Multilayered Poroelasticity Interacting with Stokes Flow

TL;DR

This paper establishes the existence of weak solutions for a complex fluid-structure interaction involving multilayered poroelastic materials and Stokes flow, addressing mathematical challenges with nonlinear and linear Biot models.

Contribution

It proves the existence of weak solutions for coupled Stokes and multilayered poroelastic systems, including nonlinear Biot models with permeability depending on fluid content.

Findings

Existence of weak solutions for linear Biot models.

Existence of weak solutions for nonlinear quasi-static Biot models.

Conditions under which weak solutions are also strong solutions.

Abstract

We consider the interaction between an incompressible, viscous fluid modeled by the dynamic Stokes equation and a multilayered poroelastic structure which consists of a thin, linear, poroelastic plate layer (in direct contact with the free Stokes flow) and a thick Biot layer. The fluid flow and the elastodynamics of the multilayered poroelastic structure are fully coupled across a fixed interface through physical coupling conditions (including the Beavers-Joseph-Saffman condition), which present mathematical challenges related to the regularity of associated velocity traces. We prove existence of weak solutions to this fluid-structure interaction problem with either (i) a linear, dynamic Biot model, or (ii) a nonlinear quasi-static Biot component, where the permeability is a nonlinear function of the fluid content (as motivated by biological applications). The proof is based on constructing approximate solutions through Rothe's method, and using energy methods and a version of Aubin-Lions compactness lemma (in the nonlinear case) to recover the weak solution as the limit of approximate subsequences. We also provide uniqueness criteria and show that constructed weak solutions are indeed strong solutions to the coupled problem if one assumes additional regularity.