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Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs

Published online by Cambridge University Press:  17 May 2019

Miles M. P. Couchman Open the ORCID record for Miles M. P. Couchman [Opens in a new window] ,
Sam E. Turton Open the ORCID record for Sam E. Turton [Opens in a new window]  and
John W. M. Bush
Miles M. P. Couchman
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Sam E. Turton
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 an integrated experimental and theoretical investigation of the vertical motion of millimetric droplets bouncing on a vibrating fluid bath. We characterize experimentally the dependence of the phase of impact and contact force between a drop and the bath on the drop’s size and the bath’s vibrational acceleration. This characterization guides the development of a new theoretical model for the coupling between a drop’s vertical and horizontal motion. Our model allows us to relax the assumption of constant impact phase made in models based on the time-averaged trajectory equation of Moláček and Bush (J. Fluid Mech., vol. 727, 2013b, pp. 612–647) and obtain a robust horizontal trajectory equation for a bouncing drop that accounts for modulations in the drop’s vertical dynamics as may arise when it interacts with boundaries or other drops. We demonstrate that such modulations have a critical influence on the stability and dynamics of interacting droplet pairs. As the bath’s vibrational acceleration is increased progressively, initially stationary pairs destabilize into a variety of dynamical states including rectilinear oscillations, circular orbits and side-by-side promenading motion. The theoretical predictions of our variable-impact-phase model rationalize our observations and underscore the critical importance of accounting for variability in the vertical motion when modelling droplet–droplet interactions.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 871 , 25 July 2019 , pp. 212 - 243
Copyright
© 2019 Cambridge University Press 

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