Effect of a small curvature of the surfaces on microscale lubrication of a gas for large Knudsen numbers
Effect of a small curvature of the surfaces on microscale lubrication of a gas for large Knudsen numbers
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DOI：
10.1103/physrevfluids.7.034201
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发表时间：
2022-03
影响因子：
2.7
通讯作者：
T. Doi
中科院分区：
文献类型：
--
作者：
T. Doi
基于动力学理论研究了同轴圆柱之间微尺度间隙中气体的润滑流动。静止内筒为圆周方向调节系数不均匀的麦克斯韦型边界，外筒为匀速旋转的漫反射边界。无量纲曲率（定义为间隙尺寸除以内圆柱半径）很小，并且基于间隙尺寸的努森数是任意的。使用缓慢变化的近似对玻尔兹曼方程进行分析研究，特别关注方程的特征。推导出雷诺型方程的两个宏观润滑模型：一个由平面库埃特流和泊肃叶流的解组成（平面润滑模型），另一个由圆柱库埃特流和弯曲泊肃叶流组成（改进的润滑模型）。为了评估模型，还使用混合有限差分法对 Bhatnagar-Gross-Krook-Welander 动力学方程进行了直接数值分析。结果表明，当克努森数足够大时，使用平面润滑模型会导致不可忽略的误差。该误差是由于忽略了这样一个事实而引起的：从外圆柱到达的分子数量比从内圆柱到达的分子数量多，其数量与无量纲曲率的平方根成正比。还表明，改进的润滑模型在努森数的整个范围内提供了对直接数值解的出色近似。
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.