Homogenization of a two-phase problem with nonlinear dynamic Wentzell-interface condition for connected-disconnected porous media

TL;DR

This paper derives macroscopic models for a reaction-diffusion system in a two-phase porous medium with nonlinear interface conditions, using homogenization techniques to handle the complex microscopic structure and nonlinearities.

Contribution

It introduces a homogenization approach for a two-phase porous medium with nonlinear dynamic Wentzell-interface conditions, including new results for surface diffusion and nonlinear term handling.

Findings

Derived macroscopic models via homogenization techniques.

Established strong two-scale convergence results for nonlinear terms.

Extended homogenization theory to nonlinear interface conditions.

Abstract

We investigate a reaction-diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $\epsilon$. On the interface between the components we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk-domains are given by a reaction-diffusion equation for the traces of the bulk-solutions, including nonlinear reaction-kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $\epsilon \to 0$ and derive macroscopic models, where we need homogenization results for surface diffusion. To cope with the nonlinear terms we derive strong two-scale results.