Derivation of Stokes-Plate-Equations modeling fluid flow interaction with thin porous elastic layers

TL;DR

This paper rigorously derives a simplified model for fluid flow interacting with a thin porous elastic layer, reducing the complex structure to an effective interface described by coupled Stokes and plate equations.

Contribution

The paper provides a rigorous homogenization of the porous layer, resulting in a coupled Stokes-plate interface model that captures microstructural effects in the limit as layer thickness tends to zero.

Findings

Effective interface model couples Stokes flow with a time-dependent plate equation.

Continuity of velocities at the interface with vertical movement only.

Higher order correctors for fluid flow in the thin layer.

Abstract

In this paper we investigate the interaction of fluid flow with a thin porous elastic layer. We consider two fluid-filled bulk domains which are separated by a thin periodically perforated layer consisting of a fluid and an elastic solid part. Thickness and periodicity of the layer are of order $\epsilon$, where $\epsilon$ is small compared to the size of the bulk domains. The fluid flow is described by an instationary Stokes equation and the solid via linear elasticity. The main contribution of this paper is the rigorous homogenization of the porous structure in the layer and the reduction of the layer to an interface $\Sigma$ in the limit $\epsilon \to 0$ using two-scale convergence. The effective model consists of the Stokes equation coupled to a time dependent plate equation on the interface $\Sigma$ including homogenized elasticity coefficients carrying information about the micro structure of the layer. In the zeroth order approximation we obtain continuity of the velocities at the interface, where only a vertical movement occurs and the tangential components vanish. The tangential movement in the solid is of order $\epsilon$ and given as a Kirchhoff-Love displacement. Additionally, we derive higher order correctors for the fluid in the thin layer.