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.