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We consider the numerical approximations of a two-phase hydrodynamics coupledphase-field model that incorporates the variable densities, viscosities andmoving contact line boundary conditions. The model is a nonlinear, coupledsystem that consists of incompressible Navier--Stokes equations with thegeneralized Navier boundary condition, and the Cahn--Hilliard equations withmoving contact line boundary conditions. By some subtle explicit--implicittreatments to nonlinear terms, we develop two efficient, unconditionally energystable numerical schemes, in particular, a linear decoupled energy stablescheme for the system with static contact line condition, and a nonlinearenergy stable scheme for the system with dynamic contact line condition. Anefficient spectral-Galerkin spatial discretization is implemented to verify theaccuracy and efficiency of proposed schemes. Various numerical results showthat the proposed schemes are efficient and accurate.

Numerical Approximations for a Phase-Field Moving Contact Line Model with Variable Densities and Viscosities