Effect of heat-flux boundary conditions on the Rayleigh-Bénard instability in a rarefied gas

We consider the effect of heat-flux boundary conditions, replacing the previously studied isothermal wall conditions, on the Rayleigh-Bénard instability in a rarefied gas. The problem is investigated in the limit of small Knudsen numbers, by means of a linear stability analysis of a slip flow model, and the direct simulation Monte Carlo method. In the latter, a noniterative algorithm is applied to implement the heat-flux conditions. The results delineate the instability domain in the parameters plane of the Knudsen number (Kn), Froude number (Fr), and walls reference temperature ratio. The heat-flux conditions result in a significant destabilizing effect, extending instability to larger Knudsen numbers. At large Fr, the Boussinesq limit is recovered, and transition to instability is governed by a critical value of the Rayleigh number. With decreasing Fr, gas compressibility becomes dominant, confining the convection layer to the vicinity of the upper cold wall. Asymptotic analysis of the low-Fr limit is carried out, to highlight the impact of difference in thermal conditions. The Monte Carlo scheme is applied to investigate system instability at supercritical states, where walls heat-flux conditions lead to elevated shear stresses. Nonmonotonic variations in the walls shear stress with Kn are observed and discussed.