Surface-tension-driven flow on a moving curved surface

The leading-order equations governing the flow of a thin viscous film over a moving curved substrate are derived using lubrication theory. Three possible distinguished limits are identified. In the first, the substrate is nearly flat and its curvature enters the lubrication equation for the film thickness as a body force. In the second, the substrate curvature is constant but an order of magnitude larger; this introduces an extra destabilising term to the equation. In the final regime, the radius of curvature of the substrate is comparable to the lengthscale of the film. The leading-order evolution equation for the thin film is then hyperbolic, and hence can be solved using the method of characteristics. The solution can develop finite-time singularities, which are regularised by surface tension over a short lengthscale. General inner solutions are found for the neighbourhoods of such singularities and matched with the solution of the outer hyperbolic problem. The theory is applied to two special cases: flow over a torus, which is the prototype for flow over a general curved tube, and flow on the inside of a flexible axisymmetric tube, a regime of interest in modelling pulmonary airways.