Abstract
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.
Similar content being viewed by others
References
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)
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)
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)
Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Springer, Berlin (2002)
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). https://doi.org/10.1137/120861333
Ciarlet, Ph. G.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)
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)
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)
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)
Grün, G.: Partiell gleichmäßige Konvergenz finiter Elemente bei quasikonvexen Variationsintegralen. Diploma Thesis (Universität Bonn), Bonn (1991)
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)
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). https://doi.org/10.4208/cicp.scpde14.39s
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)
Grün, G., Rumpf, M.: Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87, 113–152 (2000). MR 1800156 (2002h:76108)
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)
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)
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)
Metzger, S.: On numerical schemes for phase-field models for electrowetting with electrolyte solutions. PAMM 15(1), 715–718 (2015). https://doi.org/10.1002/pamm.201510346
Metzger, S.: Diffuse interface models for complex flow scenarios: modeling, analysis and simulations. Ph.D. thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen (2017)
Minjeaud, S.: An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier-Stokes model. Num. Methods PDE 29(2), 584–618 (2013)
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)
Qian, T., Wang, X., Sheng, P.: A variational approach to the moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360 (2006)
Werner, H., Arndt, H.: Gewöhnliche Differentialgleichungen. Springer, Berlin–Heidelberg (1991)
Funding
This research has been supported by Deutsche Forschungsgemeinschaft (German Science Foundation) through the Priority Programme 1506 “Transport processes at fluidic interfaces”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Metzger, S. On stable, dissipation reducing splitting schemes for two-phase flow of electrolyte solutions. Numer Algor 80, 1361–1390 (2019). https://doi.org/10.1007/s11075-018-0530-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-018-0530-2
Keywords
- Electrolyte solutions
- Phase-field model
- Navier–Stokes equations
- Cahn–Hilliard equation
- Nernst–Planck equations
- Finite element scheme
- Splitting scheme