An efficient Bubnov-Galerkin finite element formulation is employed to solve the Navier-Stokes and continuity equations in three-dimensions for the case of surface-tension dominated film flow over substrate topography, with the free-surface location obtained using the method of spines. The computational challenges encountered are overcome by employing a direct parallel multi-frontal method in conjunction with memory-efficient out-of-core storage of matrix co-factors. Comparison is drawn with complementary computational and experimental results for low Reynolds number flow where they exist, and a range of new benchmark solutions provided. These, in turn, are compared with corresponding solutions, for non-zero Reynolds number, from a simplified model based on the long-wave approximation; the latter is shown to produce comparatively acceptable results for the free-surface disturbance experienced, when the underpinning formal restrictions on geometry and capillary number are not exceeded.
Veremieiev, S., Thompson, H., & Gaskell, P. (2015). Free-surface film flow over topography: full three-dimensional finite element solutions. Computers and Fluids, 122, 66-82. https://doi.org/10.1016/j.compfluid.2015.08.016