Kinetic Theory Description of Plane, Compressible Couette Flow

Abstract

By utilizing the two-stream Maxwellian in Maxwell's integral
\nequations of transfer we are able to find a closed-form solution of the problem of compressible plane Couette flow over the whole range of gas density from free molecule flow to atmospheric. The ratio of shear
\nstress to the product of ordinary viscosity and velocity gradient, which is unity for a Newtonian fluid, here depends also on the gas density, the
\nplate temperatures and the plate spacing. For example, this ratio decreases rapidly with increasing plate Mach number when the plate temperatures are fixed. On the other hand, at a fixed Mach number based on the temperature of one plate, this ratio approaches unity as
\nthe temperature of the other plate increases. Similar remarks can be made for the ratio of heat flux to the product of ordinary heat conduction
\ncoefficient and temperature gradient.
\n
\nThe effect of gas density on the skin friction and heat transfer coefficients is described in terms of a single rarefaction parameter, which amounts to evaluating gas properties at a certain "kinetic temperature"
\ndefined in terms of plate Mach number and plate temperature ratio. One interesting result is the effect of plate temperature on velocity "slip".
\nIn the Navier-Stokes regime most of the gas follows the hot plate, because the gas viscosity is larger there. As the gas density decreases the situation is reversed, because the velocity slip is larger at the hot plate
\nthan at the cold plate. In the limiting case of a highly rarefied gas most of the gas follows the cold plate.