Effective interface laws of Navier-slip-type involving the elastic displacement for Stokes flow through a thin porous elastic layer

TL;DR

This paper rigorously derives effective interface laws of Navier-slip type involving elastic displacement for Stokes flow through a thin, periodically structured porous elastic membrane, revealing how different elastic stress scalings influence the macroscopic behavior.

Contribution

It introduces a novel two-scale convergence approach for thin elastic porous membranes and derives new effective interface conditions coupling fluid flow and elastic displacement.

Findings

Effective interface laws of Navier-slip type with elastic displacement are established.

Different elastic stress scalings lead to membrane or plate-type effective models.

Mass exchange between fluid regions is incorporated into the interface conditions.

Abstract

This paper presents a rigorous derivation of an effective model for fluid flow through a thin elastic porous membrane separating two fluid bulk domains. The microscopic setting involves a periodically structured porous membrane composed of a solid phase and fluid-filled pores, with thickness and periodicity of order $\varepsilon$, small compared to the size of the bulk regions. The microscopic model is governed by a coupled fluid-structure interaction system: instationary Stokes equations for the fluid and linear elasticity for the solid, with two distinct scalings of the elastic stress tensor yielding different effective behaviors. Using two-scale convergence techniques adapted to thin domains and oscillatory microstructures, the membrane is reduced to an effective interface across which transmission conditions are derived. The resulting macroscopic model couples the bulk fluid domains via effective interface laws of Navier-slip-type including the dynamic displacement. The character of this coupling depends critically on the choice of the scaling in the elastic stress tensor, leading to either a membrane equation or a Kirchhoff-Love plate equation for the effective displacement. The resulting interface conditions naturally admit mass exchange between the adjacent fluid regions. In the analytical framework, a new two-scale compactness theorem for the symmetric gradient is established, underpinning the passage to the limit in the coupled system. Moreover, cell problem techniques are employed systematically to construct admissible test functions and to rigorously extract the effective macroscopic coefficients.