Part I. Kinetic theory description of plane, compressible Couette flow. Part II. Kinetic theory description of conductive heat transfer from a fine wire

Abstract

PART I: By utilizing the two-stream Maxwellian in Maxwell's integral equations 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 stress to the product of ordinary viscosity and velocity gradient, which is unity for a Newtonian fluid, here depends also on the gas density, the plate 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 of one plate, this ratio approaches unity as the of the other plate increases. Similar remarks can be made for the ratio of heat flux to the product of ordinary heat conduction coefficient and gradient. The 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 temperature defined in terms of plate Mach number and plate ratio. One interesting result is the effect of plate on velocity slip. In 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 than at the cold plate. In the limiting case of a highly rarefied gas most of the gas follows the cold plate. Limitations of the present six-moment approximation at high plate Mach numbers are discussed and it is concluded that an eight-moment approximation would eliminate these difficulties. The results obtained in this simple geometry suggest certain conclusions about hypersonic flow over solid bodies when the surface is much lower than the kinetic temperature. PART II: The Maxwell moment method utilizing the two-sided Maxwellian distribution function is applied to the problem of conductive heat transfer between two concentric clylinders at rest. Analytical solutions are obtained for small differences between the cylinders. The predicted heat transfer agrees very well with experiments performed by Bomelburg, Schafer-Rating and Eucken. Comparison with results given by the Grad's thirteen moment equations, and with those given by Fourier's law plus Maxwell-Smoluchowski temperature-jump boundary condition shows that the two-sided character in the distribution function is a crucial factor in problems involving surface curvature.