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

We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurement of the Cauchy force on a part of the exterior boundary. We deal with the case where the fluid equations are the non stationary 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 information on the distance from a point to the obstacle.