On the Rayleigh-Bénard problem in the continuum limit: Effects of temperature differences and model of interaction

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. Considering a power-law (variable hard-sphere) model of interaction our analysis indicates that for sufficiently large Froude numbers “softer” potentials result in less unstable systems. At small Froude numbers this trend is reversed, i.e., the “softer” interaction potentials correspond to a more unstable response. These results are discussed in terms of the opposing mechanisms of thermal expansion and compressibility. We carry out an asymptotic expansion for small temperature differences and establish the principle of exchange of stabilities for this limit. A singularity appears in this limit when compressibility effects are dominant.