Strong resonance in two-dimensional non-Boussinesq convection

Abstract

Two-dimensional convection is described and analyzed in the case of strong 2:1 resonance. The density variation due to the temperature to linear order and the viscous heating as perturbations to the usual Boussinesq approximation are introduced. In addition the temperature dependence of the viscosity is taken into account. In this special setting one is able to quantify the influence of these effects on the dynamics. They break the symmetry with respect to reflections at the horizontal midplane that is artificially induced when adopting the Boussinesq approximation, and give rise to an O(2) symmetric bifurcation problem. Time-dependent solutions appear and become stable when the Prandtl number lies between two critical values, between zero and one. In this case there also exists a structurally stable heteroclinic orbit very close to the onset of motion in a small amplitude limit. A similar orbit exists outside this limit if Pr is sufficiently small and the viscosity ratio is relatively large. This behavior is in contrast to previous results obtained by adopting the Boussinesq approximation where only stationary convection patterns persist. Finally a brief survey over real fluids is given and the resulting dynamics is described.