The problem of the body of revolution of minimal resistance

TL;DR

This paper extends Newton's minimal resistance problem to nonconvex bodies, constructing a sequence of bodies that approximate the minimal resistance, which is lower than in the convex case, with specific asymptotic ratios.

Contribution

It broadens the classical problem to nonconvex bodies, providing a sequence that approaches the infimum resistance and comparing results with convex and intermediate classes.

Findings

Minimal resistance is lower in the nonconvex class than in the convex case.

The infimum resistance is approached by increasingly complex bodies.

As the length-to-width ratio varies, the minimal resistance ratio approaches 1/2 or 1/4.

Abstract

Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of {\it 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 {\it 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 {\it single impact} assumption \cite{CL1}. 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) $\to 0$, and to 1/4 as (length)/(width) $\to +\infty$.