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

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...

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Bibliographic Details
Published in:European journal of applied mathematics 2021-08, Vol.32 (4), p.683-710
Main Authors: KOVTUNENKO, V. A., ZUBKOVA, A. V.
Format: Article
Language:English
Subjects:
Applied mathematics
Convergence
Electric flux
Geometry
Homogenization
Partial differential equations
Solid phases
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Summary: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.
ISSN:0956-7925
1469-4425
DOI:10.1017/S095679252000025X