Modeling of anisotropic convection for the binary fluid in porous medium

We study an appearance of gravitational convection in a porous medium saturated by the double-diffusive fluid. The rectangle heated from below is considered with anisotropy of media properties. We analyze Darcy–Boussinesq equations for a binary fluid with Soret effect.
Resulting system for the stream function, the deviation of temperature and concentration is cosymmetric under some additional conditions for the parameters of the problem. It means that the quiescent state (mechanical equilibrium) loses its stability and a continuous family of stationary regimes branches off. We derive explicit formulas for the critical values of the Rayleigh numbers both for temperature and concentration under these conditions of the cosymmetry. It allows to analyze monotonic instability of mechanical equilibrium, the results of corresponding computations are presented.
A finite-difference discretization of a second-order accuracy is developed with preserving of the cosymmetry of the underlying system. The derived numerical scheme is applied to analyze the stability of mechanical equilibrium.
The appearance of stationary and nonstationary convective regimes is studied. The neutral stability curves for the mechanical equilibrium are presented. The map for the plane of the Rayleigh numbers (temperature and concentration) are displayed. The impact of the parameters of thermal diffusion on the Rayleigh concentration number is established, at which the oscillating instability precedes the monotonic instability. In the general situation, when the conditions of cosymmetry are not satisfied, the derived formulas of the critical Rayleigh numbers can be used to estimate the thresholds for the convection onset.