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

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.