Existence of contacts for the motion of a rigid body into a viscous incompressible fluid with the Tresca boundary conditions

Abstract

We consider a fluid-structure interaction system composed of a rigid ball immersed into a viscous incompressible fluid. The motion of the structure satisfies the Newton laws and the fluid equations are the standard Navier–Stokes system. At the boundary of the fluid domain, we use the Tresca boundary conditions, that permit the fluid to slip tangentially on the boundary under some conditions on the stress tensor. More precisely, there is a threshold determining if the fluid can slip or not and there is a friction force acting on the part where the fluid can slip. Our main result is the existence of contact in finite time between the ball and the exterior boundary of the fluid for this system in the bidimensional case, in presence of gravity and in the case of a symmetric configuration.