Homogenisation of the Stokes equations for evolving microstructure

TL;DR

This paper rigorously derives a homogenised Darcy's law for evolving porous media governed by the Stokes equations, incorporating time-dependent permeability and a source term linked to pore volume changes.

Contribution

It introduces a new homogenisation approach for time-evolving microstructures in porous media, including a Korn-type inequality and a family of divergence-inverse operators.

Findings

Derived a homogenised Darcy's law with space- and time-dependent permeability.

Established a Korn-type inequality for the two-scale transformation method.

Connected the source term to pore volume changes under no-slip boundary conditions.

Abstract

We consider the homogenisation of the Stokes equations in a porous medium which is evolving in time. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables to model a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit with the two-scale transformation method. In order to derive uniform a priori estimates, we show a Korn-type inequality for the two-scale transformation method and construct a family of $\varepsilon$-scaled operators $\operatorname{div}_\varepsilon^{-1}$, which are right-inverse to the corresponding divergences. The homogenisation result is a new version of Darcy's law. It features a time- and space-dependent permeability tensor, which accounts for the local pore structure, and a macroscopic compressibility condition, which induces a new source term for the pressure. In the case of a no-slip boundary condition at the interface, this source term relates to the change of the local pore volume.