On initial transients in Reynolds' drainage model

Abstract

The time-dependent, axisymmetric drainage equations have been solved for the pseudo-steady approach of a pair of rigid disks. Use of the Laplace transformation yields two analytical forms for the solution, one having better convergence properties for shorter times, the other for longer times. Both have been used to derive from the macroscopic balances a pair of equivalent, non-linear film-thinning equations which have been solved numerically for the film thickness as a function of time. Initial transients in velocity profiles decay within t ≈ δ 2/ν under all reasonable conditions and within ≈ δ 2/2ν for an initially quiescent film. The effects of microflow transients on the rate of film thinning cannot be discerned after t ≈ r f 2/ν, where r f is the disk radius. For thin films described with reasonable accuracy by Reynolds' model, the film-thinning rate thus quickly approaches that for steady microflow unless the applied forces are abnormally weak.