Existence of weak solutions up to collision for viscous fluid-solid systems with slip

TL;DR

This paper proves the existence of weak solutions for a viscous fluid-solid system with slip conditions, allowing the solid to collide with boundaries in three dimensions, addressing limitations of no-slip models.

Contribution

It establishes the existence of weak solutions up to collision in a fluid-solid system with slip boundary conditions, a novel result in three dimensions.

Findings

Existence of weak solutions up to collision in 3D

Slip conditions lead to velocity discontinuities at the interface

Classical no-slip models are inadequate for collision scenarios

Abstract

We study in this paper the movement of a rigid solid inside an incompressible Navier-Stokes flow, within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface, through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no-slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents from obtaining global H1 bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions unadapted.