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Research Papers

A Class of Conservative Phase Field Models for Multiphase Fluid Flows

[+] Author and Article Information
Jun Li

School of Mathematical Sciences and LPMC,
Nankai University,
Tianjin 300071, China
e-mail: nkjunli@gmail.com

Qi Wang

Department of Mathematics,
Interdisciplinary Mathematics Institute and
NanoCenter at USC,
University of South Carolina,
Columbia, SC 29208
School of Mathematics,
Nankai University,
Tianjin 300071, China
e-mail: qwang@math.sc.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 12, 2012; final manuscript received March 28, 2013; accepted manuscript posted May 7, 2013; published online September 16, 2013. Assoc. Editor: Nesreen Ghaddar.

J. Appl. Mech 81(2), 021004 (Sep 16, 2013) (8 pages) Paper No: JAM-12-1385; doi: 10.1115/1.4024404 History: Received August 12, 2012; Revised March 28, 2013; Accepted May 07, 2013

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.

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Grahic Jump Location
Fig. 1

Growth rates for phase field models for water and glycerin mixtures. The growth rate in the Cahn–Hilliard I model is higher than that in the Cahn–Hilliard II just like the situation for the Allen–Cahn models. The parameter values used in the figure: water density ρ1 = 1000 kg/m3, glycerin density ρ2 = 1261 kg/m3, water volume fraction ϕ0 = 0.7, ρ0 = ϕ0ρ1 + ρ2 (1 − ϕ0), η = 0.4503 Pa · s, ν = 0.4517 Pa · s, Γ1 = 10−5 kg/s2, Γ2 = 100 kg/s2m, water viscosity is 8.9 × 10−4 Pa · s, and glycerin viscosity is 1.499 Pa · s. (a) Two Allen–Cahn models, where λ1 = 10−4 kg/sm, λ2 = 10−3 kg/sm. The fastest growth rate is 446.2457104 at k = 1059.064423 in AC-I model and 446.2398776 at k = 1055.023079 in AC-II model. (b). Two Allen–Cahn models, where λ1 = 104 kg/sm, λ2 = 103 kg/sm. The fastest growth rate is 0.003655986108 at k = 3.031398953 in the AC-I model and 0.003439753578 at k = 2.940352436 in the AC-II model. The two Allen–Cahn models yield very similar growth behavior. (c) Two Cahn–Hilliard models, where λ1 = 10−4 m3s/kg, λ2 = 10−3 m3s/kg. The fastest growth rate is 403.3730366 at k = 1121.072073 in the CH-I model and 400.9478476 at k = 1124.400744 in the CH-II model. In general, we conclude that the phase field models predict quantitatively comparable results in near equilibrium dynamics in any parameter regimes.

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