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A cylindrical pipe is not optimal to minimize the energy dissipated by a fluid. (Une conduite cylindrique n’est pas optimale pour minimiser l’énergie dissipée par un fluide.) (French. Abridged English version) Zbl 1156.35444

Summary: We consider incompressible fluid in a three-dimensional cylindrical pipe, satisfying 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 is a solution of this over-determined system.

MSC:

35Q30 Navier-Stokes equations
49K20 Optimality conditions for problems involving partial differential equations
76D55 Flow control and optimization for incompressible viscous fluids
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