The flow of a thin layer of fluid on a solid surface is of considerable technological importance; many industrial applications, ranging from spin coating of microchips to the design of photographic film, are susceptible to often undesirable instabilities. We use a recently developed numerical scheme to analyze the problem of flow down an inclined plane, and concentrate on the case of completely wetting fluid characterized by a small precursor film height, b. In particular, we explore the role of imperfections of the solid surface, exploring the influence of the length scale of the surface roughness on the film stability and eventual pattern formation (fingers and rivulets). It is found that perturbations characterized by very short length scale do not influence considerably the film stability, in contrast to the perturbations specified by longer length scales. Further, we find that the role of imperfections becomes more pronounced at smaller precursor film thicknesses, b. We also analyze the effect of a time dependent body force on the film stability.
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Kondic, L., Bertozzi, A. Thin Liquid Films: Instabilities of Driven Coating Flows on a Rough Surface. MRS Online Proceedings Library 543, 213–218 (1998). https://doi.org/10.1557/PROC-543-213