The Diffusion Limit of Free Molecular Flow in Thin Plane Channels

• Published in: SIAM journal on applied mathematics Vol. 52; no. 4; pp. 1057 - 1075
• Main Authors: Börgers, Christoph, Greengard, Claude, Thomann, Enrique
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.08.1992
• Subjects: Applied mathematics; Boundary conditions; Central limit theorem; Density distributions; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Gas lubrication; Integers; Molecular flow; Molecules; Monte Carlo simulation; Physics; Probability distribution; Probability distributions; Random variables; Rarefied gas dynamics; Specular reflection; Trajectories; Rarefied gas flow; Free molecular flow; Limit; Diffusion; Modeling; Thin film; Parallel plate
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/0152062

The subject of this paper is free molecular flow in thin channels bounded by parallel plane surfaces on which Maxwell's boundary condition applies. With tools from probability theory, it is proved that in the limit as the domain width h tends to zero, the evolution of the density is described by a diffusion equation, on a timescale of 1/(h log h-1), and with a diffusion coefficient of $(2 - \alpha) \sqrt T/(2\alpha \sqrt \pi)$ (α is the accommodation coefficient and T is the surface temperature). The logarithmic factor in the timescale is geometry dependent; in thin cylinders of diameter h, the timescale is 1/h, as Babovsky has proved in [Journal of Statistical Physics, 44 (1986), pp. 865-878]. Numerical calculations indicate that the diffusion limit is closely approximated even at fairly large values of h.