Justification of lubrication approximation: an application to fluid/solid interactions

We consider the stationary Stokes problem in a three-dimensional fluid domain $\mathcal F$ with non-homogeneous Dirichlet boundary conditions. We assume that this fluid domain is the complement of a bounded obstacle $\mathcal B$ in a bounded or an exterior smooth container $\Omega$. We compute sharp asymptotics of the solution to the Stokes problem when the distance between the obstacle and the container boundary is small

Spectral simplicity and asymptotic separation of variables

We describe a method for comparing the real analytic eigenbranches of two families of quadratic forms that degenerate as t tends to zero. One of the families is assumed to be amenable to `separation of variables' and the other one not. With certain additional assumptions, we show that if the families are asymptotic at first order as t tends to 0, then the generic spectral simplicity of the separable family implies that the eigenbranches of the second family are also generically one-dimensional. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian is one-dimensional. We also show that for all but countably many t, the geodesic triangle in the hyperbolic plane with interior angles 0, t, and t, has simple spectrum.Comment: 53 pages, 2 figure

Asymptotic description of solutions of the exterior Navier Stokes problem in a half space

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier-Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. We focus on the case where the size of the body is small. We prove in a very general setup that the solution of this problem is unique and we compute a sharp decay rate of the solution far from the moving body and the wall

Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior

We consider a model of surface-mediated diffusion with alternating phases of pure bulk and surface diffusion. For this process, we compute the mean exit time from a disk through a hole on the circle. We develop a spectral approach to this escape problem in which the mean exit time is explicitly expressed through the eigenvalues of the related self-adjoint operator. This representation is particularly well suited to investigate the asymptotic behavior of the mean exit time in the limit of large desorption rate $\lambda$. For a point-like target, we show that the mean exit time diverges as $\sqrt{\lambda}$. For extended targets, we establish the asymptotic approach to a finite limit. In both cases, the mean exit time is shown to asymptotically increase as $\lambda$ tends to infinity. We also revise the optimality regime of surface-mediated diffusion. Although the presentation is limited to the unit disk, the spectral approach can be extended to other domains such as rectangles or spheres.Comment: 21 pages, 7 figure

Hindered Settling of Well-Separated Particle Suspensions

We consider $N$ identical inertialess rigid spherical particles in a Stokes flow in a domain $\Omega \subset \mathbb R^3$. We study the average sedimentation velocity of the particles when an identical force acts on each particle. If the particles are homogeneously distributed in directions orthogonal to this force, then they hinder each other leading to a mean sedimentation velocity which is smaller than the sedimentation velocity of a single particle in an infinite fluid. Under suitable convergence assumptions of the particle density and a strong separation assumption, we identify the order of this hindering as well as effects of small scale inhomogeneities and boundary effects. For certain configurations we explicitly compute the leading order corrections.Comment: All comments welcome

Reducing the debt : is it optimal to outsource an investment?

International audienceWe deal with the problem of outsourcing the debt for a big investment, according two situations: either the firm outsources both the investment (and the associated debt) and the exploitation to a private consortium, or the firm supports the debt and the investment but outsources the exploitation. We prove the existence of Stackelberg and Nash equilibria between the firm and the private consortium, in both situations. We compare the benefits of these contracts. We conclude with a study of what happens in case of incomplete information, in the sense that the risk aversion coefficient of each partner may be unknown by the other partner

Existence of global strong solutions to a beam-fluid interaction system

We study an unsteady non linear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoleastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure