On the Consistency of Dynamic Wetting Boundary Conditions for the Navier-Stokes-Cahn-Hilliard Equations

We investigate the limiting behavior of the Navier-Stokes-Cahn-Hilliard model for binary-fluid flows as the diffuse-interface thickness passes to zero, in the presence of fluid-fluid-solid contact lines. Allowing for motion of such contact lines relative to the solid substrate is required to adequately model multi-phase and multi-species fluid transport past and through solid media. Even though diffuse-interface models provide an inherent slip mechanism through the mobility-induced diffusion, this slip vanishes as the interface thickness and mobility parameter tend to zero in the so-called sharp-interface limit. The objective of this work is to present dynamic wetting and generalized Navier boundary conditions for diffuse-interface models that are consistent in the sharp-interface limit. We concentrate our analysis on the prototypical binary-fluid Couette-flow problems. To verify the consistency of the diffuse-interface model in the limit of vanishing interface thickness, we provide reference limit solutions of a corresponding sharp-interface model. For parameter values both at and away from the critical viscosity ratio, we present and compare the results of both the diffuse- and sharp-interface models. The close match between both model results indicates that the considered test case lends itself well as a benchmark for further research.