The problem of the body of revolution of minimal resistance

Alexander Plakhov; Alena Aleksenko

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 1, page 206-220
  • ISSN: 1292-8119

Abstract

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Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of convex axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but generally nonconvex bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton's problem and for the problem in the intermediate class of axisymmetric 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 is smaller 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) → +∞.

How to cite

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Plakhov, Alexander, and Aleksenko, Alena. "The problem of the body of revolution of minimal resistance." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 206-220. <http://eudml.org/doc/250701>.

@article{Plakhov2010,
abstract = { Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of convex axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but generally nonconvex bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton's problem and for the problem in the intermediate class of axisymmetric 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 is smaller 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) → +∞. },
author = {Plakhov, Alexander, Aleksenko, Alena},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Newton's problem; bodies of minimal resistance; calculus of variations; billiards},
language = {eng},
month = {1},
number = {1},
pages = {206-220},
publisher = {EDP Sciences},
title = {The problem of the body of revolution of minimal resistance},
url = {http://eudml.org/doc/250701},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Plakhov, Alexander
AU - Aleksenko, Alena
TI - The problem of the body of revolution of minimal resistance
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/1//
PB - EDP Sciences
VL - 16
IS - 1
SP - 206
EP - 220
AB - Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of convex axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but generally nonconvex bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton's problem and for the problem in the intermediate class of axisymmetric 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 is smaller 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) → +∞.
LA - eng
KW - Newton's problem; bodies of minimal resistance; calculus of variations; billiards
UR - http://eudml.org/doc/250701
ER -

References

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  10. I. Newton, Philosophiae naturalis principia mathematica (1686).  
  11. A.Yu. Plakhov, Newton's problem of a body of minimal aerodynamic resistance. Dokl. Akad. Nauk390 (2003) 314–317.  
  12. A.Yu. Plakhov, Newton's problem of the body of minimal resistance with a bounded number of collisions. Russ. Math. Surv.58 (2003) 191–192.  
  13. A. Plakhov and D. Torres, Newton's aerodynamic problem in media of chaotically moving particles. Sbornik: Math.196 (2005) 885–933.  
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