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Electrified viscous thin film flow over topography. (English) Zbl 1165.76061
Summary: We investigate the gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. We discuss the limiting behaviour of the film profile as the steepness of the wall geometry is increased. Using substantial numerical evidence, it is established that, as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. We also consider the effects of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
76A20 Thin fluid films
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