Non-Boussinesq effects on transitions in Hele-Shaw convection

Mode jumping between buoyancy-induced convection patterns in a slender Hele-Shaw cell (height > width) is studied with a center manifold projection technique at a critical aspect ratio, where both the single-roll and double-roll convection patterns destabilize simultaneously. In contradiction to some experimental observations, the Boussinesq model with insulated or isothermal side wall conditions indicates that the single-roll mode never surrenders its stability and yields to the double-roll mode as the Rayleigh number is increased. By adding the temperature dependence of fluid density and conductivity to the heat equation, it is shown that the single-roll mode can now undergo secondary bifurcations. Hysteretic jumps to the double-roll mode and smooth transitions to a mixed mode, and even to a time-periodic convection pattern, are then possible.