Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T10:27:22.486Z Has data issue: false hasContentIssue false
  • English
  • Français

The problem of the body of revolutionof minimal resistance

Published online by Cambridge University Press:  19 December 2008

Alexander Plakhov  and
Alena Aleksenko
Alexander Plakhov
Affiliation:
Aberystwyth University, Aberystwyth SY23 3BZ, UK. [email protected] On leave from Department of Mathematics, Aveiro of University, Aveiro 3810-193, Portugal.
Alena Aleksenko
Affiliation:
Department of Mathematics, Aveiro of University, Aveiro 3810-193, Portugal.
Get access

Abstract

Newton's problem of the body of minimal aerodynamic resistance is traditionallystated in the class of convex axially symmetric bodies withfixed length and width. We state and solve the minimal resistanceproblem in the wider class of axially symmetric but generallynonconvex bodies. The infimum in this problem is not attained. Weconstruct a sequence of bodies minimizing the resistance. Thissequence approximates a convex body with smooth front surface, whilethe surface of approximating bodies becomes more and morecomplicated. The shape of the resulting convex body and the value ofminimal resistance are compared with the corresponding results forNewton's problem and for the problem in the intermediate class ofaxisymmetric bodies satisfying the single impact assumption[Comte and Lachand-Robert, J. Anal. Math.83 (2001) 313–335]. In particular, the minimal resistance in our class issmaller than in Newton's problem; the ratio goes to 1/2 as(length)/(width of the body) → 0, and to 1/4 as(length)/(width) → +∞.

Keywords

Newton's problembodies of minimal resistancecalculus of variationsbilliards
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 206 - 220
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brock, F., Ferone, V. and Kawohl, B., A symmetry problem in the calculus of variations. Calc. Var. 4 (1996) 593–599. CrossRef
Buttazzo, G. and Guasoni, P., Shape optimization problems over classes of convex domains. J. Convex Anal. 4 (1997) 343–351.
Buttazzo, G. and Kawohl, B., Newton's, On problem of minimal resistance. Math. Intell. 15 (1993) 7–12. CrossRef
Buttazzo, G., Ferone, V. and Kawohl, B., Minimum problems over sets of concave functions and related questions. Math. Nachr. 173 (1995) 71–89. CrossRef
Comte, M. and Lachand-Robert, T., Newton's problem of the body of minimal resistance under a single-impact assumption. Calc. Var. 12 (2001) 173–211. CrossRef
Comte, M. and Lachand-Robert, T., Existence of minimizers for Newton's problem of the body of minimal resistance under a single-impact assumption. J. Anal. Math. 83 (2001) 313–335. CrossRef
Lachand-Robert, T. and Oudet, E., Minimizing within convex bodies using a convex hull method. SIAM J. Optim. 16 (2006) 368–379. CrossRef
Lachand-Robert, T. and Peletier, M.A., Newton's problem of the body of minimal resistance in the class of convex developable functions. Math. Nachr. 226 (2001) 153–176. 3.0.CO;2-2>CrossRef
Lachand-Robert, T. and Peletier, M.A., An example of non-convex minimization and an application to Newton's problem of the body of least resistance. Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001) 179–198. CrossRef
I. Newton, Philosophiae naturalis principia mathematica (1686).
Plakhov, A.Yu., Newton's problem of a body of minimal aerodynamic resistance. Dokl. Akad. Nauk 390 (2003) 314–317.
Plakhov, A.Yu., Newton's problem of the body of minimal resistance with a bounded number of collisions. Russ. Math. Surv. 58 (2003) 191–192. CrossRef
Plakhov, A. and Torres, D., Newton's aerodynamic problem in media of chaotically moving particles. Sbornik: Math. 196 (2005) 885–933. CrossRef
Tikhomirov, V.M., Newton's aerodynamical problem. Kvant 5 (1982) 11–18 [in Russian].