Bénard convection in a binary mixture with a nonlinear density-temperature relation

B\'enard convection of a two-component liquid in a porous medium is considered. The mixture displays Soret effects and shows a nonlinear density-temperature relation, i.e., non-Boussinesq properties. A two-parameter perturbation analysis is used to determine the effects on the stability of the basic state and the finite-amplitude convection. Linear theory demonstrates the nonlinear density-temperature relation to have a destabilizing effect; the critical Rayleigh numbers for the onset of oscillatory and steady-state convection are decreased. For the case of two-dimensional convection, traveling waves and steady-state solutions are considered. They are determined up to fifth order of the amplitude parameter. In the vicinity of the codimension-two point, a stable branch of traveling wave solutions exists near the onset. If the Rayleigh number is increased the wave motion vanishes and a transition to steady-state convection occurs. Due to the symmetry of the two-dimensional solutions this bifurcation is not affected by the non-Boussinesq properties of the mixture. However, for the case of three-dimensional convection the nonlinear density-temperature relation leads to an unfolding of the bifurcation of steady-state hexagonal solutions. As a consequence the branch of the oscillatory solution terminates at an isolated point in the parameter plane.