An approximate analytical solution of the hydrodynamic problem associated with an advancing liquid-gas contact line

Abstract

An alternative to perturbation techniques or full numerical solution is developed for the free-surface problem associated with an advancing liquid-gas contact line. The method, which makes use of a local-wedge approximation to obtain the free-surface pressure variation, leads to a second-order ordinary differential equation for the meniscus shape. Both analytical considerations and comparisons with available full numerical solutions for capillary tubes confirm the validity of the meniscus equation up to values of the capillary number (Ca) of order 10 −1. At higher Ca the equation retains its validity in the wall region, for which an approximate analytical solution is derived. Matching of this solution with the central circular meniscus profile leads to an analytical approximation for the advancing contact angle, which compares excellently with available data (Ca up to an order 1). In contrast with preceding analyses, the classical approximations—in particular, that of no slip—are assumed to retain their validity up to the order of a molecular dimension from the wall, at which point the true contact angle is reached. While this angle is again supposed to be equal to the static value, this assumption is not critical to the dynamic angle predicted.