8 pages
English
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Control of some fluid structure interactions

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8 pages
English

Description

Niveau: Supérieur, Doctorat, Bac+8
Titles and abstracts Mini-course Control of some fluid-structure interactions by Marius Tucsnak – Universite Henri Poincare, Nancy, France Abstract: In many practical problems a fluid interacts with a solid structure, exerting stresses that may cause deformation in the structure and, thus, alter the flow of the fluid itself. These phenomena are usually called fluid-structure interactions and they occur, for instance, in aerodynamics (flow around an aircraft), medicine (blood flow in vessels), zoology (swimming of aquatic animals). The mathematical study of these problems rises several challenges, the main one being due to the fact that the domain filled by the fluid is one of the unknowns of the problem. Another difficulty which has to be tackled is that the dynamics of the system couples the ordinary or partial differential equations modeling the solid with the partial differential equations modeling the fluid. Within this work we focus on the coupled motion of a collection of solids and of a fluid surround- ing them. The solids are either rigid or they have a prescribed deformation law. According to the fact that the fluid ideal, viscous (compressible or incompressible) or non-Newtonian, the corresponding mathematical models are given by systems of partial and ordinary differential equations of increasing complexity. The common features of these models are the facts that the equations for the fluid and for the solids are coupled via the boundary conditions and that the equations for the fluid hold in a spatial domain which is variable with respect to time.

  • relative velocity

  • fluid-structure system

  • equation governed

  • navier-stokes fluid

  • system unboundedness

  • when both

  • approximate controllability

  • numerical results

  • equation


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Nombre de lectures 20
Langue English

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Titles and abstracts
Mini-course Control of some fluid-structure interactions
byMsTucariuUinnskatie´evsroiiPnrHe,N´earncarF,ycnaecn
Abstract: In many practical problems a fluid interacts with a solid structure, exerting stresses that may cause deformation in the structure and, thus, alter the flow of the fluid itself. These phenomena are usually called fluid-structure interactions and they occur, for instance, in aerodynamics (flow around an aircraft), medicine (blood flow in vessels), zoology (swimming of aquatic animals). The mathematical study of these problems rises several challenges, the main one being due to the fact that the domain filled by the fluid is one of the unknowns of the problem. Another difficulty which has to be tackled is that the dynamics of the system couples the ordinary or partial differential equations modeling the solid with the partial differential equations modeling the fluid. Within this work we focus on the coupled motion of a collection of solids and of a fluid surround-ing them. The solids are either rigid or they have a prescribed deformation law. According to the fact that the fluid ideal, viscous (compressible or incompressible) or non-Newtonian, the corresponding mathematical models are given by systems of partial and ordinary differential equations of increasing complexity. The common features of these models are the facts that the equations for the fluid and for the solids are coupled via the boundary conditions and that the equations for the fluid hold in a spatial domain which is variable with respect to time. This domain is, in most of the cases, an unknown of the problem, so that we have to tackle a free boundary value problem. However, the mathematical analysis of these problems is expected to be simpler than in the general free boundary case since for particulate flows the free boundary has a finite number of degrees of freedom, namely 6m, wheremis the number of rigid bodies (in the case of a three dimensional flow). The emphasis in this lecture will be given to controllability issues. Two types of questions will be considered:
the fluid and of the solid by acting on a part of the fluid or of the exteriorControl of both boundary:This question will be tackled for a the system modeling a rigid body immersed in an incompressible Navier-Stokes fluid. A one dimensional toy model will be used to illustrate the main techniques.
A control theoretical approach of self-propelling of solids in a fluid:This question will be tackled for various models. The input function is either the shape of the solid in the case of deformable bodies or the relative velocity of the fluid with respect to the solid in the case of rigid bodies. The considered models are highly depending on the flow regime. We emphasis the case of low Reynolds numbers (corresponding to the swimming
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