Commun. Comput. Phys., 13 (2013), pp. 1045-1065.

Mass and Volume Conservation in Phase Field Models for Binary Fluids

Jie Shen 1, Xiaofeng Yang 2, Qi Wang 3*

1 Department of Mathematics, Purdue University, West Lafayette, IN 46907, USA.
2 Department of Mathematics and NanoCenter at USC, University of South Carolina, Columbia, SC 29028, USA
3 Department of Mathematics and NanoCenter at USC, University of South Carolina, Columbia, SC 29028, USA; School of Mathematics, Nankai University, Tianjin 300071, P.R. China; and Beijing Computational Science Research Center, Beijing 100084, P.R. China.

Received 30 July 2011; Accepted (in revised version) 16 February 2012
Available online 21 September 2012


The commonly used incompressible phase field models for non-reactive, binary fluids, in which the Cahn-Hilliard equation is used for the transport of phase variables (volume fractions), conserve the total volume of each phase as well as the material volume, but do not conserve the mass of the fluid mixture when densities of two components are different. In this paper, we formulate the phase field theory for mixtures of two incompressible fluids, consistent with the quasi-compressible theory [28], to ensure conservation of mass and momentum for the fluid mixture in addition to conservation of volume for each fluid phase. In this formulation, the mass-average velocity is no longer divergence-free (solenoidal) when densities of two components in the mixture are not equal, making it a compressible model subject to an internal constraint. In one formulation of the compressible models with internal constraints (model 2), energy dissipation can be clearly established. An efficient numerical method is then devised to enforce this compressible internal constraint. Numerical simulations in confined geometries for both compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions. Numerical comparisons show that (a) predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin; and (b) predictions differ significantly in binary fluid mixtures undergoing mixing with a large mixing zone. The numerical study delineates the limitation of the commonly used incompressible phase field model using volume fractions and thereby cautions its predictive value in simulating well-mixed binary fluids.

AMS subject classifications: 65M70, 76M22, 76T99

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Key words: Phase field model, compressibility, multiphase fluid flows, spectral methods.

*Corresponding author.
Email: (J. Shen), (X. Yang), (Q. Wang)

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