Mathematical modeling of a discontinuous solution of the generalized Poisson--Nernst--Planck problem in a two-phase medium

Victor A. Kovtunenko; Anna V. Zubkova

doi:10.3934/krm.2018007

Article Contents

2018, Volume 11, Issue 2: 1-17. Doi: 10.3934/krm.2018007 open access

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Mathematical modeling of a discontinuous solution of the generalized Poisson--Nernst--Planck problem in a two-phase medium

Victor A. Kovtunenko^1,2, and
Anna V. Zubkova^3, ,

1.
Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstraße 36,8010 Graz, Austria
2.
Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia
3.
Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstraße 36,8010 Graz, Austria

* Corresponding author:anna.zubkova@uni-graz.at
* Corresponding author:anna.zubkova@uni-graz.at

Received: October 31, 2016

Revised: February 28, 2017

Published: March 2018

The authors are supported by the Austrian Science Fund (FWF) Project P26147-N26: "Object identification problems: numerical analysis" (PION)

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Abstract

In this paper a mathematical model generalizing Poisson--Nernst--Planck system is considered. The generalized model presents electrokinetics of species in a two-phase medium consisted of solid particles and a pore space. The governing relations describe cross-diffusion of the charged species together with the overall electrostatic potential. At the interface between the pore and the solid phases nonlinear electro-chemical reactions are taken into account provided by jumps of field variables. The main advantage of the generalized model is that the total mass balance is kept within our setting. As the result of the variational approach, well-posedness properties of a discontinuous solution of the problem are demonstrated and supported by the energy and entropy estimates.

Keywords:

Generalized Poisson--Nernst--Planck model,
mass balance,
nonlinear boundary reaction,
interface jump,
energy and entropy estimates.

Mathematics Subject Classification: Primary:35D05;Secondary:35M10, 37B25, 82C24.

Citation:

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Figure 1. An example domain $\Omega = Q \cup \omega \cup \partial\omega$ with two phases $Q$ and $\omega$, the boundary $\partial\Omega$, and two faces $\partial\omega^+$ and $\partial\omega^-$ of the interface $\partial\omega$ shown in zoom.

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