Identification of immersed obstacles via boundary measurements (original) (raw)

On the identifiability of a rigid body moving in a stationary viscous fluid

Inverse Problems, 2012

This paper is devoted to a geometrical inverse problem associated with a fluidstructure system. More precisely, we consider the interaction between a moving rigid body and a viscous and incompressible fluid. Assuming a low Reynolds regime, the inertial forces can be neglected and, therefore, the fluid motion is modelled by the Stokes system. We first prove the well posedness of the corresponding system. Then we show an identifiability result: with one measure of the Cauchy forces of the fluid on one given part of the boundary and at some positive time, the shape of a convex body and its initial position are identified.

On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

Journal of Inverse and Ill-posed Problems, 2017

We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. We deal with the case where the fluid equations are the nonstationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid-structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid-structure system with free boundary. Using complex spherical waves, we obtain some partial information on the distance from a point to the obstacle.

Detection of a moving rigid solid in a perfect fluid, in "Inverse Problems

2016

On the identification of a single body immersed in a Navier-Stokes fluid

European Journal of Applied Mathematics, 2007

In this work we consider the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Navier-Stokes equations. It is assumed that friction forces are known on a part of the outer boundary. We first prove a uniqueness result. Then, we establish a formula for the observed friction forces, at first order, in terms of the deformation of the rigid body. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs we use unique continuation and regularity results for the Navier-Stokes equations and domain variation techniques.

On the detection of a moving obstacle in an ideal fluid by a boundary measurement

Inverse Problems, 2008

In this paper, we investigate the problem of the detection of a moving obstacle in a perfect fluid occupying a bounded domain in R 2 from the measurement of the velocity of the fluid on one part of the boundary. We show that when the obstacle is a ball, we may identify the position and the velocity of its centre of mass from a single boundary measurement. Linear stability estimates are also established by using shape differentiation techniques.

Detection of a moving rigid solid in a perfect fluid

Inverse Problems, 2010

In this paper, we consider a moving rigid solid immersed in a potential fluid. The fluid-solid system fills the whole two dimensional space and the fluid is assumed to be at rest at infinity. Our aim is to study the inverse problem, initially introduced in [3], that consists in recovering the position and the velocity of the solid assuming that the potential function is known at a given time. We show that this problem is in general ill-posed by providing counterexamples for which the same potential corresponds to different positions and velocities of a same solid. However, it is also possible to find solids having a specific shape, like ellipses for instance, for which the problem of detection admits a unique solution. Using complex analysis, we prove that the well-posedness of the inverse problem is equivalent to the solvability of an infinite set of nonlinear equations. This result allows us to show that when the solid enjoys some symmetry properties, it can be partially detected. Besides, for any solid, the velocity can always be recovered when both the potential function and the position are supposed to be known. Finally, we prove that by performing continuous measurements of the fluid potential over a time interval, we can always track the position of the solid. * C. Conca thanks the MICDB for partial support through Grant ICM P05-001-F, Fondap-Basal-Conicyt, and the French & Chilean Governments through Ecos-Conicyt Grant C07E05. † Author supported by ANR CISIFS and ANR GAOS.

Size estimates of an obstacle in a stationary Stokes fluid

Inverse Problems

In this work we are interested in estimating the size of a cavity D immersed in a bounded domain Ω ⊂ R d , d = 2, 3, filled with a viscous fluid governed by the Stokes system, by means of velocity and Cauchy forces on the external boundary ∂Ω. More precisely, we establish some lower and upper bounds in terms of the difference between the external measurements when the obstacle is present and without the object. The proof of the result is based on interior regularity results and quantitative estimates of unique continuation for the solution of the Stokes system.

Motion of a rigid body in a viscous fluid

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1999

We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary. 0 Acadkmie des Sciences/Elsevier, Paris Mouvement d'un solide rigide dans un jluide visqueux R&sum& On introduit un concept de solution faible pour un probEme des conditions aux limites qui mode'lise le mouvement d'un solide rigide dans un fluide visqueux. L..a variation du domainej-luide n'est pas connue a priori, done il s'agit din probEme de front&-e libre. Le thtfor2me principal donne l'existence d'une solution faible. Cette solution est globale en temps tant que le solide ne touche pas le bord. 0 AcadCmie des SciencesElsevier. Paris Version frangaise abrt@e Soit A c Ws un ouvert born6 qui repr&ente le domaine occupC par le fluide et le solide rigide. Par simplicid, on suppose que le solide est une boule de rayon 1 et que le fluide est homogbne de densit 1. On note par z(t) la position du centre de la boule et par B(t), Q(t), les domaines occup6s par le corps solide et par le fluide B l'instant t, respectivement ; voir (1) et (2). Le mod?le mathkmatique rkgissant leurs mouvements est d&-it par les Cquations (8)-(15). Dans (18)-(20) se trouve la definition de solutions faibles pour ce modkle. Le thkortime principal de cette Note (thCor&me 3) assure l'existence d'au moins une solution faible. Sa preuve consiste A introduire un nouveau problitme dit p&nalisC (cJ ), pour lequel l'existence d'une solution est classique, et de passer en_suite g la limite lorsque le paramktre de pknalisation tend vers l'infini. L'existence de la fonction h(t) (lemme 4) dkcoule de l'estimation a priori de I'Cnergie (~6 (22)). On se sert enfin d'un r&.&at de compacitk qui nous permet de conclure que la limite de la suite de solutions pCnalisCes satisfait les iquations du syst&me fluide-solide rigide.

Identification of immersed obstacles via boundary measurements (original) (raw)

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