Abstract
The paper provides a rigorous homogenization of the Poisson–Nernst–Planck problem stated in an inhomogeneous domain composed of two, solid and pore, phases. The generalized PNP model is constituted of the Fickian cross-diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials, and Darcy's flow model. At the interface between two phases inhomogeneous boundary conditions describing electrochemical reactions are considered. The resulting doubly non-linear problem admits discontinuous solutions caused by jumps of field variables. Using an averaged problem and first-order asymptotic correctors, the homogenization procedure gives us an asymptotic expansion of the solution which is justified by residual error estimates.
Highlights
The paper is devoted to the mathematical study of homogenization of a non-linear diffusion model in a two-phase domain
We consider cross-diffusion of multiple charged species coupled with an overall electrostatic potential
We describe a two-phase medium with a micro-structure consisting of solid and pore phases which are separated by a thin interface
Summary
The paper is devoted to the mathematical study of homogenization of a non-linear diffusion model in a two-phase domain. A homogenization procedure in a two-phase domain for steady-state Poisson–Boltzmann equations and homogeneous Neumann boundary conditions was investigated in [24]. We describe a discontinuous prolongation from the perforated domain inside solid particles following the approach of [28] In this respect, the two-phase homogenization procedure differs from a perforated domain case. As the result of homogenization of the PNP model, we obtain an averaged model consisting of linear parabolic-elliptic equations and supported by first-order correctors. In order to justify cell problems we use the periodic unfolding technique It is based on the unfolding operator and the averaging operator, which were defined for perforated domains in [29]. The averaged problem is formulated and supported by error estimates of the corrector terms
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