Abstract
The transition to convection in the Rayleigh–Bénard problem at small Knudsen numbers is studied via a linear temporal stability analysis of the compressible “slip-flow” problem. No restrictions are imposed on the magnitudes of temperature difference and compressibility-induced density variations. The dispersion relation is calculated by means of a Chebyshev collocation method. The results indicate that occurrence of instability is limited to small Knudsen numbers (Kn≲0.03) as a result of the combination of the variation with temperature of fluid properties and compressibility effects. Comparison with existing direct simulation Monte Carlo and continuum nonlinear simulations of the corresponding initial-value problem demonstrates that the present results correctly predict the boundaries of the convection domain. The linear analysis thus presents a useful alternative in studying the effects of various parameters on the onset of convection, particularly in the limit of arbitrarily small Knudsen numbers.
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