We consider the inverse problem of the detection of a single body, immersed in a bounded container filled with a fluid which obeys the stationary Stokes or Navier Stokes equations, from a single measurement of force and velocity on a portion of the boundary. Under appropriate a priori hypotheses we obtain an estimate of stability of log-log type for both cases. We then present a numerical method for the reconstruction of the body using a boundary elements representation of the solutions, combined with the iteratively regularized Gauss-Newton method, and present some partial numerical results in this direction.
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