Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve

In this work we study the asymptotic behavior of solutions of the incompressible two-dimensional Euler equations in the exterior of a single smooth obstacle when the obstacle becomes very thin tending to a curve. We extend results by Iftimie, Lopes Filho and Nussenzveig Lopes, obtained in the context of an obstacle tending to a point, see [Comm. PDE, {\bf 28} (2003), 349-379]

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 $\varepsilon$ separated by distances $d_\varepsilon$ and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of $\frac{d_{\varepsilon}}\varepsilon$ when $\varepsilon$ goes to zero. If $\frac{d_{\varepsilon}}\varepsilon\to \infty$, 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 $\frac{d_{\varepsilon}}\varepsilon\to 0$, 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 $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}}$ where $\gamma\in (0,\infty]$ is related to the geometry of the lateral boundaries of the obstacles. If $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}} \to \infty$, 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 $\varepsilon^3$ for balls

Small moving rigid body into a viscous incompressible fluid

We consider a single disk moving under the influence of a 2D viscous fluid and we study the asymptotic as the size of the solid tends to zero.If the density of the solid is independent of $\varepsilon$, the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This will be achieved thanks to the optimal $L^p-L^q$ decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument. Next, we will deduce the convergence to the solution of the Navier-Stokes equations in $\R^2$

The vanishing viscosity limit in the presence of a porous medium

We consider the flow of a viscous, incompressible, Newtonian fluid in a perforated domain in the plane. The domain is the exterior of a regular lattice of rigid particles. We study the simultaneous limit of vanishing particle size and distance, and of vanishing viscosity. Under suitable conditions on the particle size, particle distance, and viscosity, we prove that solutions of the Navier-Stokes system in the perforated domain converges to solutions of the Euler system, modeling inviscid, incompressible flow, in the full plane. That is, the flow is not disturbed by the porous medium and becomes inviscid in the limit. Convergence is obtained in the energy norm with explicit rates of convergence

Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex

We prove uniqueness for the vortex-wave system with a single point vortex introduced by Marchioro and Pulvirenti in the case where the vorticity is initially constant near the point vortex. Our method relies on the Eulerian approach for this problem and in particular on the formulation in terms of the velocity

The Two Dimensional Euler Equations on Singular Exterior Domains

This paper is a follow-up of article [Gerard-Varet and Lacave, ARMA 2013], on the existence of global weak solutions to the two dimensional Euler equations in singular domains. In [Gerard-Varet and Lacave, ARMA 2013], we have established the existence of weak solutions for a large class of bounded domains, with initial vorticity in $L^p$ ($p>1$). For unbounded domains, we have proved a similar result only when the initial vorticity is in $L^p_{c}$ ($p>2$) and when the domain is the exterior of a single obstacle. The goal here is to retrieve these two restrictions: we consider general initial vorticity in $L^1\cap L^p$ ($p>1$), outside an arbitrary number of obstacles (not reduced to points)

Two Dimensional Incompressible Ideal Flow Around a Small Curve

We study the asymptotic behavior of solutions of the two dimensional incompressible Euler equations in the exterior of a curve when the curve shrinks to a point. This work links two previous results: [Iftimie, Lopes Filho and Nussenzveig Lopes, Two Dimensional Incompressible Ideal Flow Around a Small Obstacle, Comm. PDE, 28 (2003), 349-379] and [Lacave, Two Dimensional Incompressible Ideal Flow Around a Thin Obstacle Tending to a Curve, Ann. IHP, Anl, 26 (2009), 1121-1148]. The second goal of this work is to complete the previous article, in defining the way the obstacles shrink to a curve. In particular, we give geometric properties for domain convergences in order that the limit flow be a solution of Euler equations

Two Dimensional Incompressible Viscous Flow Around a Thin Obstacle Tending to a Curve

International audienceIn [Lacave, IHP, ana, to appear (2008)] the author considered the two dimensional Euler equations in the exterior of a thin obstacle shrinking to a curve and determined the limit velocity. In the present work, we consider the same problem in the viscous case, proving convergence to a solution of the Navier-Stokes equations in the exterior of a curve. The uniqueness of the limit solution is also shown