Oscillatory convection in binary mixtures

Abstract

Two-dimensional oscillatory convection in a binary fluid mixture in an infinite plane porous layer heated from below is studied. Small-amplitude nonlinear solutions in the form of standing and traveling waves are found and their relative stability is established. Stable traveling waves are preferred near onset. The interaction of the two types of wave with steady overturning convection is also studied. As the Rayleigh number is increased the period of each type of wave approaches infinity, standing waves as -ln(${R}_{c}^{\mathrm{SW}\mathrm{\ensuremath{-}}\mathrm{R}}$) and traveling waves as 1/(${R}_{c}^{\mathrm{TW}\mathrm{\ensuremath{-}}\mathrm{R}}$), where ${R}_{c}$ is the critical Rayleigh number at which the transition to finite amplitude overturning convection occurs. This transition is hysteretic. The presence of modulated traveling waves (i.e., waves with two distinct frequencies) is also predicted. These predictions are made on the basis of analyses of multiple bifurcations in the presence of O(2) symmetry. This symmetry is present in two-dimensional problems with periodic boundary conditions and a reflection symmetry in a vertical plane. The relevance of the results to recent experiments on binary fluids, both in bulk mixtures and in a porous medium, is discussed.