Homogenization of a Flow in a Periodic Channel of Small Section
Kazuo Aoki, Pierre Degond · Multiscale Modeling and Simulation · 2003
In this paper, we derive a convection-diffusion model for a rarefied gas flow in a channel of small section. The flow is driven by temperature gradients along the channel walls. The channel itself is made of segments of different sections. Both the channel section and the temperature profile have periodic variations along the channel axis. This situation is that of an experiment reported in, e.g., [Y. Sone and K. Sato, Phys. Fluids, 12 (2000), pp. 1864--1868]. The model is obtained through a three-step asymptotic procedure. First, a diffusion model for the gas density in a channel of constant and small section is derived from a kinetic description of the gas, using the methodology established in [H. Babovsky, C. Bardos, and T. Platkowski, Asymptot. Anal., 3 (1991), pp. 265--289]. Second, connection conditions for the gas density at the junction between two segments of different sections are obtained. Third, by using homogenization techniques, a convection-diffusion equation is deduced for the periodic channel when the number of periods is large.