Dynamics of levitation during rolling over a thin viscous film

TL;DR

This paper develops a mathematical model for the levitation dynamics of a wheel rolling over a thin viscous film, analyzing asymptotic limits and finite-width effects, and compares predictions with experimental data.

Contribution

It introduces a combined model using Reynolds lubrication and motion equations, exploring asymptotic limits and finite-width effects for the first time.

Findings

Steady planing states can be achieved under certain conditions.

Flooding by back flow occurs with high flux or heavy loading.

Theoretical predictions align with experimental observations.

Abstract

A mathematical model is derived for the dynamics of a cylinder, or wheel, rolling over a thin viscous film. The model combines the Reynolds lubrication equation for the fluid with an equation of motion for the wheel. Two asymptotic limits are studied in detail to interrogate the dynamics of levitation: an infinitely wide wheel and a relatively narrow one. In both cases the front and back of the fluid-filled gap are either straight or nearly so. To bridge the gap between these two asymptotic limits, wheels of finite width are considered, introducing a further simplying approximation: although the front and back are no longer expected to remain straight for a finite width, the footprint of the fluid-filled gap is still taken to be rectangular, with boundary conditions imposed at the front and back in a wheel-averaged sense. The Reynolds equation can then be solved by separation of variables. For wider wheels, with a large amount of incoming flux or a relatively heavy loading of the wheel, the system is prone to flooding by back flow with fluid unable to pass underneath. Otherwise steady planing states are achieved. Both lift-off and touch-down are explored for a wheel rolling over a film of finite length. Theoretical predictions are compared with a set of experimental data.