Nonlinear rivulet dynamics during unstable wetting flows

The growth and propagation of nonlinear rivulets is studied. Lubrication theory and a Navier type slip model are used to establish the thin film equation for the nonlinear evolution of the height of liquid in the vicinity of a driven contact line. This equation is then solved using a semi-implicit numerical scheme, adapted to handle the moving boundary nature of the problem. The problem involves two parameters, alpha, a dimensionless slip parameter and the dimensionless contact slope C = (3Ca)(-1/3)tan(theta), where theta is the contact angle and Ca is the capillary number, mu U/sigma. A parametric study establishes both the shapes and the dynamics of nonlinear rivulet propagation. The shapes are found to be relatively independent of the slip parameter and are primarily determined by the contact slope, while the rivulet speeds are dependent on the level of slip, as expected. The computed shapes. including the occurrence of a capillary bulge near the advancing front, are in excellent agreement with experiments. Chevron-shaped steady traveling wave solutions, and both chevron and straight-sided convectively propagating shapes are obtained, with the traveling wave solutions occurring for small contact slope and the straight-sided solutions for large slope. Complete coating of the substrate in the presence of rivulet instabilities occurs only for C > 1.0. The predictions an in excellent agreement with experiments and suggest that simple lubrication theory and slip models are all that is necessary to describe the observed shapes and dynamics. (C) 1999 Elsevier Science Ltd. All rights reserved.