Efficient energy stable numerical schemes for a phase field moving contact line model

In this paper, we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line. The model is a coupled system that consists of incompressible Navier-Stokes equations with a generalized Navier boundary condition and Cahn-Hilliard equation in conserved form. In both schemes the projection method is used to deal with the Navier-Stokes equations and stabilization approach is used for the non-convex Ginzburg-Landau bulk potential. By some subtle explicit-implicit treatments, we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions. An efficient spectral-Galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed schemes are very efficient and accurate.