The Onset of Double Diffusive Convection in a Viscoelastic Fluid-Saturated Porous Layer with Non-Equilibrium Model

The onset of double diffusive convection in a viscoelastic fluid-saturated porous layer is studied when the fluid and solid phase are not in local thermal equilibrium. The modified Darcy model is used for the momentum equation and a two-field model is used for energy equation each representing the fluid and solid phases separately. The effect of thermal non-equilibrium on the onset of double diffusive convection is discussed. The critical Rayleigh number and the corresponding wave number for the exchange of stability and over-stability are obtained, and the onset criterion for stationary and oscillatory convection is derived analytically and discussed numerically.


Introduction
The problem of double diffusive convection in porous media has attracted considerable interest during the past few decades because of its wide range of applications, including the disposal of the waste material, high quality crystal production, liquid gas storage and others.
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. The double-diffusive convection instabilities in a horizontal porous layer was studied primarily by Nield [1,2] on the basis of linear stability theory for various thermal and solutal boundary conditions. Then the analysis is extended by Taunton [3] et al., Turner [4][5][6], Huppert and Turner [7]. Platten and Legros [8] reported excellent reviews about these studies, using subject of extensive theoretical and experimental investigations. Recently, Pritchard and Richardson [9] discussed how the dissolution or precipitation of the solute effect the onset of convection.
On the other hand, viscoelastic fluid flow in porous media is of interest for many engineering fields. Unfortunately, the convective instability problem for a binary viscoelastic fluid in the porous media has not been given much attention. Wang and Tan [10,11] performed the stability analysis of double diffusive convection of Maxwell fluid in a porous medium, and they pointed out that the relaxation time of Maxwell fluid enhances the instability of the system. Double-diffusive convection of Oldroyd-B fluid in the porous media is studied by Malashetty and co-workers [12][13][14].
In present research, we perform the linear stability of double diffusive convection in a viscoelastic fluid-saturated porous layer, with the assumption that the fluid and solid phases are not in local thermal equilibrium (LTE). The effects of parameters of the system on the onset of convection are discussed analytically and numerically. The critical Rayleigh number, wave number and frequency for exchange of stability are determined.

Basic Equations
We consider an infinite horizontal porous layer of depth d, saturated with a Maxwell fluid mixture heated and salted from below, with the vertically downward gravity force g acting on it. The lower surface is held at a temperature T 1 and concentration S 1 , the upper one is kept at a lower temperature T 2 and concentration S 2 . Moreover, T 1 wT 2 ,S 1 wS 2 : Assuming slow flows in porous media, the momentum balance equation can be linearized as where r is the density, q~(u,w) is the volume average velocity obtained by using a volume averaging technique and g is the acceleration due to gravity, p is the pressure. For general viscoelastic fluids, the constitutive relations between stress tensort t and strain tensorD D is given by Delenda et al [15] 1zl 1 L Lt where m is the viscosity, l 1 and l 2 are relaxation time and retardation time, respectively. When the viscoelastic fluid is Maxwell model, l 2~0 . Substituting Eq.(2) into (1), then we get the modified Darcy-Maxwell model to describe the flow in the porous media, neglecting the Soret and Dufour effects between temperature T and concentration S [11,16] + : q~0 ð3Þ where K and e are the permeability and porosity of the medium while k is the effective solutal diffusivity of the medium. We assume that the diffusion of temperature obeys the following equations, which is a non-equilibrium model between the solid and fluid phases, suggested by [2,14,17] where c is the specific heat, k is the thermal conductivity with the subscripts f and s denoting fluid and solid phase respectively, h is the inter-phase heat transfer coefficient. The inter-phase heat transfer coefficient h depends on the nature of the porous matrix and the saturating fluid, and the small values of h gives rise the relatively strong thermal non-equilibrium effects. In Eqs.(6)-(7), T f and T s are intrinsic average of the temperature fields and this allows one to set T f~Ts~Tb , whenever the boundary of the porous medium is maintained at the temperature T b . The onset of double diffusive convection can be studied under the Boussinesq approximation and an assumption that the fluid r depends linearly on the temperature T and solute concentration S where r f and r o are the densities at the current and reference state, respectively. The quantities b T and b S are the coefficients for thermal and solute expansion, respectively. Because of the Boussinesq approximation, which states that the effect of compressibility is negligible everywhere in the conservations except in the buoyancy term, is assumed to hold.

Basic State
The basic state is assumed to be quiescent and we superimpose a small perturbation on it. We eliminate the pressure from the momentum transport equation (4) and define stream function y by  Then the following dimensionless variables are defined as Here the symbol ''Ã'' means dimensionless, and h, w are nondimensional temperatures of fluid and solid phase, respectively. Q is non-dimensional concentration of solute in porous medium. Substituting the above dimensionless variables in the system yields the following non-dimensional governing equations (for simplicity, the dimensionless mark ''*'' will be neglected hereinafter) where Lz 2 is the two-dimensional Laplacian operator, and the non-dimensional variables that appear in the above equations are defined as where the Ra is the thermal Rayleigh number, Rs is the solute Rayleigh number, l is the relaxation parameter, Pr is the Prandtl number, Da is the Darcy number, Va is the Vadasz number, g is the normalized porosity, n is the kinematic viscosity, Le is the Lewis number, a is the diffusive ratio, l is the porosity modified conductivity ratio, H is the non-dimensional interphase heat transfer coefficient. When H??, the solid and fluid phase have h~Q~w~0, on z~0 and 1:

Linear Stability Theory
In this section, we discuss the linear stability of the system. According to the normal mode analysis, the Eqs.(10)-(13) is solved using the time dependent periodic disturbances in a horizontal plane. We assume that the amplitudes are small enough, so the perturbed quantities can be expressed as follows Where a is the horizontal wavenumber, and s is the growth rate. Substitution of Eq.(15) into the linearized version of Eqs.(10)- (13), yields the following equation: The growth rate s is in general a complex quantity such that s~v r ziv i . The system with v r v0 is always stable, while for v r w0, it will unstable. For the neutral stability state v r~0 , we set where  Since Ra is a physical quantity, it must be real. Hence, from Eq.(18) it follows that either v i~0 (steady onset) or D 2~0 (v i =0, oscillatory onset).

Stationary Convection
The steady onset corresponds to v i~0 and reduces the Eq. (18) to This result is obtained by Banu and Rees [18] in the case of a Darcy porous medium with thermal non-equilibrium model. When H??, in the case of local thermal equilibrium Eq.(17) takes the form Further Eq. (20) can be written as In the absence of the solute effect, Eq.(21) reduces to which is the classical result, obtained by Horton and Rogers [19]. The value of Rayleigh number Ra given by Eq.(17) can be minimized with respect to the wavenumber a by setting LRa La 2 and solve the equation    where a 0 is the critical wavenumber for the LTE case,we obtain a 0~p from the Eq.(21). Substituting Eq.(26) into the Eq.(25), and rearranging the terms and then equating the coefficients of same powers of H will allow us to obtain the a 1 and a 2 , we get Substituting these values of a 0 , a 1 and a 2 into the Eq.(25), we can obtain the critical Rayleigh number for small H.
Letting LRa=La 2~0 , we obtain the following expression Similarly, we expand a in power series of H as Then, substituting these values of a 0 , a 1 and a 2 into the Eq.(28), we can obtain the critical Rayleigh number for large H.

Oscillatory Convection
For oscillatory onset v i is non-zero, which requires D 2~0 in (18), giving    Fig. 3 and 4. Moreover, the larger the heat transfer coefficient H is, the faster the heat transfer enabling the viscoelastic fluid to attain greater percolation velocity. Therefore large heat transfer coefficient favors onset of convection. From Figs. 4, we observe that the effect of increasing c decreases the minimum of the Rayleigh number for stationary mode, indicating that the effect of the porosity modified conductivity ratio is to advance the onset of convection.

Numerical Results and Discussion
The variation of conductivity ratio on the critical Rayleigh number for stationary mode with the heat transfer coefficient for different values of conductivity ratio is shown in Fig. 5. We find that the critical Rayleigh number is independent of c for small values of H, but for large H, the critical Rayleigh number decreases with increasing c. Moreover, for very large c( §10), the critical Rayleigh number is independent of H. Thus, we can draw the conclusion that the presence of non-equilibrium of heat transfer between the viscoelastic fluid and solid make the system instable. Fig. 6-13 present the neutral curves for different values of the relaxation parameter c, Vadasz number, heat transfer coefficient H, normalized porosity parameter g, solute Rayleigh number Rs, porosity modified conductivity ratio c, Lewis number Le and diffusivity ratio a, respectively. As can be seen from the figures, these parameters has significant effects upon the neutral curves.
The effect of relaxation time on the neutral curves is shown in Fig. 6. It is shown in Fig. 6a, i.e., for local thermal non-equilibrium case, the minimum of the Rayleigh number is smaller when c is larger, which makes the onset of convection easier. Based on the theory of Maxwell fluid model, a fluid relaxation or characteristic time, c, is defined to quantify the viscoelastic behavior [20]. So we draw a conclusion that the physical mechanism is the increasing relaxation time increases the elasticity of a viscoelastic fluid thus causing instability. As a result, the elasticity of the Maxwell fluid has a destabilizing effect on the fluid layer in the porous media, and the oscillatory convection is easy to occur for viscoelastic fluid. And this result agrees with the result given by Wang and Tan [11],  where they studied the double diffusive convection problem with thermal equilibrium, as shown in Fig. 6b.
From Fig. 7, We find that an increase in the value of the Vadasz number decreases the oscillatory Rayleigh number, indicating that the Vadasz number advances the onset of double diffusive convection, which is in agreement with the literature by Malashetty and Biradar [16].
The stationary Rayleigh number increases with an increase in the value of heat transfer coefficient H, as shown in Fig. 8, indicating that the effect of heat transfer coefficient is to enhance the stability of the system. At the same time, the same effect of H upon the oscillatory Rayleigh number can be observed in this figure. Comparing with the curve for local thermal equilibrium model, it can be seen that the the oscillatory convection is easy to occur for thermal non-equilibrium case.
In Fig. 9, we note that the effect of normalized porosity parameter is to advance the onset of oscillatory convection. From Fig. 10, we find that the increasing Rs has a stabilizing effect on the onset of double diffusive convection. The neutral stability curves for stationary and oscillatory modes for different values of porosity modified conductivity ratio is shown in Fig. 11, which leads us to the conclusion that the increasing porosity modified conductivity ratio has a destabilizing effect for the system.
The effect of Lewis number Le on the critical oscillatory Rayleigh number is shown in Fig. 12. From the figure, it can be found that increasing of Lewis number decreases the critical oscillatory Rayleigh number indicating that the Lewis number destabilizes the system in oscillatory mode. The physical interpre-tation has been given by Malashetty and Biradar [16], when Lew1, the diffusivity of heat is more than that of solute, and therefore, destabilizing solute gradient augments the onset of oscillatory convection. From Fig. 13, we observe that the diffusivity ratio a has little effect on the onset of double diffusive convection.

Conclusion
The onset of double diffusive convection in a binary Maxwell fluid, which is heated and salted from below, is studied analytically using using a thermal non-equilibrium model. Based on the normal mode technique, the linear stability has been studied analytically. The effects of relaxation time, heat transfer coefficient, normalized porosity parameter and other parameters on the stationary and oscillatory convection are discussed and shown graphically. It is found that the increasing relaxation time increases the elasticity of a viscoelastic fluid thus causing instability. The asymptotic solutions for both small and large values of H were obtained. In general, this work showed how the relaxation time and non-equilibrium model affects the double-diffusive convection in porous media, and it may be useful in some applications which contains heat and mass transfer.