Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer

Abstract

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid. Both the mechanical equilibrium solution and the monocellular flow obtained for particular ranges of the physical parameters of the problem are considered. The porous cavity, bounded by horizontal infinite or finite boundaries, is heated from below or from above. The two horizontal plates are maintained at different constant temperatures while no mass flux is imposed. The influence of the governing parameters and more particularly the role of the separation ratio, ψ, characterizing the Soret effect and the normalized porosity, ε, are investigated theoretically and numerically. From the linear stability analysis, we find that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. The role of the porosity is important, when it decreases, the stability of the equilibrium solution is reinforced. For a cell heated from below, the equilibrium solution loses its stability via a stationary bifurcation when the separation ratio ψ>ψ0(Le,ε), while for ψ<ψ0(Le,ε), it loses stability via a Hopf subcritical bifurcation. The oscillatory solution is unstable and becomes stationary. For a cell heated from above, the equilibrium solution is linearly stable if ψ>0, while a stationary or an oscillatory bifurcation occurs if ψ<0. The results obtained from the linear stability analysis are widely corroborated by direct 2D numerical simulations. In the case of long-wave disturbances, for ψ<0 and for ψ higher than a particular value called ψmono, we observe that the monocellular flow leads to a separation of the species between the two ends of the cell. First, we determined the velocity, temperature, and concentration fields analytically for monocellular flow. Then we studied the stability of this flow. For a cell heated from below and for ψ>ψmono the monocellular flow loses stability via a Hopf bifurcation. As the Rayleigh number increases, the resulting oscillatory solution evolves to a stationary multicellular flow. For a cell heated from above and ψ<0, the monocellular flow remains linearly stable. We verified numerically that this problem admits other stable multicellular stationary solutions for this range of parameters.