Vortex dynamics on a M\"obius strip

TL;DR

This paper explores the unique vortex dynamics of an ideal fluid on a M"obius strip, revealing how non-orientability affects conservation laws and demonstrating phenomena like vortex translation and turbulence through numerical simulations.

Contribution

It derives the Euler equations for a fluid on a M"obius strip using exterior calculus and highlights the impact of non-orientability on conservation properties and vortex behavior.

Findings

Vortices translate along boundaries in the M"obius strip.

Shear instability is observed in the flow.

Turbulence decays over time in simulations.

Abstract

We consider the dynamics of a two-dimensional incompressible perfect fluid on a M\"obius strip embedded in $\mathbb{R}^3$. The vorticity-streamfunction formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the M\"obius strip and the distinction between forms and pseudo-forms this introduces lead to unusual properties: a boundary condition is provided by the conservation of circulation along the single boundary of the strip, and there is no integral conservation for the vorticity or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the M\"obius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability, and decaying turbulence.