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What is the optimal shape of a pipe? (English) Zbl 1304.76022

Summary: 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 define the first order optimality condition, thanks to the 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.

MSC:

76D55 Flow control and optimization for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76M30 Variational methods applied to problems in fluid mechanics
49J20 Existence theories for optimal control problems involving partial differential equations
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[1] Allaire, G.: Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, vol. 146. Springer, New York, 2002 · Zbl 0990.35001
[2] Arumugam G., Pironneau O.: On the problems of riblets as a drag reduction device. Optim. Control Appl. Methods 10(2), 93–112 (1989) · Zbl 0667.49002
[3] Boyer, F., Fabrie, P.: Eléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles. Mathématiques & Applications, vol. 52. Springer, Berlin, 2006 · Zbl 1105.76003
[4] Chenais D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52, 189–289 (1975) · Zbl 0317.49005
[5] Delfour, M., Zolésio, J.P.: Shapes and geometries. Analysis, Differential Calculus, and Optimization. Advances in Design and Control. SIAM, Philadelphia, 2001 · Zbl 1002.49029
[6] Dogĝan, G., Morin, P., Nochetto, R.H., Verani, M.: Discrete gradient flows for shape optimization and applications. In: Computer methods in Applied Mechanics and Engineering, 2007 · Zbl 1173.49307
[7] Feireisl E.: Shape optimization in viscous compressible fluids. Appl. Math. Optim. 47(1), 59–78 (2003) · Zbl 1013.49029
[8] Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vols. 1 and 2. Springer Tracts in Natural Philosophy, vol. 38, 1998
[9] Henrot, A., Pierre, M.: Variation et Optimisation de formes, coll. Mathématiques et Applications, vol. 48. Springer, Heidelberg, 2005 · Zbl 1098.49001
[10] Henrot, A., Privat, Y.: Une conduite cylindrique n’est pas optimale pour minimiser l’énergie dissipée par un fluide. C. R. Acad. Sci. Paris Sér. I Math 346, 19–20, 1057–1061 (2008) · Zbl 1156.35444
[11] Mohammadi B., Pironneau O.: Applied Shape Optimization for Fluids. Clarendon Press, Oxford (2001) · Zbl 0970.76003
[12] Morrey C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966) · Zbl 0142.38701
[13] Murat, F., Simon, J.: Sur le contrôle par un domaine géométrique. Publication du Laboratoire d’Analyse Numérique de l’Université Paris 6, vol. 189, 1976
[14] Pironneau O.: Optimal Shape Design for Elliptic Systems. Springer Series in Computational Physics. Springer, New York (1984) · Zbl 0534.49001
[15] Pironneau, O., Arumugam, G.: On Riblets in Laminar Flows. Control of boundaries and stabilization (Clermont-Ferrand, 1988). Lecture Notes in Control and Inform. Sci., vol. 125, pp. 53–65. Springer, Berlin, 1989
[16] Plotnikov P., , : Domain dependence of solutions to compressible Navier–Stokes equations. SIAM J. Control Optim. 45(4), 1165–1197 (2006) · Zbl 1122.35098
[17] Privat, Y.: Quelques problèmes d’optimisation de formes en sciences du vivant. Ph.D. Thesis of the University of Nancy, October 2008
[18] Sokolowski et, J., Zolesio, J.P.: Introduction to Shape Optimization Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin, 1992 · Zbl 0761.73003
[19] Temam R.: Navier–Stokes Equations. North-Holland, Amsterdam (1979) · Zbl 0426.35003
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