Chaotic Geodesics

TL;DR

This paper explores how fluid particle trajectories over curved substrates can exhibit chaotic behavior due to the geodesic equations governing their motion, revealing chaos depends on substrate shape.

Contribution

It demonstrates that shallow inviscid fluid flows over curved surfaces can produce chaotic geodesic trajectories, linking geometry to fluid dynamics.

Findings

Chaotic trajectories occur for certain substrate shapes.

Geodesic equations exhibit chaos depending on substrate curvature.

Fluid particle paths can be unpredictable on curved surfaces.

Abstract

When a shallow layer of inviscid fluid flows over a substrate, the fluid particle trajectories are, to leading order in the layer thickness, geodesics on the two-dimensional curved space of the substrate. Since the two-dimensional geodesic equation is a two degree-of-freedom autonomous Hamiltonian system, it can exhibit chaos, depending on the shape of the substrate. We find chaotic behaviour for a range of substrates.