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Effect of weak gravitation on the plane Poiseuille flow of a highly rarefied gas

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Abstract

Plane Poiseuille flow of a highly rarefied gas that flows horizontally in the presence of weak gravitation is studied based on the Boltzmann equation for a hard sphere molecular gas and the diffuse reflection boundary condition. The behavior of the solution in the regime of large mean free path and small strength of gravity is studied numerically based on the one-dimensional Boltzmann equation derived by means of the asymptotic analysis for a slow variation in the flow direction. It is clarified that the effect of weak gravity on the flow is not negligible when the gas is so rarefied that the mean free path is comparable to the maximum range that the molecules travel along the parabolic path within the channel. When the mean free path is much larger than this range, the effect of gravity that makes the molecules fall plays the dominant role in determining the distribution function, and thus the over-concentration in the distribution function as well as the flow velocity does not increase further even if the mean free path is increased. The upper bound of the flow velocity and the mass flow rate of the gas are obtained as a function of the gravitational acceleration.

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  1. Department of Applied Mathematics and Physics, Graduate School of Engineering, Tottori University, 680–8552, Tottori, Japan

    Toshiyuki Doi

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  1. Toshiyuki Doi
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Doi, T. Effect of weak gravitation on the plane Poiseuille flow of a highly rarefied gas. Z. Angew. Math. Phys. 63, 1091–1102 (2012). https://doi.org/10.1007/s00033-012-0213-0

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  • DOI: https://doi.org/10.1007/s00033-012-0213-0

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