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Asymptotic Behavior for Rayleigh Problem Based on Kinetic Theory

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Abstract

We investigate the dynamics of the gas bounded by an infinite flat plate which is initially in equilibrium and set at some instant impulsively into uniform motion in its own plane. We use the Boltzmann equation to describe intermolecular collisions and assume the diffuse reflection to describe the interaction of the gas with the boundary. The Mach number of the plate is assumed to be small so that we can linearize the Boltzmann equation as well as the boundary condition. We show that the asymptotic behavior of the gas represents a perturbation to the free molecular gas when the time is much less than the mean free time. On the other hand, if the time is much greater than the mean free time, we show that the gas dynamics is governed by the linearized Navier–Stokes equation with a slip flow on the boundary and establish a boundary layer correction with thickness of the order of the mean free path. We also establish the singularity of velocity distribution function along the particle trajectory near the boundary.

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Acknowledgements

We thank Professor Tai-Ping Liu for his interest and discussions. We also thank Professor Kazuo Aoki for his insightful suggestions. This work was supported by MOST Grant 104-2115-M-006-010-MY2. It was also in part supported by the Ministry of Education, Taiwan, R.O.C. The Aim for the Top University Project to the National Cheng Kung University.

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  1. Department of Mathematics and Research Center for Theoretical Sciences, National Cheng Kung University, Tainan, 70101, Taiwan

    Hung-Wen Kuo

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  1. Hung-Wen Kuo
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Kuo, HW. Asymptotic Behavior for Rayleigh Problem Based on Kinetic Theory. J Stat Phys 166, 1247–1275 (2017). https://doi.org/10.1007/s10955-017-1717-y

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  • DOI: https://doi.org/10.1007/s10955-017-1717-y

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