Computing inertial two- and three-dimensional thin film flow on planar surfaces featuring topography

A range of problems is investigated, involving the gravity-driven inertial flow of a thin viscous liquid film over a planar surface containing topographical features, modelled via a depth-averaged form of the governing unsteady Navier-Stokes equations. The discrete analogue of the resulting coupled equation set, employing a staggered mesh arrangement for the dependent variables, is solved accurately using an efficient Full Approximation Storage (FAS) algorithm and Full Multigrid (FMG) technique together with adaptive time- stepping and proper treatment of the nonlinear convective terms. A unique, comprehensive set of results is presented for both one- and two-dimensional topographical features, and errors quantified via detailed comparisons drawn with complementary experimental data and predictions from finite element analyses where they exist. It is found in the case of one-dimensional (spanwise) topography that for small Reynolds number and shallow/short features the depth-averaged form produces results that are in close agreement with corresponding finite element solutions of the full free-surface problem. For the case of flow over two-dimensional (localised) topography the free-surface disturbance observed is influenced significantly by the presence of inertia. It leads, as in the case of spanwise topography, to an increase in the magnitude and severity of the capillary ridge/trough patterns which form.