A Mesh-Dependent Model For Applying Dynamic Contact Angles To VOF Simulations

Typical VOF algorithms rely on an implicit slip that scales with mesh refinement, to allow contact lines to move along no-slip boundaries. As a result, solutions of contact line phenomena vary continuously with mesh spacing; this study presents examples of that variation, when applying both no-slip and Navier-slip boundary conditions. A mesh-dependent dynamic contact angle model is then presented, that is based on fundamental hydrodynamics and serves as a more appropriate boundary condition at a moving contact line. This new boundary condition eliminates the stress singularity at the contact line; the resulting problem is thus well-posed and yields solutions that converge with mesh refinement. This scaling relationship is then used as a means to evaluate the contact angle boundary condition as a function of the apparent contact angle, $\theta_{app}$, the capillary number, $\mathbf{Ca}$, and the mesh size, that yields mesh-independent solutions of dynamic contact line phenomena. Numerical results are presented of a solid plate withdrawing from a fluid pool, and of spontaneous droplet spread at small capillary and Reynolds numbers.