  We consider the numerical approximations of a two-phase hydrodynamics coupled
phase-field model that incorporates the variable densities, viscosities and
moving contact line boundary conditions. The model is a nonlinear, coupled
system that consists of incompressible Navier--Stokes equations with the
generalized Navier boundary condition, and the Cahn--Hilliard equations with
moving contact line boundary conditions. By some subtle explicit--implicit
treatments to nonlinear terms, we develop two efficient, unconditionally energy
stable numerical schemes, in particular, a linear decoupled energy stable
scheme for the system with static contact line condition, and a nonlinear
energy stable scheme for the system with dynamic contact line condition. An
efficient spectral-Galerkin spatial discretization is implemented to verify the
accuracy and efficiency of proposed schemes. Various numerical results show
that the proposed schemes are efficient and accurate.
