Analysis of Ultra-Thin Gas Film Lubrication Based on Linearized Boltzmann Equation: First Report—Derivation of a Generalized Lubrication Equation Including Thermal Creep Flow

TLDR

The study numerically analyzes the generalized lubrication equation at high Knudsen numbers to uncover key findings. A generalized Reynolds‑type lubrication equation for arbitrary Knudsen numbers is derived from the linearized Boltzmann equation by semi‑numerically evaluating Poiseuille, Couette, and thermal‑creep flow rates. The analysis shows that first‑order slip overestimates load capacity while second‑order slip underestimates it, that Couette flow dominates at large bearing numbers driving results toward a common asymptote, and that thermal‑creep flow can generate additional load capacity when temperature gradients exist along the boundaries.

Abstract

A generalized Reynolds-type lubrication equation valid for arbitrary Knudsen numbers, defined as the ratio of the molecular mean free path to the film thickness, is derived from a linearized Boltzmann equation by semi-numerically calculating the flow rates of fundamental flows in the lubrication film: Poiseuille flow, Couette flow, and thermal creep flow. Numerical analysis of the equation for high Knudsen numbers reveals three principal results. First, Burgdorfer’s modified Reynolds equation featuring the first-order velocity slip boundary condition overestimates load carrying capacities, while the approximation equation including both the first- and second-order velocity slip boundary condition underestimates them. Second, since the flow rate of the Couette flow, which is independent of Knudsen numbers, becomes dominant as the bearing number increases, all the lubrication equation results tend toward the same asymptotic value for an infinite bearing number. Third, a new kind of load carrying capacity caused by thermal creep flow occurs if temperature gradients at the boundaries exist in the flow direction.