Title: Linear, Second order and Unconditionally Energy Stable schemes for a phase-field moving contact line Model,
Abstract: In this paper, we consider the numerical approximations for solving a
hydrodynamics coupled phase field model consisting of incompressible
Navier-Stokes equations with generalized Navier boundary conditions, and the
Cahn-Hilliard equation with dynamic moving contact line boundary conditions.
The main challenging issue for solving this model numerically is the time
marching problem, i.e., how to develop suitable higher order temporal schemes
while preserving the unconditional energy stability at the discrete level. We
solve this issue by developing two linear, second-order schemes based on the
"Invariant Energy Quadratization" method for the nonlinear terms in the bulk
and on the boundary, the projection method for the Navier-Stokes equations, and
a subtle implicit-explicit treatment for the stress and convective terms.
Rigorous proofs of the well-posedness of the linear system and the
unconditional energy stabilities are provided. A spectral-Galerkin spatial
discretization is implemented and various numerical results are presented to
verify the second order accuracy and the efficiency of the proposed schemes.