On the detection of a moving rigid solid in a perfect fluid

Carlos Conca (1) , Muslim Malik (1) , Alexandre Munnier (2, 3)

1 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]
2 CORIDA - Robust control of infinite dimensional systems and applications
3 IECN - Institut Élie Cartan de Nancy

In this paper, we consider a moving rigid solid immersed in a potential fluid. The fluid-solid system fills the whole space and the fluid is assumed to be at rest at infinity. Our aim is to study the inverse problem, initially introduced in \cite{Conca:2008aa}, 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 several counterexamples for which the same potential corresponds to different positions and velocities of a given solid. However, for solids having specific shapes, like ellipses for instance, this phenomena can not occur and 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 approach allows us to show that when the solid enjoys some symmetry properties, it can be {\it partially} detected. Further, for any solid, the velocity can always be recovered when both the potential function and the position are supposed to be known. At last, we prove that by performing continuous measurements of the fluid potential over a time interval, we can always track the position of the solid.

Domaines: Equations aux dérivées partielles [math.AP]

Carlos Conca, Muslim Malik, Alexandre Munnier. On the detection of a moving rigid solid in a perfect fluid. Inverse Problems, 2010, 26 (9), ⟨10.1088/0266-5611/26/9/095010⟩. ⟨inria-00468480v2⟩