Film flow on a rotating disk

Abstract

Unsteady liquid film flow on a rotating disk is analyzed by asymptotic methods for low and high Reynolds numbers. The analysis elucidates how a film of uniform thickness thins when the disk is set in steady rotation. In the low Reynolds number analysis two time scales for the thinning film are identified. The long-time-scale analysis ignores the initial acceleration of the fluid layer and hence is singular at the onset of rotation. The singularity is removed by matching the long-time-scale expansion for the transient film thickness with a short-time-scale expansion that accounts for fluid acceleration during spinup. The leading order term in the long-time-scale solution for the transient film thickness is shown to be a lower bound for film thickness for all time. A short-time analysis that accounts for boundary layer growth at the disk surface is also presented for arbitrary Reynolds number. The analysis becomes invalid either when the boundary layer has a thickness comparable to that of the thinning film, or when nonlinear effects become important.