An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

Minjeaud, Sébastian

doi:10.1002/num.21721

Cited by 105 publications

(60 citation statements)

References 26 publications

(40 reference statements)

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“…The above scheme is constructed by combining several effective approaches in approximation of Allen-Cahn equation [42], Navier-Stokes equations [16], and phasefield models [43,5,37].…”

Section: Where H(d) Is the Hessian Matrix Of G(d)mentioning

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.

106

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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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“…The above scheme is constructed by combining several effective approaches in approximation of Allen-Cahn equation [42], Navier-Stokes equations [16], and phasefield models [43,5,37].…”

Section: Where H(d) Is the Hessian Matrix Of G(d)mentioning

confidence: 99%

“…• Inspired by [5,37], which deal with a phase-field model of three-phase Newtonian fluids, we introduce new, explicit, convective velocities u n and u n in the phase equation and director equation. u n and u n can be computed directly from (3.22) and (3.24), i.e.,…”

Section: Where H(d) Is the Hessian Matrix Of G(d)mentioning

confidence: 99%

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

Shen¹,

Yang²

2014

SIAM J. Sci. Comput.

106

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“…a These schemes have been extended to NSCH systems with matched densities 31,39 and for quasiincompressible NSCH systems with a solenoidal mixture-velocity field. 15,22,44,47,48 However, non-solenoidal quasi-incompressible NSCH systems have only received scant consideration so far. The reason for this is that existing techniques for solenoidal systems can not be straightforwardly extended to non-solenoidal systems (which apart from being non-solenoidal also have auxiliary pressure terms).…”

Section: Lowengrub and Truskinovskymentioning

confidence: 99%

Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

Roudbari

Şimşek

Brummelen

et al. 2018

Math. Models Methods Appl. Sci.

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While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier-Stokes Cahn-Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman-Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with nonsolenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and break-up dynamics of droplets including pinching due to gravity.

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“…The main ingredient for the presented partially decoupled scheme is the choice of the transport velocities u φ and u ωi,M (i = 1, ..., M ). Using the idea presented in [6], [7] and [8], we update u n by a discrete time integration of the force densities. Hence we are able to postpone the computation of u n+1 until the very end without losing stability.…”

Section: Discrete Schemementioning

confidence: 99%

On numerical schemes for phase‐field models for electrowetting with electrolyte solutions

Metzger

2015

Proc Appl Math and Mech

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We present an energy-stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two-phase flow with general mass densities. This model couples momentum and Cahn-Hilliard type phase-field equations with Nernst-Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential. The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase-field equation, the equations for ion transport and electrostatic potential, and the Navier-Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme.

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An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

Cited by 105 publications

References 26 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

Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

On numerical schemes for phase‐field models for electrowetting with electrolyte solutions

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