Energy stable numerical scheme for the viscous Cahn-Hilliard-Navier-Stokes equations with moving contact line

In the present article we study the numerics of the viscous Cahn-Hilliard-Navier-Stokes model, endowed with dynamic boundary conditions which allow us to take into account the interaction between the fluids interface and the moving walls of the physical domain. In what follows, we propose an energy stable temporal scheme for the problem and we prove the stability and the unconditional solvability of the discretization proposed. We also propose a fully discrete scheme for which we prove the stability and the unconditional solvability. Numerical simulations are presented to illustrate the theoretical results.