On inertial effects in the Moffatt–Pukhnachov coating-flow problem

Abstract

The effects are investigated of including inertial terms, in both small- and large-surface-tension limits, in a remodelling of the influential and fundamental problem first formulated by Moffatt and Pukhnachov in 1977: that of viscous thin-film free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field.An analysis of the non-dimensionalizations of previous related literature is made and the precise manner in which different rescalings lead to the asymptotic promotion or demotion of pure-inertial flux terms over gravitational-inertial terms is highlighted. An asymptotic mass-conserving evolution equation for a perturbed-film thickness is derived and solved using two-timescale asymptotics with a strained fast timescale. By using an algebraic manipulator to automate the asymptotics to high orders in the small expansion parameter of the ratio of the film thickness to the cylinder radius, consistent a posteriori truncations are obtained.Via two-timescale and numerical solutions of the evolution equation, new light is shed on diverse effects of inertia in both small- and large-surface-tension limits, in each of which a critical Reynolds number is discovered above which the thin-film evolution equation has no steady-state solution due to the strength of the destabilizing inertial centrifugal force. Extensions of the theory to the treatment of thicker films are discussed.