Boundary conditions for free surface inlet and outlet problems

Abstract

AbstractWe investigate and compare the boundary conditions that are to be applied to free-surface problems involving inlet and outlets of Newtonian fluid, typically found in coating processes. The flux of fluid is a priori known at an inlet, but unknown at an outlet, where it is governed by the local behaviour near the film-forming meniscus. In the limit of vanishing capillary number $\mathit{Ca}$ it is well known that the flux scales with ${\mathit{Ca}}^{2/ 3} $, but this classical result is non-uniform as the contact angle approaches $\lrm{\pi} $. By examining this limit we find a solution that is uniformly valid for all contact angles. Furthermore, by considering the far-field behaviour of the free surface we show that there exists a critical capillary number above which the problem at an inlet becomes over-determined. The implications of this result for the modelling of coating flows are discussed.