Linear and Non-Linear Stability Analysis for Thermal Convection in A Bidispersive Porous Medium with Thermal Non-Equilibrium Effects

The linear instability and nonlinear stability analyses are performed for the model of bidispersive local thermal non-equilibrium flow. The effect of local thermal nonequilibrium on the onset of convection in a bidispersive porous medium of Darcy type is investigated. The temperatures in the macropores and micropores are allowed to be different. The effects of various interaction parameters on the stability of the system are discussed. In particular, the effects of the porosity modified conductivity ratio parameters, p  and s  , with the inter-phase momentum transfer parameters 1  and 2  , on the onset of thermal Convection are also considered. Furthermore, the nonlinear stability boundary is found to be below the linear instability threshold. The numerical results are presented for free-free boundary conditions.


Introduction
Thermal convection in a bidispersive porous medium is one of much current interest due to its practical applications in various fields, such as in heat pipes technologies, catalytic chemistry, methane recovery from coal deposits, and thermal insulation see e.g., Szczygieł [1], Lin et al. [2], Shi and Durucan [3], Nield and Bejan [4], Straughan [5], and the references  Other nondimensional variables are required and these are given by 22 (4) 2 , and then analysis the linear system. The linear system arising from equation (4) is where * 2 2 2 2 = / / xy       is the horizontal Laplacian. This is an eigenvalue problem for  to be solved subject to the boundary conditions in equation (5).
To analyze (6) and (5) where 2 2 2 = na   . Then, one may consider the following two cases. (6). The stationary convection boundary is given by

Stationary Convection
With the coefficients  to obtain the lowest instability boundary.

Oscillatory Convection
To study oscillatory convection put = I i  in equation (6) (8) and (9) over 2 a , as will be presented in the numerical results section.

Nonlinear Stability Analysis
Let V be a period cell for the solution to (4) and (5), and let  , ( , )  be the norm and inner product on 2 () LV . We commence by multiplying (4) 1 by f i u , (4) 2 by p i u , We also multiply (4) 3 by s  , (4) 4 by f  , (4) 5 by p  and integrate each over . V Thus, we derive the following equations: = 2 To determine the parameter 0   we now form 345 (10) (10) (10)     , to obtain an energy identity of form Equation (11) is the same as the expression was report by Straughan [5]. Now put 1 =, max whereis the space of admissible functions which are given by Then from (12), we deduce and then, from the Poincaré's inequality on D , we have To obtain the decay of f u and p u , we have to employ the arithmetic geometric mean inequality in (10) 1 and (10) where the decay of f u and p u follow. We now turn our attention to the maximization problem In order to determine E R , we have to derive the Euler-Lagrange equations and to maximize in the coupling parameter  . The Euler-Lagrange equations arising from equation (12) are Let us define , and  is a positive constant. Furthermore, after some integrations by parts and using the boundary conditions, we find that ] .
where f  and p  are Lagrange multipliers. Further progress is made by taking curl of equation 1 (14) and equation 2 (14) , we obtain ** 1 ** 2 We again use a normal mode representation as for the linear stability analysis in section 3. These results are to solve the determinant equation

Numerical results
In this section, we present new numerical computations for the linear and nonlinear stability analyses. Our analysis supports the work of Franchi et at [19] by computing the stationary convection instability threshold equation (8), and the oscillatory convection threshold equation (9). Both cases are studied by using golden section search to minimise  (8), equation (9) and equation (18) numerically by using Matlab routines. For the parameters that are employed in this article, we have to be very careful when minimize 2 R in equation (8) and equation (9) Figures 3 and 4 that as the value of p  decreases, the difference between the linear instability and nonlinear stability thresholds increases. As a result, the region of potential subcritical instabilities between the linear and nonlinear stability thresholds considerable. It is also noted that as p  increases the linear instability and nonlinear stability thresholds become closer. This is, in a sense, the best possible agreement between the two thresholds since the region of potential subcritical instabilities decreases. Figure 5 and Table 5  S . It is found that the effect of increasing 1 S is to increases the critical Rayleigh number Ra . However, Figure 5 shows that as s  and 2  increase with the decreasing 1  , Ra increases. Therefore, the parameter s  have a stabilizing effect on the stability of the system.

Conclusion
The onset of convection in a fluid saturated bidsperse porous medium is investigated when the temperature of the fluid and solid phases is in local thermal non-equilibrium. The linear instability threshold and nonlinear one is derived analytically in case of free surfaces. The stationary convection boundary, , is similar to that found by Nield and Kwznetsove for particular values of the porosity modified interaction coefficient and the porosity modified conductivity ratio, , and . Through investigation we found that the onset of convection is by stationary convection for various interaction parameters. It can be argued from the results that the critical Rayleigh numbers Ra and the critical wave numbers L a are greater in the case of and increases. Also, we observed that the subcritical instability region decreases as the parameter increases. The effects of increasing and as well as increasing the value of were seen to stabilize the system. Also, we observed that for small values of and , the effect of increasing is to stabilize the system. This indicates that the thermal convection occurs more easily.