Quelques résultats d'existence, de contrôlabilité et de stabilisation pour des systèmes couplés fluide-structure

Dans cette thèse nous étudions des systèmes couplés fluide-structure. Ces systèmes peuvent modéliser un écoulement sanguin dans un vaisseau large ou un problème d'aéroélasticité. La vitesse et la pression du fluide sont décrites par les équations de Navier-Stokes incompressibles et le déplacement de la structure frontière est régi par une équation de poutre/plaque/membrane (selon la dimension du modèle et la nature de la structure). Dans la première partie, nous montrons l'existence de solutions fortes pour de tels systèmes en deux ou trois dimensions, soit pour des conditions initiales petites (existence globale en temps), soit pour des conditions initiales quelconques (existence locale en temps). Dans une seconde partie, nous étudions d'abord la contrôlabilité à zéro d'un système couplant les équations de Navier-Stokes à une équation de structure correspondant à une approximation de dimension finie des modèles de poutres ou de plaques. Nous étudions ensuite la stabilisation (pour tout taux de décroissance), locale au voisinage de la solution nulle, d'un système couplant les équations de Navier-Stokes à deux équations de poutres, par deux contrôles de dimension finie agissant dans l'équation de la structure et dans la deuxième condition au bord pour la vitesse. Le second contrôle ne dépend que du temps.In this thesis, we are interested in the study of fluid-structure systems. These systems may model blood flows in large vessels or aeroelasticy problems. The velocity and the pressure of the blood are described by the incompressible Navier-Stokes equations and the displacement of the structure boundary satisfies a beam/plate/membrane equation (it depends on the dimension of the model and of the nature of the structure). In the fist part, we prove the exitence and uniqueness of strong solutions to the kind of systems in two or three dimensions, either for small initial data (global in time existence) or for any initial data (local in time existence). In the second part, we study on one hand the null controllability of a system coupling the Navier-Stokes equations with a structure equation corresponding with a finite dimensional approximation of the beam or plate equation. On the other hand, we study the stabilization (for any decay rate) local around the stationary null solution of a system coupling the Navier-Stokes equations with two beam equations with two finite dimension controls acting on the structure equation and in the second boundary condition for the velocity. The second control only depends on time

Quelques résultats d'existence, de contrôlabilité et de stabilisation pour des systèmes couplés fluide - structure

In this thesis, we are interested in the study of fluid-structure systems. These systems may model blood flows in large vessels. The velocity and the pressure of the blood are sdescribed by the incompressible Navier-Stokes equations and the displacement of the mobile part of the boundary satisfies a beam/plate equation (it depends on the dimension of the model). In the fist part, we prove the exitence and uniqueness of strong solutions to two systems (they correspond with the zero or nonzero value of a certain paramter) in two and three dimensions. More precisely, we prove the following alternative. We have either global existence for small initial data or local existence for any initial data. In the second part, we study on one hand the null controllability of a system coupling the Navier-Stokes equations with a finite dimensional beam equation for small initial data in two dimensions. On the other hand, we prove the stabilzation (for any decay rate) of a system coupling the Navier-Stokes equations with two beam equations with two controls in the periodic setting for small initial data. In this case, the controls are of finite dimension.Nous nous intéressons dans cette thèse à l'étude de systèmes couplés fluide-structure. Ces systèmes peuvent modéliser l'écoulement du sang dans un vaisseau large. La vitesse et la pression du sang sont alors décrites par les équations de Navier-Stokes incompressibles et le déplacement de la partie mobile de la frontière vérifie une équation des poutres/ plaques (selon la dimension du modèle). Dans la première partie, nous montrons l'existence de solutions fortes à deux systèmes (correspondant à un paramètre nul ou non) en deux ou trois dimensions. Plus précisément, nous prouvons l'alternative suivante. Nous avons soit l'existence globale pour des conditions initiales petites, soit l'existence locale pour des conditions initiales quelconques. Dans une seconde partie, nous étudions d'une part la contrôlabilité à zéro d'un système couplant les équations de Navier-Stokes à une équation différentielle ordinaire pour des conditions initiales petites en deux dimensions. D'autre part, nous montrons la stabilisation (pour tout taux de décroissance) d'un système couplant les équations de Navier-Stokes et deux équations des plaques par deux contrôles dans le cadre périodique pour des conditions initiales petites. Dans ce cas, les contrôles sont de dimension finie

Vorticity and stream function formulations for the 2d Navier-Stokes equations in a bounded domain

International audienceThe main purpose of this work is to provide a Hilbertian functional framework for the analysis of the planar Navier-Stokes (NS) equations either in vorticity or in stream function formulation. The fluid is assumed to occupy a bounded possibly multiply connected domain. The velocity field satisfies either homogeneous (no-slip boundary conditions) or prescribed Dirichlet boundary conditions. We prove that the analysis of the 2D Navier-Stokes equations can be carried out in terms of the so-called nonprimitive variables only (vorticity field and stream function) without resorting to the classical NS theory (stated in primitive variables, i.e. velocity and pressure fields). Both approaches (in primitive and nonprimitive variables) are shown to be equivalent for weak (Leray) and strong (Kato) solutions. Explicit, Bernoulli-like formulas are derived and allow recovering the pressure field from the vorticity fields or the stream function. In the last section, the functional framework described earlier leads to a simplified rephrasing of the vorticity dynamics, as introduced by Maekawa in [52]. At this level of regularity, the vorticity equation splits into a coupling between a parabolic and an elliptic equation corresponding respectively to the non-harmonic and harmonic parts of the vorticity equation. By exploiting this structure it is possible to prove new existence and uniqueness results, as well as the exponential decay of the palinstrophy (that is, loosely speaking, the $H^1$ norm of the vorticity) for large time, an estimate which was not known so far

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

An energy stable one-field monolithic arbitrary Lagrangian-Eulerian formulation for fluid-structure interaction

In this article we present a one-field monolithic finite element method in the Arbitrary Lagrangian–Eulerian (ALE) formulation for Fluid–Structure Interaction (FSI) problems. The method only solves for one velocity field in the whole FSI domain, and it solves in a monolithic manner so that the fluid solid interface conditions are satisfied automatically. We prove that the proposed scheme is unconditionally stable, through energy analysis, by utilising a conservative formulation and an exact quadrature rule. We implement the algorithm using both F-scheme and D-scheme, and demonstrate that the former has the same formulation in two and three dimensions. Finally several numerical examples are presented to validate this methodology, including combination with remesh techniques to handle the case of very large solid displacement

Existence of strong solutions to a fluid-structure system

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Weak Solutions for a System Modeling the Movement of a Piston in a Viscous Compressible Gas

International audienceWe first study the global-in-time existence of strong solutions to a one-dimensional system modeling the movement of a piston in a viscous compressible gas. Moreover, we prove the asymptotic stability of the solution toward a chosen constant state (in the sense that we can impose the final position of the piston, the final densities being fixed by the conservation of mass and the choice of the final position) thanks to a constant force acting in the equation of the point mass whose expression depends explicitly of the chosen final position. The norm of the solution in the function space of the initial data decays exponentially toward this constant state. Then, we prove the existence of weak solutions to this system for initial velocity in the energy state and for the initial density with bounded total variation. The weak solution is unique and also decay exponentially toward the chosen constant state thanks to the same constant force acting on the point mass. We use the result of existence of strong solutions to prove the existence of weak solutions, whereas the result on exponential decay of weak solution is independent of the one for the strong solutions

Existence of strong solutions for a system coupling the Navier-Stokes equations and a damped wave equation

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Existence of local strong solutions to fluid-beam and fluid-rod interaction systems

International audienceWe study an unsteady nonlinear fluid–structure interaction problem. 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 wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. We prove existence of a unique local-in-time strong solution. In the case of a damped beam this is an alternative proof (and a generalization) of the result that can be found in [19]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. One key point, is to use the fluid dissipation to control, in appropriate function spaces, the structure velocity