An Interface Capturing Method for Two–phase Flow with Moving Contact Lines

This paper presents a conservative numerical method capable of capturing contact line movement for the simulation of capillary dominated flow of two immiscible, incompressible fluids. The interface between the two fluids is represented implicitly by the 0.5–level set of a function varying smoothly from zero to one. The flow of each phase is governed by the incompressible Navier–Stokes equations with no–slip boundary condition at solid walls. The key idea is to drive the contact line movement by prescribing the normal vector at the boundary to be the static normal vector corresponding to the equilibrium contact angle. Our model is formulated as a system of partial differential equations and takes into account differences in density and viscosity, gravity, and surface tension effects. We discretize using the finite element method which allows for use of the weak form of the equations. The surface tension forces are included as line integrals along the interfaces. We eliminate the second order derivatives in these forces, coming from the curvature term, by applying partial integration of the Laplace–Beltrami operator. Numerical experiments of capillary dominated flows in two space dimensions are presented and include tests of convergence rates and comparison with other numerical methods. Our numerical scheme show good accuracy and our results compare well with available theory.