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 the two previous results [5 Iftimie , D. , Lopes Filho , M.C. , Nussenzveig Lopes , H.J. ( 2003 ). Two dimensional incompressible ideal flow around a small obstacle . Comm. Part. Diff. Eqs. 28 : 349 – 379 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 9 Lacave , C. ( 2009 ). Two dimensional incompressible ideal flow around a thin obstacle tending to a curve . Annales de l'IHP, Anl. 26 : 1121 – 1148 . [Google Scholar]]. 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.     

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.     

THE VORTEX BLOB METHOD AS A SECOND-GRADE NON-NEWTONIAN FLUID

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity and initial data in the space of Radon measures M (R ²). The solutions of this regularized PDE, also known as the isotropic Lagrangian averaged Euler or Euler-α equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in M (R ²) such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes as are common in practical vortex computations.     

Homogenization in general periodically perforated domains by a spectral approach

In this article we study the asymptotic behaviour as tends to 0 of the Neumann problem $-\Delta u_\epsilon+u_\epsilon=\epsilon$-periodic bounded open set of . The period cell of is equal to where is a regular open subset of the d-dimensional torus. We prove that if there exists a smallest integer such that the n-th non-zero eigenvalue of the spectral problem in satisfies , the limiting problem is a linear system of second order p.d.e.'s, of size n. By this spectral approach we extend in the periodic framework a result due to Khruslov without making strong geometrical assumptions on the perforated domain . Received: 20 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001     

A corrector for the sverdrup solution for a domain with islands

In this paper we look at the influence of the Coriolis force on the quasi-geostrophic equations on a domain with islands. We prove that asymptotically we obtain the solution of the Sverdrup equation with homogeneous Dirichlet conditions on the inward boundary plus a corrector function which takes into account the presence of the islands. This work is motivated by the fact that in oceanography most of the surfaces are not simply connected. This is the case for example for the North Pacific with the Japanese islands. At our knowledge, in all the previous mathematical works, just simply connected domains have been considered. Finally we will give some simple numerical simulations related to the Stommel model to see the importance of the corrector.