Fast Escape in Incompressible Vector Fields

TL;DR

This paper presents a mathematical principle for efficiently escaping from incompressible flows, showing that combining orthogonal and original flow directions can significantly reduce escape length, with implications for understanding fluid dynamics and escape strategies.

Contribution

It introduces a novel method combining flow directions to improve escape efficiency in incompressible vector fields, along with a new quantitative Poincaré-Bendixson theorem.

Findings

Combining flow and orthogonal flow improves escape length.

Escape length of combined flow is bounded by rom the abstract.

The orthogonal flow can enable fast escape when the original flow has large escape length.

Abstract

Swimmers caught in a rip current flowing away from the shore are advised to swim orthogonally to the current to escape it. We describe a mathematical principle in a similar spirit. More precisely, we consider flows $\gamma$ in the plane induced by incompressible vector fields $\textbf{v}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ satisfying $ c_1 < \|v\| < c_2.$ The length $\ell$ a flow curve $\dot \gamma(t) = \textbf{v}(\gamma(t))$ until $\gamma$ leaves a disk of radius 1 centered at the initial position can be as long as $\ell \sim c_2/c_1$. The same is true for the orthogonal flow $\textbf{v}^{\perp} = (-\textbf{v}_2, \textbf{v}_1)$. We show that a combination does strictly better: there always exists a curve flowing first along $\textbf{v}^{\perp}$ and then along $\textbf{v}$ which escapes the unit disk before reaching the length $ \sqrt{4\pi c_2 / c_1}$. Moreover, if the escape length of $\textbf{v}$ is uniformly $\sim c_2/c_1$, then the escape length of $\textbf{v}^{\perp}$ is uniformly $\sim 1$ (allowing for a fast escape from the current). We also prove an elementary quantitative Poincar\'{e}-Bendixson theorem that seems to be new.