On the Moffatt-Pukhnachov problem

The classic coating-flow problem first studied experimentally by Moffat and asymptotically by Pukhnachov in 1977 is reconsidered in the framework of multiple-timescale asymptotics. Two-timescale approximations of the height of the thin film coating a rotating horizontal circular cylinder are obtained from an asymptotic analysis, in terms of small gravitational and capillary parameters, of Pukhnachov's nonlinear evolution for the film thickness. The transition, as capillary effects are reduced, from smooth to shock-like solutions is examined, and interesting large-time dynamics in this case are determined via a multiple-timescale analysis of a Kuramoto-Sivashinsky equation. A pseudo-three-timescale method is proposed and demonstrated to improve the accuracy of the smooth solutions, and an asymptotic analysis of a modified Pukhnachov's equation, one augmented by inertial terms, leads to an expression for the critical Reynolds number above which the steady states first analysed by Moffatt and Pukhnachov cannot be realised. As part of this analysis, some interesting implications of the effects of different scalings on inertial terms is discussed. All theoretical results are validated by either spectral or extrapolated numerics.