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Faraday pilot-wave dynamics in a circular corral

Published online by Cambridge University Press:  18 March 2020

Matthew Durey Open the ORCID record for Matthew Durey [Opens in a new window] ,
Paul A. Milewski Open the ORCID record for Paul A. Milewski [Opens in a new window]  and
Zhan Wang Open the ORCID record for Zhan Wang [Opens in a new window]
Matthew Durey*
Affiliation:
Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
Paul A. Milewski
Affiliation:
Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK
Zhan Wang
Affiliation:
Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing100190, PR China
*
†Email address for correspondence: [email protected]
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Abstract

A millimetric droplet of silicone oil may bounce and self-propel on the free surface of a vertically vibrating fluid bath due to the droplet’s interaction with its accompanying Faraday wave field. This hydrodynamic pilot-wave system exhibits many dynamics that were previously thought to be peculiar to the quantum realm. When the droplet is confined to a circular cavity, referred to as a ‘corral’, a range of dynamics may occur depending on the details of the geometry and the decay time of the subcritical Faraday waves. We herein present a theoretical investigation into the behaviour of subcritical Faraday waves in this geometry and explore the accompanying pilot-wave dynamics. By computing the Dirichlet-to-Neumann map for the velocity potential in the corral geometry, we can evolve the quasi-potential flow between successive droplet impacts, which, when coupled with a simplified model for the droplet’s vertical motion, allows us to derive and implement a highly efficient discrete-time iterative map for the pilot-wave system. We study the onset of the Faraday instability, the emergence and quantisation of circular orbits and simulate the exotic dynamics that arises in smaller corrals.

JFM classification

Waves/Free-surface Flows: Faraday waves Drops and Bubbles: Drops Nonlinear Dynamical Systems: Bifurcation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 891 , 25 May 2020 , A3
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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