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On Fully Decoupled, Convergent Schemes for Diffuse Interface Models for Two-Phase Flow with General Mass Densities

Part of: Equations of mathematical physics and other areas of application Two-phase and multiphase flows Incompressible viscous fluids Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 May 2016

Günther Grün ,
Francisco Guillén-González  and
Stefan Metzger
Günther Grün*
Affiliation:
Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, Cauerstr. 11, 91058 Erlangen, Germany
Francisco Guillén-González*
Affiliation:
Universidad Sevilla, Dpto. Ecuaciones Diferenciales y Análisis Numérico, Instituto de Matemáticas, Aptdo. 1160, 41080, Sevilla, Spain
Stefan Metzger*
Affiliation:
Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, Cauerstr. 11, 91058 Erlangen, Germany
*
*Corresponding author. Email addresses:gruen@am.uni-erlangen.de (G. Grün), guillen@us.es (F. Guillén-Gonzalez), stefan.metzger@fau.de (S. Metzger)
*Corresponding author. Email addresses:gruen@am.uni-erlangen.de (G. Grün), guillen@us.es (F. Guillén-Gonzalez), stefan.metzger@fau.de (S. Metzger)
*Corresponding author. Email addresses:gruen@am.uni-erlangen.de (G. Grün), guillen@us.es (F. Guillén-Gonzalez), stefan.metzger@fau.de (S. Metzger)
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Abstract

In the first part, we study the convergence of discrete solutions to splitting schemes for two-phase flow with different mass densities suggested in [Guillen-Gonzalez, Tierra, J.Comput.Math. (6)2014]. They have been formulated for the diffuse interface model in [Abels, Garcke, Grün, M3AS, 2012, DOI:10.1142/S0218202511500138] which is consistent with thermodynamics. Our technique covers various discretization methods for phase-field energies, ranging from convex-concave splitting to difference quotient approaches for the double-well potential. In the second part of the paper, numerical experiments are presented in two space dimensions to identify discretizations of Cahn-Hilliard energies which are ϕ-stable and which do not reduce the acceleration of falling droplets. Finally, 3d simulations in axial symmetric geometries are shown to underline even more the full practicality of the approach.

Keywords

Two-phase flowCahn-Hilliard equationdiffuse interface modelconvergence of finite-element schemesnumerical simulation

MSC classification

Secondary: 35Q35: PDEs in connection with fluid mechanics 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M22: Solution of discretized equations 65M12: Stability and convergence of numerical methods 76D05: Navier-Stokes equations 76T10: Liquid-gas two-phase flows, bubbly flows
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1473 - 1502
Copyright
Copyright © Global-Science Press 2016 

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References

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