## Mass-Conserved Phase Field Models for Binary Fluids

**A 2011 Preprint by
J. Shen,
X. Yang, and
Q. Wang
**

- 2011:05
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, conserve the total volume of each phase as well as the material volume, but do not conserve the mass of the fluid mixture when the 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 the conservation of mass and momentum for the fluid mixture as well as the volume for each fluid phase. In this formulation, the mass-average velocity is no longer divergence-free (solenoidal) when the densities of two components in the mixture are not equal, making it a compressible model subject to an internal constraint. An efficient numerical method is then devised to enforce this compressible internal constraint. Numerical simulations in confined geometries for both the compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions. Numerical comparisons show that (a) the predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin; and (b) the 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 and thereby cautions its predictive value in simulating well-mixed binary fluids.