In this talk, we are interested in the optimal shape of a pipe (inlet and outlet are fixed and the volume is prescribed). We consider an incompressible fluid, subject to the Navier-Stokes equations with classical boundary conditions on the boundary of the cylinder (velocity profile given on the inlet, no slip condition on the lateral boundary and outlet-pressure condition on the outlet). We are interested in the optimal shape of the pipe for the criterion "energy dissipated by the fluid"? In this talk, we will consider the following questions:

• Does there exist a minimizer?
• Is the cylinder the optimal shape? We prove that it is not the case.

For that purpose, we explicit the first order optimality condition, thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system.

This is joint work with Yannick Privat.