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Existence and two-scale convergence of the generalised Poisson–Nernst–Planck problem with non-linear interface conditions

Part of: Partial differential equations Equations and systems of special type Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  03 August 2020

V. A. KOVTUNENKO Open the ORCID record for V. A. KOVTUNENKO [Opens in a new window]  and
A. V. ZUBKOVA
V. A. KOVTUNENKO
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstraße 36, 8010 Graz, Austria Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia email: victor.kovtunenko@uni-graz.at
A. V. ZUBKOVA
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Mozartgasse 14, 8010 Graz, Austria email: anna.zubkova@uni-graz.at
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Abstract

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The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.

Keywords

Generalised Poisson–Nernst–Planck modeltwo-phase interface conditionhomogenisationtwo-scale convergence

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure
Secondary: 35M10: Equations of mixed type 82C24: Interface problems; diffusion-limited aggregation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 4 , August 2021 , pp. 683 - 710
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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