Computing and Visualization in Science

, Volume 14, Issue 8, pp 385–400 | Cite as

Numerical investigation of homogenized Stokes–Nernst–Planck–Poisson systems

Article

Abstract

We consider charged transport within a porous medium, which at the pore scale can be described by the non-stationary Stokes–Nernst–Planck–Poisson (SNPP) system. We state three different homogenization results using the method of two-scale convergence. In addition to the averaged macroscopic equations, auxiliary cell problems are solved in order to provide closed-form expressions for effective coefficients. Our aim is to study numerically the convergence of the models for vanishing microstructure, i. e., the behavior for \(\varepsilon \rightarrow 0\), where \(\varepsilon \) is the characteristic ratio between pore diameter and size of the porous medium. To this end, we propose a numerical scheme capable of solving the fully coupled microscopic SNPP system and also the corresponding averaged systems. The discretization is performed fully implicitly in time using mixed finite elements in two space dimensions. The averaged models are evaluated using simulation results and their approximation errors in terms of \(\varepsilon \) are estimated numerically.

Keywords

Homogenization Two-scale convergence Porous media Stokes/Darcy–Nernst–Planck–Poisson system Colloidal transport Numerical simulation  Mixed finite elements 

Mathematics Subject Classification

35B27 76M10 76M50 76Sxx 76Rxx 76Wxx 

Notes

Acknowledgments

The authors want to thank N. Marheineke, K. U. Totsche, T. van Noorden, and V. Aizinger for helpful discussions. F. F. was partially funded by the German Research Foundation (DFG) within the framework of the Project To 184/7-1,2. N. R. was partially funded by the Deutsche Telekom Foundation.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Chair of Applied Mathematics I, Department of MathematicsFriedrich–Alexander University of Erlangen–NurembergErlangenGermany

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