什么是管道的最佳形状？
What is the optimal shape of a pipe?
摘要：
考虑三维管道中的不可压缩流体，遵循经典边界条件的Navier-Stokes方程组。我们感兴趣的问题是：是否有任何最佳形状的标准“能量耗散的流体”？此外，圆柱形是最佳形状吗？我们证明了在合理的容许域类中存在最优形状，但圆柱不是最优的。为此，我们明确了一阶最优性条件，由于伴随状态，我们证明了这是不可能的，伴随状态是这个超定系统的解决方案时，域是圆柱。最后，我们给出了该问题的一些数值模拟。
英文摘要：
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.
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