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A pinned or free-floating rigid plate lying on the free surface of a thinfilm of viscous fluid, which itself lies on top of a horizontal substrate thatis moving to the right at a constant speed is considered. The focus of thepresent work is to describe how the competing effects of the speed of thesubstrate, surface tension, viscosity, and, in the case of a pinned plate, theprescribed pressure in the reservoir of fluid at its upstream end, determinethe possible equilibrium positions of the plate, the free surface, and the flowwithin the film. The present problems are of interest both in their own rightas paradigms for a range of fluid-structure interaction problems in whichviscosity and surface tension both play an important role, and as a first steptowards the study of elastic effects.

A pinned or free-floating plate on a thin viscous film. Part 1: A rigid plate