Onset of thermal convection of a weakly rarefied Maxwellian gas: A continuum-slip approach

Abstract

The onset of the Rayleigh–Bénard convection for a monatomic rarefied gas of Maxwellian type is investigated at small Knudsen numbers. Density variations due to compressibility have been considered without any restriction imposed on the magnitude of temperature difference. The steady-state conduction problem has been solved both analytically and numerically using a continuum approach with velocity-slip and temperature-jump conditions at the solid boundaries. The influence of the Froude number and the Knudsen number is comprehensively studied. To examine the onset of thermal convection, a linear temporal stability analysis is performed for the conduction state and the dispersion relation is calculated by a Chebyshev collocation method. The neutral stability curve obtained in the Froude–Knudsen number plane marks transition to convection from a pure conduction state. The critical wave number for the onset of convection is found to be close to 3.14, agreeing well with the existing findings. The convection only occurs for Knudsen numbers smaller than 0.027. A comparison between the Maxwellian model and the hard-sphere model is given and discussed. The encouraging results suggest the use of more realistic gas models in future studies.