Cahn-Hilliard vs Singular Cahn-Hilliard Equations in Phase Field Modeling (Communications in Computational Physics, 2018)
The Cahn-Hilliard equation is often used to describe evolution of phaseboundaries in phase field models for multiphase fluids. In this paper, we comparethe use of the Cahn-Hilliard equation (of a constant mobility) for the phase variablewith that of the singular or modified Cahn-Hilliard equation (of a variable mobility) inthe context of physical derivation of the transport equation and numerical simulationsof immiscible binary fluids. We show numerically that (i). both equations work finefor interfaces of small to moderate curvature in short to intermediate time scales; (ii)the Cahn-Hilliard equation renders strong dissipation in simulations of small dropletsleading to dissolution of small droplets into the surrounding fluid and/or absorptionof small droplets by larger droplets nearby, an artifact for immiscible binary fluids;whereas, the singular Cahn-Hilliard equation can significantly reduce the numericaldissipation around small droplets to yield physically acceptable results in intermediatetime scales; (iii) the size of droplets that can be simulated by the Cahn-Hilliardequations scale inversely with the strength of the mixing free energy. Since the intermediatetimescale is the time scale of interest in most transient fluid simulations, thesingular Cahn-Hilliard equation proves to be the more accurate phase transportingequation for immiscible binary fluids.