Matthieu Hillairet

Collisions in Three-Dimensional Fluid Structure Interaction Problems

This paper deals with a system composed of a rigid ball moving into a viscous incompressible fluid over a fixed horizontal plane. The equations of motion for the fluid are the Navier–Stokes equations, and the equations for the motion of the rigid ball are obtained by applying Newtons laws. We show that for any weak solution of the corresponding system satisfying the energy inequality, the rigid ball never touches the plane. This result is the extension of that obtained in [M. Hillairet, Comm. Partial Differential Equations, 32 (2007), pp. 1345–1371] in the two-dimensional setting.

Asymptotic Description of Solutions of the Planar Exterior Navier–Stokes Problem in a Half Space

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in a planar exterior domain in a half space with appropriate boundary conditions on the wall, the body, and at infinity. We focus on the case where the size of the body is small. We prove in a very general setup that the solution of this problem is unique and we compute a sharp decay rate of the solution far from the moving body and the wall.

Long-Time Behavior for the Two-Dimensional Motion of a Disk in a Viscous Fluid

In this article, we study the long-time behavior of solutions of the two-dimensional fluid-rigid disk problem. The motion of the fluid is modeled by the two-dimensional Navier–Stokes equations, and the disk moves under the influence of the forces exerted by the viscous fluid. We first derive Lp−Lq decay estimates for the linearized equations and compute the first term in the asymptotic expansion of the solutions of the linearized equations. We then apply these computations to derive time-decay estimates for the solutions to the full Navier–Stokes fluid-rigid disk system.

Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies

In this paper, we review recent results devoted to the interactions between a collection of rigid bodies \((\mathcal{B}_{i})_{i=1,\ldots,n}\) and a surrounding viscous fluid \(\mathcal{L}\), the whole system filling a container \(\Omega \). We assume that the motion of \(\mathcal{L}\) (resp. the rigid bodies \(\mathcal{B}_{i}\)) is governed by the incompressible Navier Stokes equations (resp. Newton laws), and that velocities and stress tensors are continuous at the fluid/body interfaces. Our concern is the well-posedness of the associated Cauchy problem, with a specific eye towards the handling of contact between bodies or between one body and the container boundary.