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Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization

Published online by Cambridge University Press:  13 March 2014

Anand U. Oza ,
Daniel M. Harris ,
Rodolfo R. Rosales  and
John W. M. Bush
Anand U. Oza
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Daniel M. Harris
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Rodolfo R. Rosales
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
John W. M. Bush*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

We present the results of a theoretical investigation of droplets walking on a rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an integro-differential equation that incorporates the influence of its propulsive wave force. Predictions for the dependence of the orbital radius on the bath’s rotation rate compare favourably with experimental data and capture the progression from continuous to quantized orbits as the vibrational acceleration is increased. The orbital quantization is rationalized by assessing the stability of the orbital solutions, and may be understood as resulting directly from the dynamic constraint imposed on the drop by its monochromatic guiding wave. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 744 , 10 April 2014 , pp. 404 - 429
Copyright
© 2014 Cambridge University Press 

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