In this paper, we derive a thermodynamically consistent phase-field model for flows containing three (or more) liquid components. The model is based on a Navier-Stokes (NS) and Cahn-Hilliard system (CH) which accounts for surface tension among the different components and three-phase contact lines. We develop a stable conservative, second order accurate fully implicit discretization of the NS and three-phase (ternary) CH system. We use a nonlinear multigrid method to efficiently solve the discrete ternary CH system at the implicit time-level and then couple it to a multigrid/projection method that is used to solve the NS equation. We demonstrate convergence of our scheme numerically and perform numerical simulations to show the accuracy, flexibility, and robustness of this approach. In particular, we simulate a three interface contact angle resulting from a spreading liquid lens on an interface, a buoyancy-driven compound drop, and the Rayleigh-Taylor instability of a flow with three partially miscible components.
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Junseok Kim, John Lowengrub, Phase field modeling and simulation of three-phase flows. Interfaces Free Bound. 7 (2005), no. 4, pp. 435–466DOI 10.4171/IFB/132