A cylindrical pipe is not optimal to minimize the energy dissipated by a fluid

We consider an incompressible fluid in a three-dimensional cylindrical pipe, following the Navier-Stokes system with classical boundary conditions on the boundary of the cylinder. We are interested in the following question: is the cylinder the optimal shape for the criterion "energy dissipated by the fluid"? 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. To cite this article: A. Henrot, Y. Prival, C. R. Acad. Sci. Paris, Ser. I 346 (2008). (C) 2008 Academie des sciences. Publie par Elsevier Masson SAS. Tous droits reserves.