What is the optimal shape of a pipe?

TL;DR

This paper investigates the optimal shape of a pipe to minimize energy dissipation in fluid flow, proving the cylinder is not optimal and identifying conditions for optimality through mathematical analysis and simulations.

Contribution

It establishes the existence of an optimal pipe shape, derives first-order optimality conditions, and demonstrates that the cylinder shape is not optimal for energy dissipation.

Findings

The optimal shape exists within a broad class of domains.

The cylinder is proven not to be the optimal shape.

Numerical simulations support the theoretical results.

Abstract

We consider an incompressible fluid in a three-dimensional pipe, following the Navier-Stokes system with classical boundary conditions. We are interested in the following question: is there any optimal shape for the criterion "energy dissipated by the fluid"? Moreover, is the cylinder the optimal shape? We prove that there exists an optimal shape in a reasonable class of admissible domains, but the cylinder is not optimal. 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 when the domain is the cylinder. At last, we show some numerical simulations for that problem.