Effective slip in numerical calculations of moving-contact-line problems

Abstract

For many coating flows, the profile thickness h, near the front of the coating film, is governed by a third-order ordinary differential equation of the form h ^m = f(h), for some given f(h). We consider here the case of dry wall coating which allows for slip in the vicinity of the moving contact-line. For this case, one such model equation, due to Greenspan, f h = -1 + (1 + α)/(h ₂ + α), where α is the slip coefficient. The equation is solved using a finite difference scheme, with a contact angle boundary condition prescribed at the moving contact-line. Using the maximum thickness of the profile as the control parameter, we show that there is a direct relationship between the effective Greenspan slip coefficient and the grid-spacing of the numerical scheme used to solve the model equation. In doing so, we show that slip is implicitly built into the numerical scheme through the finite grid-spacing. We also show why converged results with finite film thickness cannot be obtained if slip is ignored.