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newton's problem

ESAIM: Control, Optimisation and Calculus of Variations (1)

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The problem of the body of revolutionof minimal resistance
Alexander Plakhov, Alena Aleksenko
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 16 / Issue 1 / January 2010

Published online by Cambridge University Press:

19 December 2008, pp. 206-220

Print publication:

January 2010
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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) → +∞.