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

Abstract

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.

Some inverse and control problems for fluids

“…Then, in order to achieve the exact Lagrangian controllability property between γ 0 and γ 1 at time T , it is enough to construct a map 14) where ν denotes the unit outwards normal to φ ∇xψ (t, 0, γ 0 ).…”

“…For many other questions and results, the reader is referred for instance to the general references [28,29,34,35,37,38]; see also [5,14,49].…”

On the Detection of Small Moving Disks in a Fluid

“…In Conca, Malik, and Munnier [16], the authors consider a moving rigid solid immersed in a potential fluid and provide examples of detectable (ellipses for instance) and undetectable shapes. Conca, Schwindt, and Takahashi obtained in [17] an identifiability result in the case of a rigid solid immersed in a viscous fluid. In this work, we restrict the analysis to the case where the solids are small disks (see Figure 1) We answer this question in several steps.…”

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

“…Let us remark that the above system is not linear since S(t) is not given. This system is studied in [7] where the identifiability of the rigid body is obtained through the measurement of the Cauchy forces on the boundary. Like system (1.24)-(1.35), the solid moves through the action of f on this system.…”

“…They develop an integral method in order to recover the structure. The identifiability result of [1] is extended in [7] to the case of a moving rigid body, but only in the case of the stationary Stokes system. In the case of a potential fluid (thus inviscid), one can use, in 2D, complex analysis ( [5], [6]) to detect a moving rigid body of particular shape (ball, ellipse) if the fluid fills the exterior of the structure domain.…”