Kepler orbits of settling discs

TL;DR

This paper experimentally investigates the complex settling behavior of two discs in a viscous fluid, revealing Kepler-like bound orbits and transitions to scattering, modeled through an effective Hamiltonian analogy.

Contribution

It introduces a novel Hamiltonian framework for describing the dynamics of settling discs, linking their behavior to the Kepler problem and uncovering new orbit classes.

Findings

Identification of two classes of bound periodic orbits.

Discovery of Kepler-like orbital dynamics in viscous settling.

Observation of transitions from bound states to scattering.

Abstract

The collective dynamics of objects moving through a viscous fluid is complex and counterintuitive. A key to understanding the role of nontrivial particle shape in this complexity is the interaction of a pair of sedimenting spheroids. We report experimental results on two discs settling at negligible Reynolds number ($\simeq 10^{-4}$), finding two classes of bound periodic orbits, each with transitions to scattering states. We account for these dynamics, at leading far-field order, through an effective Hamiltonian in which gravitational driving endows orientation with the properties of momentum. This leads to a precise correspondence with the Kepler problem of planetary motion for a wide range of initial conditions, and also to orbits with no Keplerian analogue. This notion of internal degrees of freedom manifesting themselves as an effective inertia is potentially a more general tool in Stokesian driven systems.