Volume 13, Issue 4
Mass and Volume Conservation in Phase Field Models for Binary Fluids

Jie Shen, Xiaofeng Yang & Qi Wang

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

Published online: 2013-08

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  • Abstract

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(volumefractions), conservethe total volume ofeachphaseaswell asthe 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. Inoneformulationofthecompressiblemodels withinternalconstraints(model 2), energy dissipation can be clearly established. An efficient numerical method is then devised toenforcethis compressible internal constraint. Numericalsimulations inconfined 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.


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@Article{CiCP-13-1045, author = {Jie Shen, Xiaofeng Yang and Qi Wang}, title = {Mass and Volume Conservation in Phase Field Models for Binary Fluids}, journal = {Communications in Computational Physics}, year = {2013}, volume = {13}, number = {4}, pages = {1045--1065}, abstract = {

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(volumefractions), conservethe total volume ofeachphaseaswell asthe 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. Inoneformulationofthecompressiblemodels withinternalconstraints(model 2), energy dissipation can be clearly established. An efficient numerical method is then devised toenforcethis compressible internal constraint. Numericalsimulations inconfined 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.


}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.300711.160212a}, url = {http://global-sci.org/intro/article_detail/cicp/7263.html} }
TY - JOUR T1 - Mass and Volume Conservation in Phase Field Models for Binary Fluids AU - Jie Shen, Xiaofeng Yang & Qi Wang JO - Communications in Computational Physics VL - 4 SP - 1045 EP - 1065 PY - 2013 DA - 2013/08 SN - 13 DO - http://dor.org/10.4208/cicp.300711.160212a UR - https://global-sci.org/intro/cicp/7263.html KW - AB -

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(volumefractions), conservethe total volume ofeachphaseaswell asthe 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. Inoneformulationofthecompressiblemodels withinternalconstraints(model 2), energy dissipation can be clearly established. An efficient numerical method is then devised toenforcethis compressible internal constraint. Numericalsimulations inconfined 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.


Jie Shen, Xiaofeng Yang & Qi Wang. (1970). Mass and Volume Conservation in Phase Field Models for Binary Fluids. Communications in Computational Physics. 13 (4). 1045-1065. doi:10.4208/cicp.300711.160212a
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