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The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.

The linear stability of a viscoelastic fluid saturated densely packed horizontal porous layer heated from below and cooled from above is investigated by considering the Oldroyd-B type fluid. A generalized Darcy model, which takes into account the viscoelastic properties, is employed as momentum equation and a two-field model is used for energy equation each representing solid and fluid phases separately. Linear stability analysis suggests that, there is a competition between the processes of viscous relaxation and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Analytical expression for the occurrence of oscillatory onset is obtained, and it is found that the necessary condition for the existence of the same is < 1. Besides, the effect of viscoelastic parameters and the thermal non-equilibrium on the stability of the system is analyzed.

Abstract-The linear and non-linear stability of rotating double-diffusive convection in a sparsely packed porous medium is investigated considering a non-Darcy equation. In the case of linear theory both marginal and overstable motions are discussed. In the former case it is shown that the effect of Taylor number and porous parameter is to make the system more stable. In the latter case, however, it is shown that the bottom-heavy solute gradient and rotation destabilize the system under certain conditions. By drawing the stability boundaries in the Rayleigh number plane it is shown that the effect of rotation and porous parameter is to decrease the region of instabilities. Using the theory of self-adjoint operator the variation of critical eigenvalue with physical and boundary parameters is studied. In the case of non-linear theory, both steady and unsteady cases have been considered.In the unsteady case the transient behaviour concerning the variation of Nusselt number with time has been investigated, by solving numerically a seventh-order Lorenz model using the RungeKutta-Gill method. Interesting results are obtained by comparing these results with those of the steady case. Finally, the effect of porous parameter on streamfunction, isotherms, isohalines and zonal velocity is studied.

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The weakly nonlinear stability of the triple diffusive convection in a Maxwell fluid saturated porous layer is investigated. In some cases, disconnected oscillatory neutral curves are found to exist, indicating that three critical thermal Darcy-Rayleigh numbers are required to specify the linear instability criteria. However, another distinguishing feature predicted from that of Newtonian fluids is the impossibility of quasi-periodic bifurcation from the rest state. Besides, the co-dimensional two bifurcation points are located in the Darcy-Prandtl number and the stress relaxation parameter plane. It is observed that the value of the stress relaxation parameter defining the crossover between stationary and oscillatory bifurcations decreases when the Darcy-Prandtl number increases. A cubic Landau equation is derived based on the weakly nonlinear stability analysis. It is found that the bifurcating oscillatory solution is either supercritical or subcritical, depending on the choice of the physical parameters. Heat and mass transfers are estimated in terms of time and area-averaged Nusselt numbers.