Knudsen diffusivity in random billiards: spectrum, geometry, and computation

TL;DR

This paper presents an analytical and numerical framework to determine the self-diffusivity of rarefied gases in channels, linking microstructure geometry, spectral properties, and surface scattering effects, especially for low roughness surfaces.

Contribution

It introduces a universal form of the Markov transition operator for weak surface scattering and connects geometric, spectral, and scattering parameters to diffusivity.

Findings

The Markov operator has a universal form related to the Legendre differential operator.

Spectral gap and microstructure geometry influence diffusivity.

Numerical examples illustrate the theoretical relationships.

Abstract

We develop an analytical framework and numerical approach to obtain the coefficient of self-diffusivity for the transport of a rarefied gas in channels in the limit of large Knudsen number. This framework provides a method for determining the influence of channel surface microstructure on the value of diffusivity that is particularly effective when the microstructure exhibits relatively low roughness. This method is based on the observation that the Markov transition (scattering) operator determined by the microstructure, under the condition of weak surface scattering, has a universal form given, up to a multiplicative constant, by the classical Legendre differential operator. We also show how characteristic numbers of the system -- namely geometric parameters of the microstructure, the spectral gap of a Markov operator, and the tangential momentum accommodation coefficient of a commonly used model of surface scattering -- are all related. Examples of microstructures are investigated to illustrate the relation of these quantities numerically and analytically.