Inertially induced cyclic solutions in thin-film free-surface flows

New mechanisms are discovered regarding the effects of inertia in the transient Moffatt–Pukhnachov problem (J. Méc., vol. 187, 1977, pp. 651–673) on the evolution of the free surface of a viscous film coating the exterior of a rotating horizontal cylinder. Assuming two-dimensional evolution of the film thickness (i.e. neglecting variation in the axial direction), a multiple-timescale procedure is used to obtain explicitly parameterized high-order asymptotic approximations of solutions of the

spatio-temporal evolution equation. Novel, hitherto-unexplained transitions from stability to instability are observed as inertia is increased. In particular, a critical Reynolds number Rec is predicted at which occurs a supercritical pitchfork bifurcation in wave amplitude that is fully explained by the new asymptotic theory. For Re<Rec, free-surface profiles converge algebraically-cum-exponentially to a steady state and, for Re > Rec, stable temporally periodic solutions with leading-order amplitudes proportional to .Re Rec/1=2 are found, i.e. in the régime in which previous related literature predicts exponentially divergent instability. For ReDRec, stable solutions are found that decay algebraically to a steady state. A model solution is proposed that not only captures qualitatively the interaction between fundamental and higher-order wave modes but also offers an explanation for the formation of the lobes observed in Moffatt’s original experiments. All asymptotic theory is convincingly corroborated by numerical integrations that are spectrally accurate in space and eighth/ninth-order accurate in time.