The Moffatt-Pukhnachev flow: a new twist on an old problem

Barcenas Luque, A.J. (Antonio) and Blyth, M.G. (Mark) (2025) The Moffatt-Pukhnachev flow: a new twist on an old problem. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences. ISSN 1364-5021 (In Press)

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Abstract

The flow of a thin viscous film on the outside of a horizontal circular cylinder, whose angular velocity is time-periodic with specified frequency and amplitude, is investigated. The constant angular velocity problem was originally studied by Moffatt [1] and Pukhnachev [2]. Surface tension is neglected. The evolution equation for the film thickness is solved numerically for a range of oscillation amplitudes and frequency. A blow-up map charted in amplitude- frequency space reveals highly intricate fractal-like structures exhibiting self-similarity. For a general initial condition numerical computations indicate that the film surface reaches a slope singularity at a finite time and tends to overturn. The high-frequency and low-frequency limits are examined asymptotically using a multiple-scales approach. At high frequency the analysis suggests that an appropriate choice of initial profile can substantially delay the overturning time, and even yield a time-periodic solution. In the low-frequency limit it is possible to construct a quasi-periodic solution that does not overturn if the oscillation amplitude lies below a threshold value. Above this value the solution tends inexorably toward blow-up. It is shown how solutions exhibiting either a single-shock or a double-shock may be constructed in common with the steadily rotating cylinder problem.

Item Type:	Article
Faculty \ School:	Faculty of Science > School of Engineering, Mathematics and Physics
UEA Research Groups:	Faculty of Science > Research Groups > Fluids & Structures
Depositing User:	LivePure Connector
Date Deposited:	24 Nov 2025 12:30
Last Modified:	24 Nov 2025 12:30
URI:	https://ueaeprints.uea.ac.uk/id/eprint/101089
DOI:	issn:1364-5021

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