12 research outputs found
A Kato type Theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body
The issue of the inviscid limit for the incompressible Navier-Stokes
equations when a no-slip condition is prescribed on the boundary is a famous
open problem. A result by Tosio Kato says that convergence to the Euler
equations holds true in the energy space if and only if the energy dissipation
rate of the viscous flow in a boundary layer of width proportional to the
viscosity vanishes. Of course, if one considers the motion of a solid body in
an incompressible fluid, with a no-slip condition at the interface, the issue
of the inviscid limit is as least as difficult. However it is not clear if the
additional difficulties linked to the body's dynamic make this issue more
difficult or not. In this paper we consider the motion of a rigid body in an
incompressible fluid occupying the complementary set in the space and we prove
that a Kato type condition implies the convergence of the fluid velocity and of
the body velocity as well, what seems to indicate that an answer in the case of
a fixed boundary could also bring an answer to the case where there is a moving
body in the fluid
Measure-valued solutions and weak-strong uniqueness for the incompressible inviscid fluid-rigid body interaction
We consider a coupled system of partial and ordinary differential equations
describing the interaction between an isentropic inviscid fluid and a rigid
body moving freely inside the fluid. We prove the existence of measure-valued
solutions which is generated by the vanishing viscosity limit of incompressible
fluid-rigid body interaction system under some physically constitutive
relations. Moreover, we show that the measure-value solution coincides with
strong solution on the interval of its existence. This relies on the
weak-strong uniqueness analysis.Comment: 22 page
Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid
The point vortex system is usually considered as an idealized model where the
vorticity of an ideal incompressible two-dimensional fluid is concentrated in a
finite number of moving points. In the case of a single vortex in an otherwise
irrotational ideal fluid occupying a bounded and simply-connected
two-dimensional domain the motion is given by the so-called Kirchhoff-Routh
velocity which depends only on the domain. The main result of this paper
establishes that this dynamics can also be obtained as the limit of the motion
of a rigid body immersed in such a fluid when the body shrinks to a massless
point particle with fixed circulation. The rigid body is assumed to be only
accelerated by the force exerted by the fluid pressure on its boundary, the
fluid velocity and pressure being given by the incompressible Euler equations,
with zero vorticity. The circulation of the fluid velocity around the particle
is conserved as time proceeds according to Kelvin's theorem and gives the
strength of the limit point vortex. We also prove that in the different regime
where the body shrinks with a fixed mass the limit dynamics is governed by a
second-order differential equation involving a Kutta-Joukowski-type lift force
Existence of weak solutions for a Bingham fluid-rigid body system
International audienceWe consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body
Motion of a particle immersed in a two dimensional incompressible perfect fluid and point vortex dynamics
In these notes, we expose some recent works by the author in collaboration
with Olivier Glass, Christophe Lacave and Alexandre Munnier, establishing point
vortex dynamics as zero-radius limits of motions of a rigid body immersed in a
two dimensional incompressible perfect fluid in several inertia regimes