Influence of Vertical Vibrations on the Stability of a Binary Mixture in a Horizontal Porous Layer Subjected to a Vertical Heat Flux

Abstract

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid mixture and subjected to vertical high-frequency and small-amplitude vibrations. Two configurations have been considered and compared: an infinite horizontal layer and a bounded domain with a large aspect ratio. In both cases, the initial temperature gradient is produced by a constant uniform heat flux applied on the horizontal boundaries. A formulation using time-averaged equations is used. The linear stability of the equilibrium solution is carried out for various Soret separation ratios $$\varphi $$ , vibrational Rayleigh numbers Rv, Lewis numbers Le and normalized porosity. For an infinite horizontal layer, the critical Rayleigh number $$\mathrm{Ra}_c$$ is determined analytically. For a steady bifurcation to a one-cell solution (the critical wavenumber is zero), we obtain $$\mathrm{Ra}_{c}={12}/{({\varphi }(\mathrm{Le}+1)+1)}$$ for all Rv. When the bifurcation is a Hopf bifurcation or when the critical wavenumber is not zero, we use a Galerkin method to compute the critical values. Our study is completed by a nonlinear analysis of the bifurcation to one-cell solutions in an infinite horizontal layer that is compared to numerical simulations in bounded horizontal domains with large aspect ratio.