Schulz, Andreas The optimal shape of a pipe. (English) Zbl 1271.76083 Z. Angew. Math. Phys. 64, No. 4, 1177-1185 (2013). Summary: In shape optimization, recently the question arose, whether or not the cylindrical pipe has the optimal shape for the transport of an incompressible fluid. In this short note, a proof will be presented that a cylindrical pipe with Poiseuille’s flow inside indeed is optimal for the transportation of an incompressible fluid under the criterion “energy dissipated by the fluid“. The proof reduces the problem to the minimization of a two-dimensional Dirichlet’s integral. This simpler problem can be solved with a symmetrization argument. Cited in 1 Document MSC: 76D55 Flow control and optimization for incompressible viscous fluids Keywords:shape optimization; laminar fluid flow; Poiseuille’s flow PDFBibTeX XMLCite \textit{A. Schulz}, Z. Angew. Math. Phys. 64, No. 4, 1177--1185 (2013; Zbl 1271.76083) Full Text: DOI References: [1] Grinberg, E.L.: On the smoothness hypothesis in Sard’s theorem. The American Mathematical Monthly 92(10):733-734 JSTOR (1985) · Zbl 0633.58002 [2] Henrot A., Privat Y.: What is the optimal shape of a pipe?. Arch. Rational Mech. Anal. 196(1), 281-302 (2010) · Zbl 1304.76022 [3] Lieb, E.H., Loss, M.: Analysis 2. ed., Graduate studies in mathematics 14, American Math. Soc (2010) [4] Pironneau O., Arumugam G.: On riblets in laminar flows. Control of Boundaries and Stabilization 125, 51-65 (1989) · Zbl 0673.76039 [5] Privat, Y.: Quelques problèmes d’optimisation de formes en sciences du vivant. PhD-thesis, Université Henri Poincaré, Nancy-I (2008) [6] Schulz, A.: Über die optimale Rohrform beim Flüssigkeitstransport. diploma thesis, RWTH Aachen (2008) · Zbl 0673.76039 [7] Sokolowski J., ZolÃsio J.-P.: Introduction to Shape Optimization. Springer Series in Computational Mathematics 16. Springer, Berlin (1992) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.