Kinetic theory analysis of microscale lubrication of a gas between eccentric circular cylinders with an arbitrary temperature difference

Abstract

A microscale lubrication flow of a gas between eccentric circular cylinders with an arbitrary temperature difference is studied on the basis of kinetic theory. The dimensionless curvature, defined as the mean clearance divided by the radius of the inner cylinder, is small, whereas the temperature ratio and the Knudsen number based on the mean clearance take arbitrary values. The Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation in bipolar coordinates is studied analytically using the slowly varying approximation. The leading-order term of the perturbation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by the free molecular solution or an equilibrium solution at rest. Two macroscopic lubrication models are derived, along with a numerical database that enables one to use the models quickly. Direct numerical analysis of the BGKW equation is also conducted, and the validity of the lubrication models is assessed. The heating of either cylinder enhances both the eccentric force and the torque acting on the inner cylinder. When the Knudsen number is small, there is little difference in the eccentric force between the cases in which the rotating inner cylinder is heated and the stationary outer cylinder is heated. However, this difference becomes significant as the Knudsen number increases, with heating of the outer cylinder yielding the greater eccentric force. If the two lubrication models are applied complementarily depending on the Knudsen number, they provide a reasonable result for the eccentric force over a wide range of the Knudsen number.