An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity

In this paper, we develop an efficient numerical method for the two phase moving contact line problem with variable density, viscosity, and slip length. The physical model is based on a phase field approach, which consists of a coupled system of the Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition [1], [2] and [5]. To overcome the difficulties due to large density and viscosity ratio, the Navier–Stokes equations are solved by a splitting method based on a pressure Poisson equation [11], while the Cahn–Hilliard equation is solved by a convex splitting method. We show that the method is stable under certain conditions. The linearized schemes are easy to implement and introduce only mild CFL time constraint. Numerical tests are carried out to verify the accuracy, stability and efficiency of the schemes. The method allows us to simulate the interface problems with extremely small interface thickness. Three dimensional simulations are included to validate the efficiency of the method.