On imposing dynamic contact-angle boundary conditions for wall-bounded liquid–gas flows

We present an efficient scheme within the phase field framework for imposing dynamic contact angle boundary conditions for wall-bounded flows of two immiscible incompressible fluids with large density ratios. First, we develop an algorithm for imposing the dynamic contact angle boundary conditions to the Cahn–Hilliard equation. Our algorithm consists of two components: (i) we ignore the boundary conditions and transform the Cahn–Hilliard equation into two nominally de-coupled Helmholtz type equations; (ii) we treat the dynamic contact angle boundary conditions in such a manner that the two Helmholtz-type equations are truly de-coupled. Then, we combine this algorithm, together with a scheme for variable-density Navier–Stokes equations we developed recently, to form an efficient method for the coupled system of Navier–Stokes and Cahn–Hilliard equations for contact line problems involving large density ratios. The overall method can deal with moving contact lines under dynamic and also static contact angle boundary conditions. It is endowed with several attractive features that make the method very efficient. In particular, computations for all flow variables are completely decoupled. The resultant linear algebraic systems after discretization for all flow variables involve only constant and time-independent coefficient matrices, which can be pre-computed during pre-processing, even though the coupled Navier–Stokes/Cahn–Hilliard system involves variable density and variable viscosity. Ample numerical simulations of wall-bounded air/water two-phase flows have been presented to demonstrate the capability of the method for dealing with contact line problems under dynamic and static contact-angle conditions involving large density ratios.