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

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.Comment: 45 page

Computation of the drag force on a rough sphere close to a wall

We consider the effect of surface roughness on solid-solid contact in a Stokes flow. Various models for the roughness are considered, and a unified methodology is given to derive the corresponding asymptotics of the drag force. In this way, we recover and clarify the various expressions that can be found in the litterature

A compressible multifluid system with new physical relaxation terms

International audienceIn this paper, we rigorously derive a new compressible multifluid system from compressible Navier-Stokes equations with density-dependent viscosity in the one-dimensional in space setting. More precisely, we propose and mathematically derive a generalization of the usual one velocity Baer-Nunziato model with a new relaxation term in the PDE governing the volume fractions. This new relaxation term encodes the change of viscosity and pressure between the different fluids. For the reader's convenience, we first establish a formal derivation in the bifluid setting using a WKB decomposition and then we rigorously justify the multifluid homogenized system using a kinetic formulation via Young measures characterization

Collisions in 3D Fluid Structure interactions problems

International audienceThis paper deals with the system composed by 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 Newton's laws. We show that for any weak solutions of the corresponding system satisfying the energy inequality, the rigid ball never touches the plane

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

We consider a fluid-structure interaction system composed by a rigid ball immersed into a viscous in-compressible 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 and in presence of gravity

Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant

We prove global existence of a nonnegative weak solution to a degenerate parabolic system, which models the spreading of insoluble surfactant on a thin liquid film