Proper initial conditions for the lubrication model of the flow of a thin film of fluid

Abstract

A lubrication model describes the dynamics of a thin layer of fluid spreading over a solid substrate. But to make forecasts we need to supply correct initial conditions to the model. Remarkably, the initial fluid thickness is not the correct initial thickness for the lubrication model. Theory recently developed in \cite{Roberts89b,Roberts97b} provides the correct projection of initial conditions onto a model of a dynamical system. The correct projection is determined by requiring that the model's solution exponentially quickly approaches that of the actual fluid dynamics. For lubrication we show that although the initial free surface shape contributes the most to the model's initial conditions, the initial velocity field is also an influence. The projection also gives a rationale for incorporating miscellaneous small forcing effects into the lubrication model; gravitational forcing is given as one example.