Cahn-Hilliard vs Singular Cahn-Hilliard Equations in Phase Field Modeling
Tianyu Zhang 1, Qi Wang 2*1 Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717-2400, USA.
2 Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.
Received 5 January 2009; Accepted (in revised version) 26 March 2009
Available online 24 August 2009
The Cahn-Hilliard equation is often used to describe evolution of phase boundaries in phase field models for multiphase fluids. In this paper, we compare the use of the Cahn-Hilliard equation (of a constant mobility) for the phase variable with that of the singular or modified Cahn-Hilliard equation (of a variable mobility) in the context of physical derivation of the transport equation and numerical simulations of immiscible binary fluids. We show numerically that (i). both equations work fine for interfaces of small to moderate curvature in short to intermediate time scales; (ii) the Cahn-Hilliard equation renders strong dissipation in simulations of small droplets leading to dissolution of small droplets into the surrounding fluid and/or absorption of small droplets by larger droplets nearby, an artifact for immiscible binary fluids; whereas, the singular Cahn-Hilliard equation can significantly reduce the numerical dissipation around small droplets to yield physically acceptable results in intermediate time scales; (iii) the size of droplets that can be simulated by the Cahn-Hilliard equations scale inversely with the strength of the mixing free energy. Since the intermediate timescale is the time scale of interest in most transient fluid simulations, the singular Cahn-Hilliard equation proves to be the more accurate phase transporting equation for immiscible binary fluids.AMS subject classifications: 65M06, 76D05, 76A05, 76T30, 76Z05, 92C05
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Key words: Cahn-Hilliard equation, phase field, finite difference method, immiscible multiphase flow, singular Cahn-Hilliard equation.
Email: email@example.com (T. Zhang), firstname.lastname@example.org (Q. Wang)