Effective transmission through an interface with evolving microstructure

TL;DR

This paper derives an effective model for nonlinear reaction-diffusion-advection systems in domains with evolving microstructures, reducing complex layered geometries to lower-dimensional interfaces using homogenisation techniques.

Contribution

It extends homogenisation methods to domains with evolving microstructure and thin layers, deriving new jump conditions and effective interface models.

Findings

Derived an effective lower-dimensional interface model.

Established jump conditions involving local cell problems.

Extended homogenisation techniques to time-evolving microstructures.

Abstract

We study the asymptotic behaviour of a system of nonlinear reaction--diffusion--advection equations in a domain consisting of two bulk regions connected via microscopic channels distributed within a thin membrane. Both the width of the channels and the thickness of the membrane are of order $\varepsilon \ll 1$, and the geometry evolves in time in an a priori known way. We consider nonlinear flux boundary conditions at the lateral boundaries of the channels and critical scaling of the diffusion inside the layer. Extending the method of homogenisation in domains with evolving microstructure to thin layers, we employ two-scale convergence and unfolding techniques in thin layers to derive an effective model in the limit $\varepsilon \to 0$, in which the membrane is reduced to a lower-dimensional interface. We obtain jump conditions for the solution and the total fluxes, which involve the solutions of local, space--time-dependent cell problems in the reference channel.