On the stability of viscous free-surface flow supported by a rotating cylinder

Using an adaptive finite-element (FE) scheme developed recently by the auth ors, we shed new light on the long-standing fundamental problem of the unst eady free surface Stokes flow exterior to a circular cylinder rotating abou t its horizontal axis in a vertical gravitational field. For supportable lo ads, we observe that the steady-state is more readily attained for near-max imal fluid loads on the cylinder than for significantly sub-maximal loads. For the latter, we investigate large-time dynamics by means of a finite-dif ference approximation to the thin-film equations, which is also used to val idate the adaptive FE simulations (applied to the full Stokes equations) fo r these significantly sub-maximal loads. Conversely, by comparing results o f the two methods, we assess the validity of the thin-film approximation as either the load is increased or the rotation rate of the cylinder is decre ased. Results are presented on the independent effects of gravity, surface tension and initial film thickness on the decay to steady-state. Finally, n ew numerical simulations of load shedding are presented.