Minimal resistance of curves under the single impact assumption

TL;DR

This paper determines the shape of a hollow in a half-plane that minimizes resistance under a single impact elastic reflection condition, finding a specific parabolic shape with minimal resistance approximately 0.6435, and extends results to higher dimensions.

Contribution

It explicitly characterizes the minimal resistance shape under SIC in 2D and extends the analysis to higher dimensions, providing exact resistance values and asymptotic behavior.

Findings

Minimal resistance in 2D is approximately 0.6435.

The shape minimizing resistance is formed by two symmetric parabola arcs.

Resistance approaches 0.5 as the dimension tends to infinity.

Abstract

We consider the hollow on the half-plane $\{(x,y) : y \le 0\} \subset \mathbb{R}^2$ defined by a function $u : (-1, 1) \to \mathbb{R}$, $u(x) < 0$ and a vertical flow of point particles incident on the hollow. It is assumed that $u$ satisfies the so-called single impact condition (SIC): each incident particle is elastically reflected by graph$(u)$ and goes away without hitting the graph of $u$ anymore. We solve the problem: find the function $u$ minimizing the force of resistance created by the flow. We show that the graph of the minimizer is formed by two arcs of parabolas symmetric to each other with respect to the $y$-axis. Assuming that the resistance of $u \equiv 0$ equals 1, we show that the minimal resistance equals $\pi/2 - 2\arctan(1/2) \approx 0.6435$. This result completes the previously obtained result stating in particular that the minimal resistance of a hollow in higher dimensions equals 0.5. We additionally consider a similar problem of minimal resistance, where the hollow in the half-space $\{(x_1,\ldots,x_d, y) : y \le 0 \} \subset \mathbb{R}^{d+1}$ is defined by a radial function $U$ satisfying SIC, $U(x) = u(|x|)$, with $x = (x_1,\ldots,x_d), u(\xi) < 0$ for $0 \le \xi < 1$ and $u(\xi) = 0$ for $\xi \ge 1$, and the flow is parallel to the $y$-axis. The minimal resistance is greater than $0.5$ (and coincides with $0.6435$ when $d = 1$) and converges to $0.5$ as $d \to \infty$.