Abstract
We consider the dynamics of a two-dimensional incompressible perfect fluid on a Möbius strip embedded in $\mathbb {R}^{3}$ . The vorticity–stream function formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the Möbius 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 density or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the Möbius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability and decaying turbulence.
Highlights
We examine the dynamics of a two-dimensional incompressible inviscid fluid confined to a Möbius strip
The main motivation is a curiosity about the impact of the non-orientability of the Möbius strip on the phenomenology of vortex dynamics
The system is physically realisable: soap films, for example, can be used as proxies for two-dimensional fluids (e.g. Couder & Basdevant 1986; Couder, Chomaz & Rabaud 1989); with a suitable wire frame they take the topology of a Möbius strip (e.g. Courant 1940; Goldstein et al 2010)
Summary
We examine the dynamics of a two-dimensional incompressible inviscid fluid confined to a Möbius strip. The main motivation is a curiosity about the impact of the non-orientability of the Möbius strip on the phenomenology of vortex dynamics. The main effect of non-orientability arises from the fact that the sense of rotation of vortices is reversed when they are transported once along the entire length of the strip, that is, when θ increases by π. We will illustrate this by considering the dynamics of a single vortex propagating along the strip’s boundary.
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