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We consider the numerical approximations for a phase field model consistingof incompressible Navier--Stokes equations with a generalized Navier boundarycondition, and the Cahn-Hilliard equation with a dynamic moving contact lineboundary condition. A crucial and challenging issue for solving this modelnumerically is the time marching problem, due to the high order, nonlinear, andcoupled properties of the system. We solve this issue by developing two linear,second order accurate, and energy stable schemes based on the projection methodfor the Navier--Stokes equations, the invariant energy quadratization for thenonlinear gradient terms in the bulk and boundary, and a subtleimplicit-explicit treatment for the stress and convective terms. Thewell-posedness of the semidiscretized system and the unconditional energystabilities are proved. Various numerical results based on a spectral-Galerkinspatial discretization are presented to verify the accuracy and efficiency ofthe proposed schemes.

Efficient Second Order Unconditionally Stable Schemes for a Phase-field Moving Contact Line Model Using Invariant Energy Quadratization Approach