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A trajectory equation for walking droplets: hydrodynamic pilot-wave theory

Published online by Cambridge University Press:  27 November 2013

Anand U. Oza ,
Rodolfo R. Rosales  and
John W. M. Bush
Anand U. Oza
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: bush@math.mit.edu
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Abstract

We present the results of a theoretical investigation of droplets bouncing on a vertically vibrating fluid bath. An integro-differential equation describing the horizontal motion of the drop is developed by approximating the drop as a continuous moving source of standing waves. Our model indicates that, as the forcing acceleration is increased, the bouncing state destabilizes into steady horizontal motion along a straight line, a walking state, via a supercritical pitchfork bifurcation. Predictions for the dependence of the walking threshold and drop speed on the system parameters compare favourably with experimental data. By considering the stability of the walking state, we show that the drop is stable to perturbations in the direction of motion and neutrally stable to lateral perturbations. This result lends insight into the possibility of chaotic dynamics emerging when droplets walk in complex geometries.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves Waves/Free-surface Flows: Waves/Free-surface Flows

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 737 , 25 December 2013 , pp. 552 - 570
Copyright
©2013 Cambridge University Press 

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