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

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 (Moffatt 1977 J. Mécanique16, 651–673) and Pukhnachev (Pukhnachev 1977 J. Appl. Mech. Tech. Phys.18, 344–351). 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.