125 research outputs found
Asymptotic analysis of a Neumann problem in a domain with cusp. Application to the collision problem of rigid bodies in a perfect fluid
We study a two dimensional collision problem for a rigid solid immersed in a
cavity filled with a perfect fluid. We are led to investigate the asymptotic
behavior of the Dirichlet energy associated to the solution of a Laplace
Neumann problem as the distance between the solid and the
cavity's bottom tends to zero. Denoting by the tangency exponent at
the contact point, we prove that the solid always reaches the cavity in finite
time, but with a non zero velocity for (real shock case), and with
null velocity for (smooth landing case). Our proof is
based on a suitable change of variables sending to infinity the cusp
singularity at the contact. More precisely, for every ,
we transform the Laplace Neumann problem into a generalized Neumann problem set
on a domain containing a horizontal strip ,
where
Justification of lubrication approximation: an application to fluid/solid interactions
We consider the stationary Stokes problem in a three-dimensional fluid domain
with non-homogeneous Dirichlet boundary conditions. We assume that
this fluid domain is the complement of a bounded obstacle in a
bounded or an exterior smooth container . We compute sharp asymptotics
of the solution to the Stokes problem when the distance between the obstacle
and the container boundary is small
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
Impermeability through a perforated domain for the incompressible 2D Euler equations
We study the asymptotic behavior of the motion of an ideal incompressible
fluid in a perforated domain. The porous medium is composed of inclusions of
size separated by distances and the fluid fills
the exterior.
If the inclusions are distributed on the unit square, the asymptotic behavior
depends on the limit of when
goes to zero. If , then the limit
motion is not perturbed by the porous medium, namely we recover the Euler
solution in the whole space. On the contrary, if
, then the fluid cannot penetrate the
porous region, namely the limit velocity verifies the Euler equations in the
exterior of an impermeable square.
If the inclusions are distributed on the unit segment then the behavior
depends on the geometry of the inclusion: it is determined by the limit of
where is related to the geometry of the lateral boundaries of the
obstacles. If , then the presence of holes is not felt at the limit, whereas an
impermeable wall appears if this limit is zero. Therefore, for a distribution
in one direction, the critical distance depends on the shape of the inclusions.
In particular it is equal to for balls
Numerical hydrodynamics in general relativity
The current status of numerical solutions for the equations of ideal general
relativistic hydrodynamics is reviewed. With respect to an earlier version of
the article the present update provides additional information on numerical
schemes and extends the discussion of astrophysical simulations in general
relativistic hydrodynamics. Different formulations of the equations are
presented, with special mention of conservative and hyperbolic formulations
well-adapted to advanced numerical methods. A large sample of available
numerical schemes is discussed, paying particular attention to solution
procedures based on schemes exploiting the characteristic structure of the
equations through linearized Riemann solvers. A comprehensive summary of
astrophysical simulations in strong gravitational fields is presented. These
include gravitational collapse, accretion onto black holes and hydrodynamical
evolutions of neutron stars. The material contained in these sections
highlights the numerical challenges of various representative simulations. It
also follows, to some extent, the chronological development of the field,
concerning advances on the formulation of the gravitational field and
hydrodynamic equations and the numerical methodology designed to solve them.Comment: 105 pages, 12 figures. The full online-readable version of this
article, including several animations, will be published in Living Reviews in
Relativity at http://www.livingreviews.or
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