Fluid-Structure interactions with Navier- and full-slip boundary conditions

TL;DR

This paper proves the existence of weak solutions for a fluid-structure interaction problem involving a deforming viscoelastic solid and a viscous incompressible fluid with Navier-slip boundary conditions, addressing geometric complexities.

Contribution

It extends previous models by incorporating Navier-slip boundary conditions and adapting the weak solution framework to handle the resulting geometric and boundary normal dependencies.

Findings

Existence of weak solutions until contact occurs.

Compatibility between weak and strong formulations established.

Handling of time-dependent boundary normals in slip conditions.

Abstract

We show the existence of weak solutions to the fluid-structure interaction problem of a largely deforming viscoelastic bulk solid with a viscous fluid governed by the incompressible Navier-Stokes equations. In contrast to previous works, the fluid is allowed to slip along the solid boundary; namely, the so called Navier-slip boundary conditions are considered. Such boundary conditions naturally involve the time-changing outer normal of the fluid domain. Hence, their dependence on the varying geometry is one degree higher than in the previously considered no-slip case, which makes it necessary to adjust the concept of weak coupled solutions. Two classes of test functions are introduced: test functions that are continuous over the fluid-solid domain, and fluid-only test functions with nonzero tangential component at the boundary. The weak equations are established until the point of contact, and moreover, compatibility with the strong formulation is shown.