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

## TL;DR

This paper investigates the dynamics of a viscous film on a rotating cylinder with oscillating angular velocity, revealing complex fractal structures, conditions for overturning, and asymptotic behaviors at frequency extremes.

## Contribution

It introduces a numerical and analytical study of oscillating cylinder flows, uncovering fractal-like structures and conditions for film overturning and periodicity.

## Key findings

- Fractal-like structures in amplitude-frequency space.
- Overturning occurs at finite time for general initial conditions.
- High-frequency oscillations can delay overturning and produce periodic solutions.

## 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 (1977) and Pukhnachev (1977). 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 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.