Analysis of a Phase-Field Model for Two-Phase Compressible Fluids

Feireisl, Eduard; Petzeltová, Hana; Rocca, Elisabetta; Schimperna, Giulio

doi:10.1142/s0218202510004544

Cited by 109 publications

(62 citation statements)

References 20 publications

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“…Anderson et al proposed a slightly different approach corresponding to δ ( ϱ ) = 1 in . Such a hypothesis gives rise to a model that is mathematically tractable; see Abels and Feireisl and Feireisl et al In this present paper, we proposed a variant of Blesgen approach adapted to the ansatz δ = 1. Specifically, we consider the following system of equations:

\partial_{t} ϱ + {div}_{x} false(ϱ boldu false) = 0,

\partial_{t} false(ϱ boldu false) + {div}_{x} false(ϱ boldu \otimes boldu false) + \nabla_{x} p false(ϱ, c false) = {div}_{x} double-struckS false(\nabla_{x} boldu false) - {div}_{x} ((), \nabla_{x} c \otimes \nabla_{x} c - \frac{1}{2} false| \nabla_{x} c |^{2} double-struckI),

\partial_{t} c + boldu \cdot \nabla_{x} c = Δ_{x} c - ϱ \frac{\partial f false(ϱ, c false)}{\partial c} .

The viscous stress tensor

double-struckS

is given by the standard Newton rheological law,

S (\nabla_{x} u) = ν (\nabla_{x} {boldu}^{t} + \nabla_{x}^{t} u - \frac{2}{N} {div}_{x} u I) + λ {div}_{x} u I, ν > 0,

while the pressure p = p ( ϱ , c ) is derived from the free energy,

p (ϱ, c) = ϱ^{2} \partial f (<...>

…”

Section: Introductionsupporting

confidence: 55%

Relative energy approach to a diffuse interface model of a compressible two‐phase flow

Feireisl

Petcu

Pražák

2019

Math Methods in App Sciences

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We propose a simple model for a two‐phase flow with a diffuse interface. The model couples the compressible Navier‐Stokes system governing the evolution of the fluid density and the velocity field with the Allen‐Cahn equation for the order parameter. We show that the model is thermodynamically consistent, in particular, a variant of the relative energy inequality holds. As a corollary, we show the weak‐strong uniqueness principle, meaning any weak solution coincides with the strong solution emanating from the same initial data on the life span of the latter. Such a result plays a crucial role in the analysis of the associated numerical schemes. Finally, we perform the low Mach number limit obtaining the standard incompressible model.

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\partial_{t} ϱ + {div}_{x} false(ϱ boldu false) = 0,

\partial_{t} false(ϱ boldu false) + {div}_{x} false(ϱ boldu \otimes boldu false) + \nabla_{x} p false(ϱ, c false) = {div}_{x} double-struckS false(\nabla_{x} boldu false) - {div}_{x} ((), \nabla_{x} c \otimes \nabla_{x} c - \frac{1}{2} false| \nabla_{x} c |^{2} double-struckI),

\partial_{t} c + boldu \cdot \nabla_{x} c = Δ_{x} c - ϱ \frac{\partial f false(ϱ, c false)}{\partial c} .

The viscous stress tensor

double-struckS

is given by the standard Newton rheological law,

S (\nabla_{x} u) = ν (\nabla_{x} {boldu}^{t} + \nabla_{x}^{t} u - \frac{2}{N} {div}_{x} u I) + λ {div}_{x} u I, ν > 0,

while the pressure p = p ( ϱ , c ) is derived from the free energy,

p (ϱ, c) = ϱ^{2} \partial f (<...>

…”

Section: Introductionsupporting

confidence: 55%

Relative energy approach to a diffuse interface model of a compressible two‐phase flow

Feireisl

Petcu

Pražák

2019

Math Methods in App Sciences

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“…see [22,1,11]. Other models include [26], which proposes the Cahn-Hilliard type model under a gravitational eld.…”

Section: Introductionmentioning

confidence: 99%

Diffuse interface surface tension models in an expanding flow

Liu

Bertozzi

Kolokolnikov

2012

Communications in Mathematical Sciences

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We consider a diusive interface surface tension model under compressible ow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity eld. We prove that both model problems are well-posed. We are especially interested in the behavior of solutions with respect to droplet breakup phenomena. Numerical simulations of 1,2 and 3D all illustrate that the Cahn-Hilliard model is much more eective for droplet breakup. Using asymptotic methods we correctly predict the breakup condition for the Cahn-Hilliard model. Moreover, we prove that the Allen-Cahn model will not break up under certain circumstances due to a maximum principle. BackgroundThere is a need to develop simple computational models for surface tension in the droplet breakup phenomena. As an example, consider a piece of material that expands under a sudden pulse of energy that comes from laser fusion [21] or heavy ion fusion [12]. The material will breakup, and surface tension plays an important role in the ensuing dynamics. There are many numerical methods that deal with surface tension in twophase uids. This problem is known for its computational stiness. It contains two dierent time scales, the small surface tension time scale and the convection time scale. Three main algorithms exist for two-phase uids. The sharp interface method tracks the interface explicitly, yet it requires extensive processing when the interface splits and merges. Since droplet breakup involves mainly merging and splitting of the interface, we * W. Liu and A. Bertozzi are with the

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“…Existence of a weak solution for compressible Allen-Cahn-Navier-Stokes systems has been recently proven in [28]. This paper is devoted to the analysis of the global dynamics of solutions to system (1.1)-(1.4) in the 3D case and, in contrast with most of the quoted papers, here we allow the presence of a time-dependent external nongradient force (see, e.g., [8] for its role in coarsening processes).…”

Section: Introductionmentioning

confidence: 99%

Trajectory attractors for binary fluid mixtures in 3D

Gal

Grasselli

2010

Chin. Ann. Math. Ser. B

123

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Two different models for the evolution of incompressible binary fluid mixtures in a three-dimensional bounded domain are considered. They consist of the 3D incompressible Navier-Stokes equations, subject to time-dependent external forces and coupled with either a convective Allen-Cahn or Cahn-Hilliard equation. Such systems can be viewed as generalizations of the Navier-Stokes equations to two-phase fluids. Using the trajectory approach, the authors prove the existence of the trajectory attractor for both systems.

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Analysis of a Phase-Field Model for Two-Phase Compressible Fluids

Cited by 109 publications

References 20 publications

Relative energy approach to a diffuse interface model of a compressible two‐phase flow

Relative energy approach to a diffuse interface model of a compressible two‐phase flow

Diffuse interface surface tension models in an expanding flow

Trajectory attractors for binary fluid mixtures in 3D

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