Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T14:44:46.613Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abels, H., Garcke, H., and Grün, G., Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Mathematical Models and Methods in Applied Sciences 22 (2012), no. 3, 1150013.CrossRefGoogle Scholar
[2]Aland, S., Boden, S., Hahn, A., Klingbeil, F., Weismann, M., and Weller, S., Quantitative comparison of Taylor flow simulations based on sharp-interface and diffuse-interface models, International Journal for Numerical Methods in Fluids 73 (2013), no. 4, 344–361.Google Scholar
[3]Armero, F. and Simo, J.C., Formulation of a new class of fractional-step methods for the incompressible mhd equations that retains the long-term dissipativity of the continuum dynamical system, Fields Institute Communications, 10 (1996), 1–24.Google Scholar
[4]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer, 2002.CrossRefGoogle Scholar
[5]Campillo-Funollet, E., Grün, G., and Klingbeil, F., On modeling and simulation of electrokinetic phenomena in two-phase flow with general mass densities, SIAM Journal on Applied Mathematics 72 (2012), no. 6, 1899–1925.Google Scholar
[6]Ciarlet, P. G., The finite element method for elliptic problems, Classics in applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics, Philadelphia, US-PA, 2002.Google Scholar
[7]Ding, H., Spelt, P. D. M., and Shu, C., Diffuse interface model for incompressible two-phase flows with large density ratios, Journal of Computational Physics 226 (2007), 2078–2095.CrossRefGoogle Scholar
[8]Dohrmann, C. R. and Bochev, P. B., A stabilized finite element method for the Stokes problem based on polynomial pressure projections, International Journal for Numerical Methods in Fluids 46 (2004), 183–201.CrossRefGoogle Scholar
[9]Eck, C., Fontelos, M. A., Grün, G., Klingbeil, F., and Vantzos, O., On a phase-field model for electrowetting, Interfaces and Free Boundaries 11 (2009), 259–290.Google Scholar
[10]Ern, A. and Guermond, J., Theory and practice of finite elements, Springer Series in Applied Mathematical Sciences, vol. 159, Springer, New York, US-NY, 2004.Google Scholar
[11]Gagliardo, E., Ulteriori propriet di alcune classi difunzioni in più variabili., Ricerche di Matematica 8 (1959), 24–51.Google Scholar
[12]Grün, G., On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities, SIAM Journal on Numerical Analysis 51 (2013), no. 6, 3036–3061.Google Scholar
[13]Grün, G. and Klingbeil, F., Two-phase flow with mass density contrast: Stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model, Journal of Computational Physics 257, Part A (2014), 708–725.Google Scholar
[14]Guillén-González, F. and Tierra, G., Splitting schemes for a Navier–Stokes–Cahn–Hilliard model for two fluids with different densities, Journal of Computational Mathematics 32 (2014), no. 6, 643–664.Google Scholar
[15]Kahle, C.Garcke, H., Hinze, M., A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow, Hamburger Beiträge zur Angewandte Mathematik (2014).Google Scholar
[16]Hansbo, A., Strong stability and non-smooth data error estimates for discretizations of linear parabolic problems, BIT 42 (2002), no. 2, 351–379.Google Scholar
[17]Jr.Douglas, J., Dupont, T., and Wahlbin, L., The stability in lq of the l2-projection into finite element function spaces, Numerische Mathematik 23 (1975), 193–197.CrossRefGoogle Scholar
[18]Minjeaud, S., An unconditionally stable uncoupled scheme for a triphasic cahn–hilliard/navier-stokes model., Numerical Methods for PDE 29 (2013), no. 2, 584–618.Google Scholar
[19]Modica, L., The gradient theory of phase transitions and the minimal interface criterion, Archive for Rational Mechanics and Analysis 98 (1987), 123–142.Google Scholar
[20]Modica, L. and Mortola, S., Un esempio di γ-convergenza, Bollettino dell'Unione Matematica Italiana B 14 (1977), 285–299.Google Scholar
[21]Nirenberg, L., On elliptic partial differential equations., Annali della Scuola Normale Superiore di Pisa 13 (1959), 115–162.Google Scholar
[22]Qian, T., Wang, X., and Sheng, P., A variational approach to the moving contact line hydrodynamics, Journal of Fluid Mechanics 564 (2006), 333–360.Google Scholar
[23]Werner, H. and Arndt, H., Gewöhnliche Differentialgleichungen, Springer-Verlag, Berlin–Heidelberg, 1991.Google Scholar
[24]Yue, P., Feng, J. J., Liu, C., and Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids, Journal of Fluid Mechanics 515 (2004), 293–317.Google Scholar