A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow

Garcke, Harald; Hinze, Michael; Kahle, Christian

doi:10.1016/j.apnum.2015.09.002

Cited by 57 publications

(89 citation statements)

References 38 publications

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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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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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“…The system is described by a velocity field v, a phase field ϕ, and a chemical potential µ for the two-phase structure. The time discrete system under investigation is given by the following equations, see [2]: Two-step scheme for m > 1:…”

Section: The Time Discrete Modelmentioning

confidence: 99%

“…In [1] a thermodynamically consistent model for two-phase flow using a phase field approach is developed for which in [2] an energy stable time discretization is proposed. Here we consider the optimal control of this time discrete system using either Dirichlet boundary control or volume forces, where the control aim is to achieve a desired distribution of the two phases.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of time discrete two‐phase flow governed by a diffuse interface model

Garcke

Hinze

Kahle

2016

Proc Appl Math and Mech

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We present results on optimal control of two-phase flows. The fluid is modeled by a thermodynamically consistent diffuse interface model and allows to treat fluids of different densities and viscosities. In earlier work we proposed an energy stable time discretization for this model that we now employ to derive existence of optimal controls for a time discrete optimal control problem. The control aim is to obtain a desired distribution of the two phases in the system. For this we investigate three control actions. We use tangential Dirichlet boundary control and distributed control. We further consider the inverse problem of finding an initial distribution such that the evolution over a given time horizon starting from this value is close to a desired distribution.

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“…Convergence of numerical approximations for density independent models of two-phase ows has been shown in [16,21]. A thermodynamically consistent phase-eld model for two-phase ows has been proposed in [1]; energy preserving numerical approximations for that model have been considered in [20,17], while convergence of a numerical approximation is shown in [18], [19].…”

Section: Introductionmentioning

confidence: 99%

Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow

Baňas

Nürnberg

2017

ESAIM: M2AN

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Abstract. We present a phase-eld model for multiphase ow for an arbitrary number of immiscible incompressible uids with variable densities and viscosities. The model consists of a system of the Navier-Stokes equations coupled to multicomponent Cahn-Hilliard variational inequalities. The proposed formulation admits a natural energy law, preserves physically meaningful constraints and allows for a straightforward modelling of surface tension eects. We propose a practical fully discrete nite element approximation of the model which preserves the energy law and the associated physical constraints. In the case of matched densities we prove convergence of the numerical scheme towards a weak solution of the continuous model. The convergence of the numerical approximations also implies the existence of weak solutions. Furthermore, we propose a convergent iterative xed-point algorithm for the solution of the discrete nonlinear system of equations and present several computational studies of the proposed model.

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A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow

Cited by 57 publications

References 38 publications

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

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

Optimal control of time discrete two‐phase flow governed by a diffuse interface model

Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow

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