Bifurcation of and ghost effect on the temperature field in the Bénard problem of a gas in the continuum limit

Abstract

A gas in a time-independent state under a uniform weak gravity in a general domain is considered. The asymptotic behavior of the gas in the limit that the Knudsen number of the system tends to zero (or in the continuum limit) is investigated on the basis of the Boltzmann system for the case where the flow velocity vanishes in this limit, and the fluid-dynamic-type equations and their associated boundary conditions describing the behavior of the gas in the continuum limit are derived. The equations, different from the Navier–Stokes ones, contain thermal stress and infinitesimal velocity amplified by the inverse of the Knudsen number. The system is applied to analysis of the behavior of a gas between two parallel plane walls heated from below (Bénard problem), and a bifurcated strongly distorted temperature field is found in infinitesimal velocity and gravity. This is an example showing that the Navier–Stokes system fails to describe the correct behavior of a gas in the continuum limit.