Ghost effect of infinitesimal curvature in the plane Couette flow of a gas in the continuum limit

Abstract

The time-independent plane Couette flow of a gas in the continuum limit is studied on the basis of kinetic theory as the limit of the cylindrical Couette flow of a rarefied gas between two rotating coaxial circular cylinders when the mean free path and the curvature (or the inverse of the radius) of the inner cylinder tend to zero simultaneously, keeping the difference of the radii of the two cylinders fixed. The fluid-dynamic-type equations and their boundary conditions governing the limiting state are derived for arbitrary circumferential speeds of rotation of the cylinders and for arbitrary temperature difference of the two cylinders. The resulting equations depend on the relative speed of decay of the two parameters and contain a term due to the infinitesimal curvature of the cylinder, as well as non-Navier–Stokes stress terms, when the curvature decays not faster than some function (generally, the square) of the mean free path. The bifurcation analysis of the plane Couette flow of a linear profile is carried out when the speeds of the walls and their temperature difference are small. The bifurcation point and the bifurcated flow fields, where the linear profile is considerably deformed by infinitesimal cross flows induced by the infinitesimal curvature, are obtained.