Hydrodynamic phase field models for multiphase fluids formulated using volume fractions of incompressible fluid components do not normally conserve mass. In this paper, we formulate phase field theories for mixtures of multiple incompressible fluids, using volume fractions, to ensure conservation of mass and momentum for the fluid mixture as well as the total volume for each fluid phase. In this formulation, the mass-average velocity is nonsolenoidal when the densities of incompressible fluid components in the mixture are not equal, making it a bona fide compressible model subject to an internal constraint. Derivation of mass conservation and energy dissipation in phase field models based on both Allen–Cahn dynamics and Cahn–Hilliard dynamics are presented. One salient feature of the phase field models is that the hydrostatic pressure is coupled with the transport of the volume fractions making the momentum transport and the volume fraction transport fully coupled in light of the mass conservation. Near equilibrium dynamics are explored using a linear analysis. In the case of binary fluid mixtures, one potential growth mode is identified in all the models for a class of free energy, which has been adopted for multiphase fluids. The growth is either absent for all waves or of a longwave feature.

References

1.
Du
,
Q.
,
Liu
,
C.
,
Ryham
,
R.
, and
Wang
,
X.
,
2005
, “
Phase Field Modeling of the Spontaneous Curvature Effect in Cell Membranes
,”
Comm. Pure Appl. Anal.
,
4
, pp.
537
548
.10.3934/cpaa.2005.4.537
2.
Du
,
Q.
,
Liu
,
C.
,
Ryham
,
R.
, and
Wang
,
X.
,
2005
, “
A Phase Field Formulation of the Willmore Problem
,”
Nonlinearity
,
18
, pp.
1249
1267
.10.1088/0951-7715/18/3/016
3.
Du
,
Q.
,
Liu
,
C.
, and
Wang
,
X.
,
2004
, “
A Phase Field Approach in the Numerical Study of the Elastic Bending Energy for Vesicle Membranes
,”
J. Comput. Phys.
,
198
, pp.
450
468
.10.1016/j.jcp.2004.01.029
4.
Du
,
Q.
,
Liu
,
C.
, and
Wang
,
X.
,
2005
, “
Retrieving Topological Information for Phase Field Models
,”
SIAM J. Appl. Math.
,
65
, pp.
1913
1932
.10.1137/040606417
5.
Du
,
Q.
,
Liu
,
C.
, and
Wang
,
X.
,
2005
, “
Simulating the Deformation of Vesicle Membranes Under Elastic Bending Energy in Three Dimensions
,”
J. Comput. Phys.
,
212
, pp.
757
777
.10.1016/j.jcp.2005.07.020
6.
Shen
,
J.
, and
Yang
,
X.
,
2009
, “
An Efficient Moving Mesh Spectral Method for the Phase-Field Model of Two Phase Flows
,”
J. Comput. Phys.
,
228
, pp.
2978
2992
.10.1016/j.jcp.2009.01.009
7.
Shen
,
J.
, and
Yang
,
X.
,
2010
, “
Energy Stable Schemes for Cahn-Hilliard Phase-Field Model of Two-Phase Incompressible Flows
,”
Chin. Ann. Math., Ser. B
,
31
, pp.
743
758
.10.1007/s11401-010-0599-y
8.
Shen
,
J.
, and
Yang
,
X.
,
2010
, “
A Phase-Field Model and Its Numerical Approximation for Two-Phase Incompressible Flows With Different Densities and Viscositites
,”
SIAM J. Sci. Comput. (USA)
,
32
, pp.
1159
1179
.10.1137/09075860X
9.
Feng
,
J. J.
,
Liu
,
C.
,
Shen
,
J.
, and
Yue
,
P.
,
2005
, “
Transient Drop Deformation Upon Startup of Shear in Viscoelastic Fluids
,”
Phys. Fluids
,
17
, p.
123101
.10.1063/1.2139630
10.
Hua
,
J.
,
Lin
,
P.
,
Liu
,
C.
, and
Wang
,
Q.
,
2011
, “
Energy Law Preserving C0 Finite Element Schemes for Phase Field Models in Two-Phase Flow Computations
,”
J. Comput. Phys.
,
230
, pp.
7115
7131
.10.1016/j.jcp.2011.05.013
11.
Liu
,
C.
, and
Walkington
,
N. J.
,
2001
, “
An Eulerian Description of Fluids Containing Visco-Hyperelastic Particles
,”
Arch. Ration. Mech. Anal.
,
159
, pp.
229
252
.10.1007/s002050100158
12.
Lowengrub
,
J.
, and
Truskinovsky
,
L.
,
1998
, “
Quasi-Incompressible Cahn–Hilliard Fluids and Topological Transitions
,”
Proc. R. Soc. London, Ser. A
,
454
, pp.
2617
2654
.10.1098/rspa.1998.0273
13.
Liu
,
C.
, and
Shen
,
J.
,
2003
, “
A Phase Field Model for the Mixture of Two Incompressible Fluids and Its Approximation by a Fourier-Spectral Method
,”
Physica D
,
179
, pp.
211
228
.10.1016/S0167-2789(03)00030-7
14.
Wise
,
S. M.
,
Lowengrub
,
J. S.
,
Kim
,
J. S.
, and
Johnson
,
W. C.
,
2004
, “
Efficient Phase-Field Simulation of Quantum Dot Formation in a Strained Heteroepitaxial Film
,”
Superlattices Microstruct.
,
36
, pp.
293
304
.10.1016/j.spmi.2004.08.029
15.
Wheeler
,
A.
,
McFadden
,
G.
, and
Boettinger
,
W.
,
1996
, “
Phase-Field Model for Solidification of a Eutectic Alloy
,”
Proc. R. Soc. London, Ser. A
,
452
, pp.
495
525
.10.1098/rspa.1996.0026
16.
Yang
,
X.
,
Feng
,
J.
,
Liu
,
C.
, and
Shen
,
J.
,
2006
, “
Numerical Simulations of Jet Pinching-Off and Drop Formation Using an Energetic Variational Phase-Field Method
,”
J. Comput. Phys.
,
218
, pp.
417
428
.10.1016/j.jcp.2006.02.021
17.
Yue
,
P.
,
Feng
,
J. J.
,
Liu
,
C.
, and
Shen
,
J.
,
2004
, “
A Diffuse-Interface Method for Simulating Two-Phase Flows of Complex Fluids
,”
ASME J. Fluid Mech.
,
515
, pp.
293
317
.10.1017/S0022112004000370
18.
Yue
,
P.
,
Feng
,
J. J.
,
Liu
,
C.
, and
Shen
,
J.
,
2005
, “
Diffuse-Interface Simulations of Drop Coalescence and Retraction in Viscoelastic Fluids
,”
J. Non-Newtonian Fluid Mech.
,
129
, pp.
163
176
.10.1016/j.jnnfm.2005.07.002
19.
Zhang
,
T. Y.
,
Cogan
,
N.
, and
Wang
,
Q.
,
2008
, “
Phase Field Models for Biofilms. II. 2-D Numerical Simulations of Biofilm-Flow Interaction
,”
Comm. Comp. Phys.
,
4
(
1
), pp.
72
101
.
20.
Kim
,
J.
, and
Lowengrub
,
J.
,
2005
, “
Phase Field Modeling and Simulation of Three-Phase Flows
,”
Interface Free Boundaries
,
7
, pp.
435
466
.10.4171/IFB/132
21.
Hobayashi
,
R
.,
1993
, “
Modeling and Numerical Simulations of Dendritic Crystal Growth
,”
Physica D
,
63
, pp.
410
423
.10.1016/0167-2789(93)90120-P
22.
McFadden
,
G.
,
Wheeler
,
A.
,
Braun
,
R.
,
Coriell
,
S.
, and
Sekerka
,
R.
,
1998
, “
Phase-Field Models for Anisotropic Interfaces
,”
Phys. Rev. E
,
48
, pp.
2016
2024
.10.1103/PhysRevE.48.2016
23.
Karma
,
A.
, and
Rappel
,
W.
,
1999
, “
Phase-Field Model of Dendritic Sidebranching With Thermal Noise
,”
Phys. Rev. E
,
60
, pp.
3614
3625
.10.1103/PhysRevE.60.3614
24.
Chen
,
L. Q.
, and
Yang
,
W.
,
1994
, “
Computer Simulation of the Dynamics of a Quenched System With Large Number of Non-Conserved Order Parameters
,”
Phys. Rev. B
,
50
, pp.
15752
15756
.10.1103/PhysRevB.50.15752
25.
Chen
,
L. Q.
,
2002
, “
Phase-Field Modeling for Microstructure Evolution
,”
Annu. Rev. Mater. Res.
,
32
, pp.
113
140
.10.1146/annurev.matsci.32.112001.132041
26.
Li
,
Y.
,
Hu
,
S.
,
Liu
,
Z.
, and
Chen
,
L.
,
2001
, “
Phase-Field Model of Domain Structures in Ferroelectric Thin Films
,”
Appl. Phys. Lett.
,
78
, pp.
3878
3880
.10.1063/1.1377855
27.
Seol
,
D. J.
,
Hu
,
S. Y.
,
Li
,
Y. L.
,
Shen
,
J.
,
Oh
,
K. H.
, and
Chen
,
L. Q.
,
2003
, “
Three-Dimensional Phase-Field Modeling of Spinodal Decomposition in Constrained Films
,”
Acta Mater.
,
51
, pp.
5173
5185
.10.1016/S1359-6454(03)00378-1
28.
Lu
,
W.
, and
Suo
,
Z.
,
2001
, “
Dynamics of Nanoscale Pattern Formation of an Epitaxial Monolayer
,”
J. Mech. Phys. Solids
,
49
, pp.
1937
1950
.10.1016/S0022-5096(01)00023-0
29.
Chen
,
L. Q.
, and
Wang
,
Y.
,
1996
, “
The Continuum Field Approach to Modeling Microstructural Evolution
,”
JOM
,
48
, pp.
13
18
.10.1007/BF03223259
30.
Wang
,
Y.
, and
Chen
,
C. L.
,
1999
, “
Simulation of Microstructure Evolution
,”
Methods in Materials Research
,
John Wiley and Sons, New York
, pp.
2a3.1
2a3.23
.
31.
Tadmor
,
E.
,
Phillips
,
R.
, and
Ortiz
,
M.
,
1996
, “
Mixed Atomistic and Continuum Models of Deformation in Solids
,”
Langmuir
,
12
, pp.
4529
4534
.10.1021/la9508912
32.
Forest
,
M. G.
,
Liao
,
Q.
, and
Wang
,
Q.
,
2010
, “
2-D Kinetic Theory for Polymer Particulate Nanocomposites
,”
Comm. Comp. Phys.
,
7
, pp.
250
282
.10.4208/cicp.2009.08.204
33.
Zhang
,
T. Y.
, and
Wang
,
Q.
,
2010
, “
Cahn–Hilliard vs Singular Cahn–Hilliard Equations in Phase Field Modeling
,”
Comm. Comp. Phys.
,
7
, pp.
362
382
.10.4208/cicp.2009.09.016
34.
Lindley
,
B.
,
Wang
,
Q.
, and
Zhang
,
T.
,
2011
, “
Multicomponent Models for Biofilm Flows
,”
Discrete Contin. Dyn. Syst., Ser. B
,
15
, pp.
417
456
.10.3934/dcdsb.2011.15.417
35.
Cahn
,
J. W.
, and
Hilliard
,
J. E.
,
1959
, “
Free Energy of a Nonuniform System. I. Interfacial Free Energy
,”
J. Chem. Phys.
,
28
, pp.
258
267
.10.1063/1.1744102
36.
Cahn
,
J. W.
, and
Hilliard
,
J. E.
,
1959
, “
Free Energy of a Nonuniform System-III: Nucleation in a 2-Component Incompressible Fluid
,”
J. Chem. Phys.
,
31
, pp.
688
699
.10.1063/1.1730447
37.
Flory
,
P. J.
,
1953
,
Principles of Polymer Chemistry
,
Cornell University
,
Ithaca, NY
.
38.
Doi
,
M.
,
1996
,
Introduction to Polymer Physics
,
Clarendon Press
,
Oxford, UK
.
39.
Coates
,
D
.,
1995
, “
Polymer-Dispersed Liquid Crystals
,”
J. Mater. Chem.
,
5
, pp.
2063
2072
.10.1039/jm9950502063
40.
Larson
,
R. G.
,
1999
,
The Structure and Rheology of Complex Fluids
,
Oxford University
,
New York
.
41.
Shen
,
J.
,
Yang
,
X.
, and
Wang
,
Q.
,
2013
, “
Mass Conserved Phase Field Model for Binary Fluids
,”
Comm. Comp. Phys.
,
13
, pp.
1045
1065
.10.4208/cicp.300711.160212a
42.
Bird
,
R. B.
,
Stewart
,
W. E.
, and
Lightfoot
,
E. N.
,
2002
,
Transport Phenomena
,
Wiley
,
New York
.
43.
Beris
,
A. N.
, and
Edwards
,
B.
,
1994
,
Thermodynamics of Flowing Systems
,
Oxford University
,
Oxford, UK
.
44.
Probstein
,
R. F.
,
1994
,
Physicochemical Hydrodynamics
,
Wiley
,
New York
.
45.
Zhang
,
T. Y.
,
Cogan
,
N.
, and
Wang
,
Q.
,
2008
, “
Phase Field Models for Biofilms. I. Theory and 1-D Simulations
,”
SIAM J. Appl. Math.
,
69
(
3
), pp.
641
669
.10.1137/070691966
You do not currently have access to this content.