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Dynamics of free surface flows on rotating cylinders
[Thesis]. Manchester, UK: The University of Manchester; 2018.
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Abstract
The problem of the dynamics of a two-dimensional thin film of viscous fluid on the outer (coating) or inner (rimming) surface of a circular cylinder rotating about its horizontal axis in a vertical gravity field was first formulated by Moffatt and Pukhnachev in 1977. This seemingly simple free surface flow problem can have fascinatingly rich mathematical structures with evidence of multiple solutions, shock-like (near-discontinuous) phenomena and complex dynamics, and to this day continues to attract much attention from both theoretical/computational and experimental fronts. In this thesis, we analyse in detail the dynamics of these two free-surface flow problem using a mixture of analytic and numerical techniques. Numerical simulations of the two-dimensional free surface Stokes equations are developed using the finite-element library oomph-lib, and we find that previously unobserved stable and unstable steady solutions coexist over an intermediate range of rotation rates for sufficiently small surface tension values. Furthermore, comparisons are made with the classical (leading-order) lubrication model, a higher-order lubrication model and a new lubrication model that was formulated using a variational approach. The variational model is confirmed to be the most accurate of the three-models, but it still does not capture the full qualitative behaviour of the Stokes equations. We also study the dynamics of two-dimensional films and drops of a partially wetting liquid on a rotating cylinder using a lubrication model proposed by Thiele in 2011 that includes the effects of viscosity, gravity, rotation, surface tension and wettability. By developing novel techniques to compute and follow steady and time-periodic solutions of the one-dimensional evolution equation of the liquid film thickness in a two-dimensional structured mesh, we investigate the problem's complex bifurcation structure. We show that our numerical results are in excellent agreement with those of an independent numerical investigation. In addition, we compare the steady and time-periodic solutions of this model with those of a new lubrication model, also obtained using a variational approach. We find that although the overall picture is preserved, there are interesting changes in the bifurcation structure, including the creation/annihilation of branches and bifurcations.