Numerical Algorithms

, Volume 80, Issue 4, pp 1361–1390 | Cite as

On stable, dissipation reducing splitting schemes for two-phase flow of electrolyte solutions

  • Stefan MetzgerEmail author
Original Paper


In this paper, we are concerned with the numerical treatment of a recent diffuse interface model for two-phase flow of electrolyte solutions (Campillo-Funollet et al., SIAM J. Appl. Math. 72(6), 1899–1925, 2012) . This model consists of a Nernst–Planck-system describing the evolution of the ion densities and the electrostatic potential which is coupled to a Cahn–Hilliard–Navier–Stokes-system describing the evolution of phase-field, velocity field, and pressure. In the first part, we present a stable, fully discrete splitting scheme, which allows to split the governing equations into different blocks, which may be treated sequentially and thereby reduces the computational costs significantly. This scheme comprises different mechanisms to reduce the induced numerical dissipation. In the second part, we investigate the impact of these mechanisms on the scheme’s sensitivity to the size of the time increment using the example of a falling droplet. Finally, we shall present simulations showing ion induced changes in the topology of charged droplets serving as a qualitative validation for our discretization and the underlying model.


Electrolyte solutions Phase-field model Navier–Stokes equations Cahn–Hilliard equation Nernst–Planck equations Finite element scheme Splitting scheme 

Mathematics Subject Classification (2010)

35Q35 65M60 65M12 76D05 76T99 76W05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


Funding information

This research has been supported by Deutsche Forschungsgemeinschaft (German Science Foundation) through the Priority Programme 1506 “Transport processes at fluidic interfaces”.


  1. 1.
    Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Model. Methods Appl. Sci. 22(3), 1150013 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aland, S., Boden, S., Hahn, A., Klingbeil, F., Weismann, M., Weller, S.: Quantitative comparison of Taylor flow simulations based on sharp-interface and diffuse-interface models. Int. J. Numer. Methods Fluids 73(4), 344–361 (2013)CrossRefGoogle Scholar
  3. 3.
    Armero, F., 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 Inst Commun 10, 1–24 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Campillo-Funollet, E., Grün, G., Klingbeil, F.: On modeling and simulation of electrokinetic phenomena in two-phase flow with general mass densities. SIAM J. Appl. Math. 72(6), 1899–1925 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ciarlet, Ph. G.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  7. 7.
    Copetti, M.I.M., Elliott, C.M.: Numerical analysis of the Cahn–Hilliard equation with a logarithmic free Energy. Numer. Math. 63, 39–65 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eck, C., Fontelos, M.A., Grün, G., Klingbeil, F., Vantzos, O.: On a phase-field model for electrowetting. Interfaces Free Boundaries 11, 259–290 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fontelos, M.A., Grün, G., Jörres, S.: On a phase-field model for electrowetting and other electrokinetic phenomena. SIAM J. Math. Anal. 43(1), 527–563 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grün, G.: Partiell gleichmäßige Konvergenz finiter Elemente bei quasikonvexen Variationsintegralen. Diploma Thesis (Universität Bonn), Bonn (1991)Google Scholar
  11. 11.
    Grün, G.: On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities. SIAM J. Numer. Anal. 6, 3036–3061 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grün, G., Guillén-González, F., Metzger, S.: On fully decoupled, convergent schemes for diffuse interface models for two-phase flow with general mass densities. Commun. Comput. Phys. 19(5), 1473–1502 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grün, G., Klingbeil, F.: Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257, Part A, 708–725 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grün, G., Rumpf, M.: Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87, 113–152 (2000). MR 1800156 (2002h:76108)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guillén-González, F., Tierra, G.: Splitting schemes for a Navier–Stokes–Cahn–Hilliard model for two fluids with different densities. J. Comput. Math. 32(6), 643–664 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Garcke, C., Kahle, 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
  17. 17.
    Klingbeil, F.: On the numerics of diffuse-interface models for two-phase flow with species transport. Ph.D. thesis, friedrich-alexander-universität erlangen-nürnberg, Erlangen (2014)Google Scholar
  18. 18.
    Metzger, S.: On numerical schemes for phase-field models for electrowetting with electrolyte solutions. PAMM 15(1), 715–718 (2015). CrossRefGoogle Scholar
  19. 19.
    Metzger, S.: Diffuse interface models for complex flow scenarios: modeling, analysis and simulations. Ph.D. thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen (2017)zbMATHGoogle Scholar
  20. 20.
    Minjeaud, S.: An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier-Stokes model. Num. Methods PDE 29(2), 584–618 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nochetto, R.H., Salgado, A.J., Walker, S.W.: A diffuse interface model for electrowetting with moving contact lines. Math. Model Methods Appl. Sci. 24(1), 67–111 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Qian, T., Wang, X., Sheng, P.: A variational approach to the moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Werner, H., Arndt, H.: Gewöhnliche Differentialgleichungen. Springer, Berlin–Heidelberg (1991)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department MathematikFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

Personalised recommendations