This paper deals with shape derivative of a cost functional in non-stationary incompressible Navier-Stokes equations. In fact, after computing shape derivative of a given cost functional J(u, p) where the pair (u, p) solves Navier-Stokes equations, we build a numerical schema and an algorithm using shape derivative as descent direction. This method allows us to minimize the energy and structure from volumic constraints. The numerical results obtained confirm the efficiency of the method.

shape derivative, Navier-Stokes system, adjoint equations, numerical method.