We present a phase-field model for multiphase flow for an arbitrary number of immiscible incompressible fluids 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 effects. We propose a practical fully discrete finite 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 fixed-point algorithm for the solution of the discrete nonlinear system of equations and present several computational studies of the proposed model.
Accepté le :
DOI : 10.1051/m2an/2016048
Mots-clés : Multiphase flow, phase field model, Cahn–Hilliard equation, Navier–Stokes equations, finite element method, convergence analysis
@article{M2AN_2017__51_3_1089_0,
author = {Ba\v{n}as, L{\textquoteright}ubom{\'\i}r and N\"urnberg, Robert},
title = {Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1089--1117},
publisher = {EDP-Sciences},
volume = {51},
number = {3},
year = {2017},
doi = {10.1051/m2an/2016048},
zbl = {1371.35212},
mrnumber = {3666658},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2016048/}
}
TY - JOUR AU - Baňas, L’ubomír AU - Nürnberg, Robert TI - Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1089 EP - 1117 VL - 51 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016048/ DO - 10.1051/m2an/2016048 LA - en ID - M2AN_2017__51_3_1089_0 ER -
%0 Journal Article %A Baňas, L’ubomír %A Nürnberg, Robert %T Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1089-1117 %V 51 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016048/ %R 10.1051/m2an/2016048 %G en %F M2AN_2017__51_3_1089_0
Baňas, L’ubomír; Nürnberg, Robert. Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 1089-1117. doi : 10.1051/m2an/2016048. http://www.numdam.org/articles/10.1051/m2an/2016048/
, and , Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22 (2012) 1150013. | DOI | MR | Zbl
and , Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations. Math. Comp. 79 (2010) 1957–1999. | DOI | MR | Zbl
and , An error bound for the finite element approximation of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 16 (1996) 257–287. | DOI | MR | Zbl
and , Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math. 77 (1997) 1–34. | DOI | MR | Zbl
and , Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix. IMA J. Numer. Anal. 18 (1998) 287–328. | DOI | MR | Zbl
and , Finite element approximation of a model for phase separation of a multi-component alloy with nonsmooth free energy and a concentration dependent mobility matrix. Math. Models Methods Appl. Sci. 9 (1999) 627–663. | DOI | MR | Zbl
, and , On fully practical finite element approximations of degenerate Cahn–Hilliard systems. ESAIM: M2AN 35 (2001) 713–748. | DOI | Numdam | MR | Zbl
, and , The numerical analysis of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 16 (1996) 111–139. | DOI | MR | Zbl
and , Study of a three component Cahn-Hilliard flow model. ESAIM: M2AN 40 (2006) 653–687. | DOI | Numdam | MR | Zbl
and , Numerical schemes for a three component Cahn–Hilliard model. ESAIM: M2AN 45 (2011) 697–738. | DOI | Numdam | MR | Zbl
and , Hierarchy of consistent -component Cahn–Hilliard systems. Math. Models Methods Appl. Sci. 24 (2014) 2885–2928. | DOI | MR | Zbl
, Algorithm 832: UMFPACK V4.3-an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30 (2004) 196–199. | DOI | MR | Zbl
, An efficient algorithm for incompressible N-phase flows. J. Comput. Phys. 276 (2014) 691–728. | DOI | MR | Zbl
, Physical formulation and numerical algorithm for simulating N immiscible incompressible fluids involving general order parameters. J. Comput. Phys. 283 (2015) 98–128. | DOI | MR | Zbl
and , Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix. Phys. D 109 (1997) 242–256. | DOI | MR | Zbl
, Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44 (2006) 1049–1072. | DOI | MR | Zbl
, and , A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow. Appl. Numer. Math. 99 (2016) 151–171. | DOI | MR | Zbl
, On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities. SIAM J. Numer. Anal. 51 (2013) 3036–3061. | DOI | MR | Zbl
, and , On Fully Decoupled, Convergent Schemes for Diffuse Interface Models for Two-Phase Flow with General Mass Densities. Commun. Comput. Phys. 19 (2016) 1473–1502. | DOI | MR | Zbl
and , Two-phase flow with mass density contrast: Stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257 (2014) 708–725. | DOI | MR | Zbl
, and , Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interfaces Free Bound. 10 (2008) 15–43. | DOI | MR | Zbl
, A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys. 204 (2005) 784–804. | DOI | MR | Zbl
, A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3105–3112. | DOI | MR | Zbl
, Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12 (2012) 613–661. | DOI | MR | Zbl
and , Phase field modeling and simulation of three-phase flows. Interfaces Free Bound. 7 (2005) 435–466. | DOI | MR | Zbl
, and , A practically unconditionally gradient stable scheme for the N-component Cahn–Hilliard system. Physica A 391 (2012) 1009–1019. | DOI
and , Convergence of numerical approximations of the incompressible Navier–Stokes equations with variable density and viscosity. SIAM J. Numer. Anal. 45 (2007) 1287–1304. | DOI | MR | Zbl
, An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model. Numer. Methods Partial Differ. Eq. 29 (2013) 584–618. | DOI | MR | Zbl
, Numerical simulations of immiscible fluid clusters. Appl. Numer. Math. 59 (2009) 1612–1628. | DOI | MR | Zbl
R. Temam, Navier–Stokes Equations. Vol. 2 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 3rd edition (1984). | MR | Zbl
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