Near-Continuum Gas Flows to Second Order in Knudsen Number with Arbitrary Surface Accommodation

Abstract

Asymptotic analyses of the Boltzmann equation for near-continuum low-Mach-number gas flows predominantly assume diffuse scattering from solid surfaces, i.e., complete surface accommodation, despite gas scattering often deviating from this idealized behavior in practice. While some results for arbitrary surface accommodation exist to second order in small Knudsen number, the full theory to this order is yet to be reported. Here, we present a matched asymptotic expansion of the linearized Boltzmann–BGK equation that generalizes existing theories to Maxwell-type boundary conditions with arbitrary accommodation at solid surfaces. This is performed to second order in small Knudsen number for smooth solid surfaces, and holds for steady and unsteady flow at oscillatory frequencies far smaller than the molecular collision frequency. In contrast to diffuse scattering, we find that the second-order Knudsen layer functions vary as \(\eta \log ^2\eta \) for incomplete but arbitrary accommodation at a curved surface, where \(\eta \) is the dimensionless normal coordinate. A modified refined moment method is developed to numerically handle this spatial dependency. Analytical formulas for all velocity slip and temperature jump coefficients for the Hilbert region are reported that exhibit accuracies greater than 99.9%. This resolves conflicting literature reports on the second-order velocity slip and temperature jump coefficients.