Weakly non linear convection in a shallow horizontal cavity under uniform crossed heat fluxes

A weakly nonlinear stability analysis is developed for the unicellular natural convection pattern in a horizontal fluid layer heated by uniform crossed heat fluxes applied on the horizontal and vertical walls, the remaining walls being adiabatic. The critical Rayleigh number is determined, for different Prandtl numbers and heat flux ratios, from the analytical parallel approximation for the single cell flow. For the range of parameters considered, stable, supercritical bifurcations occur solely for longitudinal disturbances with three velocity components. Depending on the value of Pr and the heat flux ratio, the instability is oscillating and hydrodynamic at small Pr and thermal and steady for larger Pr. Computations of the Landau coefficients for the two interacting modes reveal that the hydrodynamic mode is the only stable mode at the codimension points.