Global Weak Solutions to a Diffuse Interface Model for Incompressible Two-Phase Flows with Moving Contact Lines and Different Densities

摘要

In this paper, we analyze a general diffuse interface model for incompressible two-phase flows with unmatched densities in a smooth bounded domain Omega subset of Rd (d=2,3). This model describes the evolution of free interfaces in contact with the solid boundary, namely the moving contact lines. The corresponding evolution system consists of a nonhomogeneous Navier-Stokes equation for the (volume) averaged fluid velocity v that is nonlinearly coupled with a convective Cahn-Hilliard equation for the order parameter phi. Due to the nontrivial boundary dynamics, the fluid velocity satisfies a generalized Navier boundary condition that accounts for the velocity slippage and uncompensated Young stresses at the solid boundary, while the order parameter fulfils a dynamic boundary condition with surface convection. We prove the existence of a global weak solution for arbitrary initial data in both two and three dimensions. The proof relies on a combination of suitable approximations and regularizations of the original system together with a novel time-implicit discretization scheme based on the energy dissipation law.