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

Anand U. Oza, Rodolfo R. Rosales, John W.M. Bush

Research output: Contribution to journalArticlepeer-review

95 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)552-570
Number of pages19
JournalJournal of Fluid Mechanics
StatePublished - Dec 25 2013
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


  • Faraday waves
  • drops
  • waves/free-surface flows


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