Modeling and Numerical Approximation of Two-Phase Incompressible Flows by a Phase-Field Approach

Shen, Jie

doi:10.1142/9789814360906_0003

Cited by 57 publications

(49 citation statements)

References 70 publications

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“…However, it is easy to show that d and φ are the unique minimizer of a convex functional (cf., for instance, [44,33]). …”

Section: We Deal With Terms a B C D As Follows For Term A We Appmentioning

confidence: 99%

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Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

Shen¹,

Yang²

2014

SIAM J. Sci. Comput.

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105

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Abstract. We consider in this paper numerical approximations of phase-field models for twophase complex fluids. We first reformulate the phase-field models derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish their energy laws. Then, we construct two classes, stabilized and convex-splitting, of decoupled time discretization schemes for the coupled nonlinear systems. These schemes are unconditionally energy stable and lead to decoupled elliptic equations to solve at each time step. Furthermore, these elliptic equations are linear for the stabilized version. Stability analysis and ample numerical simulations are presented.Key words. phase-field, liquid crystals, multiphase flows, Navier-Stokes, Allen-Cahn, CahnHilliard, stability AMS subject classifications. 65M12, 65M70, 65Z05 DOI. 10.1137/130921593 1. Introduction. Complex fluids have great practical utility since the microstructure can be manipulated to produce useful mechanical, optical, or thermal properties [4,23]. Numerical studies of multiphase complex fluids is a rich area for research with important applications that can be addressed. Some numerical difficulties are (i) the moving interfaces between various components; (ii) open issues regarding wellposedness of many multiphase models leading to confusion as to whether numerical pathologies are due to the model or the methods; and most important, (iii) nonlinear coupling (hydrodynamics, interface, and microstructure) makes it very difficult to design easy-to-implement, energy stable numerical schemes.For the first two issues above, the diffuse-interface/phase-field models, whose origin can be traced back to [38] and [46], have been proved efficient with much success. A particular advantage of the phase-field approach is that models can often be derived from an energy-based variational formalism (energetic variational approaches), leading to well-posed nonlinear coupled systems that satisfy thermodynamics-consistent energy dissipation laws. This makes it possible to carry out mathematical analysis and design numerical schemes which satisfy a corresponding discrete energy dissipation law. In recent years, the phase-field method has become one of the major tools to study a variety of interfacial phenomena (cf. [17,2,36,19,32,49] and recent review papers [44,22] and the references therein).For phase-field models which satisfy a dissipative energy law, it is especially desirable to design numerical schemes that preserve the energy dissipation law at the

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“…However, it is easy to show that d and φ are the unique minimizer of a convex functional (cf., for instance, [44,33]). …”

Section: We Deal With Terms a B C D As Follows For Term A We Appmentioning

confidence: 99%

“…In recent years, the phase-field method has become one of the major tools to study a variety of interfacial phenomena (cf. [17,2,36,19,32,49] and recent review papers [44,22] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

Shen¹,

Yang²

2014

SIAM J. Sci. Comput.

Self Cite

105

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show abstract

“…A particular advantage of the phase-field approach is that the derived models are usually well-posed nonlinear partial differential equations that satisfy thermodynamics-consistent energy dissipation laws, which makes it possible to carry out mathematical analysis and further design numerical schemes which satisfy corresponding discrete energy dissipation laws. Thus phase field models recently have been the subject of many theoretical and numerical investigations (cf., for instance, [5,8,[12][13][14]19,21,24,31,33,[35][36][37][38]43]). …”

Section: Introductionmentioning

confidence: 99%

Efficient energy stable numerical schemes for a phase field moving contact line model

Shen

Yang²,

Yu³

2015

Journal of Computational Physics

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109

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In this paper, we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line. The model is a coupled system that consists of incompressible Navier-Stokes equations with a generalized Navier boundary condition and Cahn-Hilliard equation in conserved form. In both schemes the projection method is used to deal with the Navier-Stokes equations and stabilization approach is used for the non-convex Ginzburg-Landau bulk potential. By some subtle explicit-implicit treatments, we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions. An efficient spectral-Galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed schemes are very efficient and accurate.

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“…The last velocity interface boundary condition is exactly the Beavers-Joseph-SaffmanJones interface boundary condition [10,12,14,15,60,61,62,63,64,65] with the slip coefficient β equal to the Beavers-Joseph-Saffman-Jones coefficient α BJSJ . The Cahn-Hilliard-Stokes system can be viewed as the low Reynolds number approximation of the better-known Cahn-Hilliard-Navier-Stokes system for two phase flow [28,35,34,36,40,66,67,68,69,70]. The derivation above indicates that the interface boundary conditions (except for the three obtained via conservation of mass consideration) are in fact variational interface boundary conditions.…”

Section: Application Of Onsager's Extremum Principlementioning

confidence: 99%

Two‐phase flows in karstic geometry

Han

Sun

Wang

2013

Math Methods in App Sciences

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Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil and gas), cloud and fog (water vapor, water and air). Multiphase flows also play an important role in many engineering and environmental science applications.In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. Despite the importance of the subject, little work has been done on multi-phase flows in karstic geometry.In this paper we present a family of phase field (diffusive interface) models for two phase flow in karstic geometry. These models together with the associated interface boundary conditions are derived utilizing Onsager's extremum principle. The models derived enjoy physically important energy laws. Uniquely solvable first and second order in time numerical schemes that preserve the associated energy law are presented as well.

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Modeling and Numerical Approximation of Two-Phase Incompressible Flows by a Phase-Field Approach

Cited by 57 publications

References 70 publications

Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

Efficient energy stable numerical schemes for a phase field moving contact line model

Two‐phase flows in karstic geometry

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