found that for both large and small inter-phase heat transfer coefficient the system behaves like an LTE model while the intermediate values have strong influence on each of stationary, oscillatory and finite amplitude modes. The presence of a stabilizing gradient of solute will inhibit the onset of double diffusive convection. The effect of porosity modified conductivity ratio, Vadasz number is to enhance the instability of system. The Lewis number stabilizes the system toward the stationary mode while destabilizes the oscillatory and finite amplitude modes. The diffusivity ratio strengthens the stabilizing effect of inter-phase heat transfer coefficient. Each of the parameters RaS , H, , Le and Ta increases the values of Nu and Sh. Acknowledgment This work is supported by UGC New Delhi, under the Special Assistance Programme DRS. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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