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Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory

Published online by Cambridge University Press:  28 June 2013

Jan Moláček  and
John W. M. Bush
Jan Moláček
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
John W. M. Bush*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
*
†Email address for correspondence: bush@math.mit.edu
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Abstract

We present the results of a combined experimental and theoretical investigation of droplets walking on a vertically vibrating fluid bath. Several walking states are reported, including pure resonant walkers that bounce with precisely half the driving frequency, limping states, wherein a short contact occurs between two longer ones, and irregular chaotic walking. It is possible for several states to arise for the same parameter combination, including high- and low-energy resonant walking states. The extent of the walking regime is shown to be crucially dependent on the stability of the bouncing states. In order to estimate the resistive forces acting on the drop during impact, we measure the tangential coefficient of restitution of drops impacting a quiescent bath. We then analyse the spatio-temporal evolution of the standing waves created by the drop impact and obtain approximations to their form in the small-drop and long-time limits. By combining theoretical descriptions of the horizontal and vertical drop dynamics and the associated wave field, we develop a theoretical model for the walking drops that allows us to rationalize the limited extent of the walking regimes. The critical requirement for walking is that the drop achieves resonance with its guiding wave field. We also rationalize the observed dependence of the walking speed on system parameters: while the walking speed is generally an increasing function of the driving acceleration, exceptions arise due to possible switching between different vertical bouncing modes. Special focus is given to elucidating the critical role of impact phase on the walking dynamics. The model predictions are shown to compare favourably with previous and new experimental data. Our results form the basis of the first rational hydrodynamic pilot-wave theory.

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 727 , 25 July 2013 , pp. 612 - 647
Copyright
©2013 Cambridge University Press 

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