Rigorous Homogenization of a Stokes-Nernst-Planck-Poisson Problem for various Boundary Conditions

TL;DR

This paper rigorously homogenizes a coupled Stokes-Nernst-Planck-Poisson system under various boundary conditions, revealing how microscopic boundary choices influence the structure of the resulting macroscopic models.

Contribution

It introduces a detailed homogenization framework for the coupled system, accounting for different boundary conditions and scalings, and derives the corresponding upscaled equations.

Findings

Different boundary conditions lead to distinct upscaled models.

Special treatment is required for the nonlinear electrostatic drift term.

Multiple classes of limit equations are obtained depending on scalings.

Abstract

We perform the periodic homogenization (i.e. $\eps\to 0$) of the non-stationary Stokes-Nernst-Planck-Poisson system using two-scale convergence, where $\eps$ is a suitable scale parameter. The objective is to investigate the influence of \textsl{different boundary conditions and variable choices of scalings in $\eps$} of the microscopic system of partial differential equations on the structure of the (upscaled) limit model equations. Due to the specific nonlinear coupling of the underlying equations, special attention has to be paid when passing to the limit in the electrostatic drift term. As a direct result of the homogenization procedure, various classes of upscaled model equations are obtained.