An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localised topographic features

Gravity-driven continuous thin film flow over a plane, containing well-defined single and grouped topographic features, is modelled as Stokes flow using lubrication theory. The associated time-dependent, nonlinear, coupled set of governing equations is solved using a full approximation storage (FAS) multigrid algorithm by employing automatic mesh adaptivity, the power, efficiency and accuracy of which is demonstrated by comparing the results with corresponding global fine-mesh solutions. These show that automatic grid refinement effectively restricts the use of fine grids to regions of rapid flow development which, for the topographies considered, includes the topography itself, the upstream Capillary ridge, downstream surge region, and the characteristic bow wave. It is shown that for the accurate solution of the associated lubrication equations, adaptive multigrid offers increased flexibility together with a significant reduction in memory requirement. This is further demonstrated by solving the problem of transient flow over a trench topography, generated by a sinusoidally varying inlet condition.