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Pilot-wave dynamics in a rotating frame: the onset of orbital instability

Published online by Cambridge University Press:  10 October 2023

Nicholas Liu Open the ORCID record for Nicholas Liu [Opens in a new window] ,
Matthew Durey Open the ORCID record for Matthew Durey [Opens in a new window]  and
John W.M. Bush Open the ORCID record for John W.M. Bush [Opens in a new window]
Nicholas Liu
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Matthew Durey
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK
John W.M. Bush*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: bush@math.mit.edu
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Abstract

We report the results of a theoretical investigation of the stability of a hydrodynamic analogue of Landau levels, specifically circular orbits arising when a millimetric droplet self-propels along the surface of a vibrating, rotating liquid bath. Our study elucidates the form of the stability diagram characterising the critical memory at which circular orbits destabilise, and the form of instability. Particular attention is given to rationalising observations reported in prior experimental works, including the prevalence of resonant wobbling instabilities, in which the instability frequency is approximately twice the orbital frequency. We also explore the physical mechanism responsible for the onset of instability. Specifically, we compare the efficacy of different heuristic arguments proposed in prior studies, including propositions that the most unstable orbits arise when their radii correspond to the zeros of Bessel functions or when their associated wave intensity is extremised. We establish a new relation between orbital stability and the mean wave field, which supersedes existing heuristic arguments and suggests a rationale for the alternate wobbling and monotonic instabilities arising at onset as the orbital radius is increased progressively.

JFM classification

Waves/Free-surface Flows: Faraday waves Drops and Bubbles: Drops
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 973 , 25 October 2023 , A4
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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