On a thermodynamically consistent diffuse interface model for two-phase incompressible flows with non-matched densities: Dynamics of moving contact lines, surface diffusion, and mass transfer

Abstract

We examine a thermodynamically consistent diffuse interface model for two-phase incompressible viscous flows in a smooth bounded domain $\Omega \subset R^d$ ($d\in \{2,3\}$). This model characterizes the evolution of free interfaces in contact with the solid boundary, specifically addressing the phenomenon of moving contact lines. The associated evolution system comprises a nonhomogeneous Navier–Stokes equation for the (volume) averaged fluid velocity v, nonlinearly coupled with a convective Cahn–Hilliard equation governing the order parameter φ. Notably, for the boundary dynamics, the current model incorporates surface diffusion, a variable contact angle between the diffuse interface and the solid boundary, as well as mass transfer between bulk and surface. This material transfer adheres to a mass conservation law encompassing both bulk and surface contributions. In the general scenario of non-matched densities, we establish the existence of global weak solutions with finite energy in both two and three dimensions.