Steady and quasi-steady thin viscous flows near the edge of a solid surface

Abstract

A new approach is proposed for the description of thin viscous flows near the edges of a solid surface. For a steady flow, the lubrication approximation and the no-slip condition are assumed to be valid on most of the surface, except for relatively small neighbourhoods of the edges, where a universality principle is postulated: the behaviour of the liquid in these regions is universally determined by flux, external conditions and material properties. The resulting mathematical model is formulated as an ordinary differential equation involving the height of the liquid film and the flux as unknowns, and analytical results are outlined. The form of the universal functions which describe the behaviour in the edge regions is also discussed, obtaining conditions of compatibility with lubrication theory for small fluxes. Finally, an ordinary differential equation is introduced for the description of intermediate asymptotic profiles of a liquid film which flows off a bounded solid surface.