Diffusion models for Knudsen compressors

Abstract

A rarefied gas in a long straight pipe with a periodic structure consisting of alternately arranged narrow and wide pipes and with periodic temperature distribution, which is known as the Knudsen compressor (or pump), is considered. Under the assumption that the pipe is much thinner than the period, a diffusion model that describes the pressure distribution and mass flux of the gas in each pipe element is derived, together with the connection conditions at the junctions of the narrow and wide pipes, from the kinetic system composed of the Boltzmann equation and its boundary condition on the pipe wall. Then, on the basis of the diffusion model and the connection conditions, a global diffusion model for the entire pipe is constructed by means of homogenization. The present work is a formal extension to the case of the Boltzmann equation of the previous work [K. Aoki and P. Degond, Multiscale Model. Simul. 1, 304 (2003)], where a simplified version of the Bhatnagar–Gross–Krook (BGK) collision model is used. Some numerical results based on the original BGK model are also presented both for the global diffusion model and for the local diffusion model with the connection conditions at the junctions.