Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T03:15:49.911Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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: [email protected]
A. V. ZUBKOVA
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Mozartgasse 14, 8010 Graz, Austria email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Check for updates
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

References

Allaire, G. (1992) Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518.CrossRefGoogle Scholar
Allaire, G., Brizzi, R., Dufrêche, J.-F., Mikelić, A. & Piatnitski, A. (2013) Ion transport in porous media: derivation of the macroscopic equations using upscaling and properties of the effective coefficients. Comp. Geosci. 17, 479–495.CrossRefGoogle Scholar
Belyaev, A. G., Pyatnitskii, A. L. & Chechkin, G. A. (2001) Averaging in a perforated domain with an oscillating third boundary condition. Mat. Sb. 192, 3–20.Google Scholar
Bensoussan, A., Lions, J.-L. & Papanicolaou, G. (1978) Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, New York.Google Scholar
Bunoiu, R. & Timofte, C. (2019) Upscaling of a diffusion problem with interfacial flux jump leading to a modified Barenblatt model. Z. Angew. Math. Mech. 99, e201901788.CrossRefGoogle Scholar
Burger, M., Schlake, B. & Wolfram, M.-T. (2012) Nonlinear Poisson–Nernst–Planck equations for ion flux through confined geometries. Nonlinearity 25, 961–990.CrossRefGoogle Scholar
Cioranescu, D., Damlamian, A., Donato, P., Griso, G. & Zaki, R. (2012) The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44, 718–760.CrossRefGoogle Scholar
Cioranescu, D., Damlamian, A. & Griso, G. (2008) The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40, 1585–1620.CrossRefGoogle Scholar
Cioranescu, D., Donato, P. & Zaki, R. (2006) The periodic unfolding method in perforated domains. Portugaliae Mathematica 63, 467–496.Google Scholar
Desvillettes, L. & Lorenzani, S. (2018) Homogenization of the discrete diffusive coagulation-fragmentation equations in perforated domains. J. Math. Anal. Appl. 467, 1100–1128.CrossRefGoogle Scholar
Donato, P. & Le Nguyen, K. H. (2015) Homogenization of diffusion problems with a nonlinear interfacial resistance. Nonlinear Differ. Equ. Appl. 22, 1345–1380.CrossRefGoogle Scholar
Donato, P., Le Nguyen, K. H. & Tardieu, R. (2011) The periodic unfolding method for a class of imperfect transmission problems. J. Math. Sci. 176, 891–927.CrossRefGoogle Scholar
Donato, P. & Monsurrò, S. (2004) Homogenization of two heat conductors with an interfacial contact resistance. Anal. Appl. 2, 247–273.CrossRefGoogle Scholar
Dreyer, W., Guhlke, C. & Müller, R. (2015) Modeling of electrochemical double layers in thermodynamic non-equilibrium. Phys. Chem. Chem. Phys. 17, 27176–27194.CrossRefGoogle ScholarPubMed
Evendiev, M. & Zelik, S. V. (2002) Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization. Ann. Instit. H. Poincare 19, 961–989.CrossRefGoogle Scholar
Fellner, K. & Kovtunenko, V. A. (2016) A discontinuous Poisson–Boltzmann equation with interfacial transfer: homogenisation and residual error estimate. Appl. Anal. 95, 2661–2682.CrossRefGoogle Scholar
Franců, J. (2010) Modification of unfolding approach to two-scale convergence. Mathematica Bohemica 135, 403–412.CrossRefGoogle Scholar
Fuhrmann, J. (2015) Comparison and numerical treatment of generalized Nernst–Planck models. Comput. Phys. Commun. 196, 166–178.CrossRefGoogle Scholar
Gagneux, G. & Millet, O. (2014) Homogenization of the Nernst–Planck–Poisson system by two-scale convergence. J. Elast. 114, 69–84.CrossRefGoogle Scholar
Gahn, M., Neuss-Radu, M. & Knabner, P. (2016) Homogenization of reaction-diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface. SIAM J. App. Math. 76, 1819–1843.CrossRefGoogle Scholar
González Granada, J. R. & Kovtunenko, V. A. (2018) Entropy method for generalized Poisson–Nernst–Planck equations. Anal. Math. Phys. 8, 603–619.CrossRefGoogle Scholar
Herz, M., Ray, N. & Knabner, P. (2012) Existence and uniqueness of a global weak solution of a Darcy–Nernst–Planck–Poisson system. GAMM–Mitt. 35, 191–208.CrossRefGoogle Scholar
Khludnev, A. M. & Kovtunenko, V. A. (2000) Analysis of Cracks in Solids. WIT Press, Southampton, Boston.Google Scholar
Khoa, V. A. & Muntean, A. (2019) Corrector homogenization estimates for a non-stationary Stokes–Nernst–Planck–Poisson system in perforated domains. Commun. Math. Sci. 17, 705–738.CrossRefGoogle Scholar
Kovtunenko, V. A., Reichelt, S. & Zubkova, A. V. (2020) Corrector estimates in homogenization of a nonlinear transmission problem for diffusion equations in connected domains. Math. Meth. Appl. Sci. 43, 1838–1856.CrossRefGoogle ScholarPubMed
Kovtunenko, V. A. & Zubkova, A. V. (2017) Solvability and Lyapunov stability of a two-component system of generalized Poisson–Nernst–Planck equations. In: Maz’ya, V., Natroshvili, D., Shargorodsky, E. and Wendland, W. L. (editors), Recent Trends in Operator Theory and Partial Differential Equations (The Roland Duduchava Anniversary Volume). Operator Theory: Advances and Applications, Vol. 258, Birkhaeuser, Basel, pp. 173–191.CrossRefGoogle Scholar
Kovtunenko, V. A. & Zubkova, A. V. (2017) On generalized Poisson–Nernst–Planck equations with inhomogeneous boundary conditions: a-priori estimates and stability. Math. Meth. Appl. Sci. 40, 2284–2299.Google Scholar
Kovtunenko, V. A. & Zubkova, A. V. (2018) Mathematical modeling of a discontinuous solution of the generalized Poisson–Nernst–Planck problem in a two-phase medium. Kinet. Relat. Mod. 11, 119–135.CrossRefGoogle Scholar
Kovtunenko, V. A. & Zubkova, A. V. (2020) Homogenization of the generalized Poisson–Nernst–Planck problem in a two-phase medium: correctors and residual error estimates. Appl. Anal. doi: 10.1080/00036811.2019.1600676.Google Scholar
Ladyzhenskaya, O. A. (1985) The Boundary Value Problems of Mathematical Physics, Vol. 49, Springer Verlag.CrossRefGoogle Scholar
Meirmanov, A. & Zimin, R. (2011) Compactness result for periodic structures and its application to the homogenization of a diffusion-convection equation. Electr. J. Differ. Equ. 115, 1–11.Google Scholar
Mielke, A., Reichelt, S. & Thomas, M. (2014) Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. J. Netw. Heterog. Media 9, 353–382.CrossRefGoogle Scholar
Nguetseng, G. (1989) A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623.CrossRefGoogle Scholar
Ray, N., Eck, Ch ., Muntean, A. & Knabner, P. (2011) Variable choices of scaling in the homogenization of a Nernst–Planck–Poisson problem. Erlangen-Nürnberg: Inst. für Angewandte Mathematik 344.Google Scholar
Reichelt, S. (2017) Corrector estimates for a class of imperfect transmission problems. Asymptotic Anal. 105, 3–26.CrossRefGoogle Scholar
Roubček, T. (2007) Incompressible ionized non-Newtonean fluid mixtures. SIAM J. Math. Anal. 39, 863–890.CrossRefGoogle Scholar
Sánchez-Palencia, E. (1980) Non-Homogeneous Media and Vibration Theory, Springer-Verlag.Google Scholar
Sazhenkov, S. A., Sazhenkova, E. V. & Zubkova, A. V. (2014) Small perturbations of two-phase fluid in pores: Effective macroscopic monophasic viscoelastic behavior. Sib. Èlektron. Mat. Izv. 11, 26–51.Google Scholar
Schmuck, M. & Bazant, M. Z. (2015) Homogenization of the Poisson–Nernst–Planck equations for ion transport in charged porous media. SIAM J. Appl. Math. 75, 1369–1401.CrossRefGoogle Scholar
Tychonoff, A. (1935) Ein Fixpunktsatz. Math. Ann. 111, 767–776.CrossRefGoogle Scholar
Visintin, A. (2006) Towards a two-scale calculus. ESAIM: COCV 12, 371–397.Google Scholar
Zhikov, V. V., Kozlov, S. M. & Oleinik, O. A. (1994) Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin.Google Scholar