In this paper, numerical sensitivity analysis with respect to the Reynolds number for the flow past obstacle problem is presented. To carry out such analysis, at each time step, we need to solve the incompressible Navier-Stokes equations on irregular domains twice, one for the primary variables; the other is for the sensitivity variables with homogeneous boundary conditions. The Navier-Stokes solver is the augmented immersed interface method for Navier-Stokes equations on irregular domains. One of the most important contribution of this paper is that our analysis can predict the critical Reynolds number at which the vortex shading begins to develop in the wake of the obstacle. Some interesting experiments are shown to illustrate how the critical Reynolds number varies with different geometric settings.

In this paper, we prove a controllabilityresult for a fluid-structure interaction problem. In dimension two,a rigid structure moves into an incompressible fluid governed byNavier-Stokes equations. The control acts on a fixed subset of thefluid domain. We prove that, for small initial data, this system isnull controllable, that is, for a given T > 0, the system can bedriven at rest and the structure to its reference configuration attime T. To show this result, we first consider a linearizedsystem. Thanks to an observability inequality obtained from aCarleman inequality, we prove an optimal controllability result witha regular control. Next, with the help of Kakutani's fixed pointtheorem and a regularity result, we pass to the nonlinear problem.