Thin film flow on surfaces containing arbitrary occlusions

The flow of a continuous liquid film, only several microns in thickness, on a plane surface containing occlusions, is modelled using lubrication theory. The resultant time-dependent non-linear equation set is discretised using both finite difference and finite element analogues; the former is solved using a very efficient full approximation storage (FAS) multigrid algorithm employing both automatic spatial and temporal adaptivity, the latter exploits the COMSOL suite of software to solve a weak form of the governing equations. The efficiencies and accuracies of both approaches are compared and a variety of problems, of increasing complexity, explored. Although the adaptive multigrid approach is clearly the most efficient, with the two sets of solutions indistinguishable, COMSOL offers an attractive alternative to the non-specialist user. The results, the first of their kind, further demonstrate: (i) the power of using adaptive multigriding, in time as well as space, for the simultaneous control of both spatial and temporal errors; (ii) the significant reduction in memory requirements and CPU time that can be achieved. It is revealed that occlusions lead to many of the features inherent in the flow of thin liquid films over fully submerged micro-scale topographic features; namely, the presence of capillary ridges linked to “bow-wave” plus “comet-tail” free-surface disturbances.