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Effect of a small curvature of the surfaces on microscale lubrication of a gas for large Knudsen numbers

Toshiyuki Doi*

  • Department of Applied Mathematics and Physics, Tottori University, Tottori 680-8552, Japan

  • *doi@tottori-u.ac.jp
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Phys. Rev. Fluids 7, 034201 – Published 7 March, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.034201

Abstract

Lubrication flow of a gas in a microscale gap between coaxial circular cylinders is studied on the basis of kinetic theory. The stationary inner cylinder is a Maxwell-type boundary with a nonuniform accommodation coefficient in the circumferential direction, and the outer cylinder is a diffuse reflection boundary rotating at a constant speed. The dimensionless curvature, defined as the gap size divided by the radius of the inner cylinder, is small, and the Knudsen number based on the gap size is arbitrary. The Boltzmann equation is studied analytically using the slowly varying approximation, with special attention being paid to the characteristics of the equation. Two macroscopic lubrication models of the Reynolds-type equations are derived: one consisting of the solutions for plane Couette and Poiseuille flows (plane lubrication model), and the other consisting of cylindrical Couette flow and a curved Poiseuille flow (improved lubrication model). For an assessment of the models, a direct numerical analysis of the flow is also conducted for the Bhatnagar-Gross-Krook-Welander kinetic equation using a hybrid finite-difference method. It is demonstrated that the use of the plane lubrication model leads to a non-negligible error when the Knudsen number is sufficiently large. This error is caused by neglect of the fact that the number of molecules arriving from the outer cylinder is greater than that from the inner one by an amount proportional to the square root of the dimensionless curvature. It is also demonstrated that the improved lubrication model provides an excellent approximation to the direct numerical solution over the whole range of the Knudsen number.

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Phys. Rev. Fluids 7, 034201– Published 7 March, 2022
  • Received 8 December 2021
  • Accepted 11 February 2022
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