Skip to main content

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

  • Anand U. Oza (a1), Rodolfo R. Rosales (a1) and John W. M. Bush (a1)

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.

Corresponding author
Email address for correspondence:
Hide All
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. A 225, 505515.
Bush, J. W. M. 2010 Quantum mechanics writ large. Proc. Natl Acad. Sci. USA 107 (41), 1745517456.
Couder, Y. & Fort, E. 2006 Single-particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97, 154101.
Couder, Y., Gautier, C.-H. & Boudaoud, A. 2005 From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94, 177801.
Crommie, M., Lutz, C. & Eigler, D. 1993 Confinement of electrons to quantum corrals on a metal surface. Science 262 (5131), 218220.
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009 Unpredictable tunneling of a classical wave–particle association. Phys. Rev. Lett. 102, 240401.
Eddi, A., Moukhtar, J., Perrard, S., Fort, E. & Couder, Y. 2012 Level splitting at macroscopic scale. Phys. Rev. Lett. 108, 264503.
Eddi, A., Sultan, E., Moukhtar, J., Fort, E., Rossi, M. & Couder, Y. 2011 Information stored in Faraday waves: the origin of path memory. J. Fluid Mech. 675, 433463.
Eddi, A., Terwagne, D., Fort, E. & Couder, Y. 2008 Wave propelled ratchets and drifting rafts. Europhys. Lett. 82, 44001.
Faraday, M. 1831 On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299340.
Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. 2010 Path-memory induced quantization of classical orbits. Proc. Natl Acad. Sci. USA 107 (41), 1751517520.
Harris, D. M. & Bush, J. W. M. 2013 Droplets walking in a rotating frame: from quantized orbits to multinodal statistics. J. Fluid Mech. (submitted).
Harris, D. M., Moukhtar, J., Fort, E., Couder, Y. & Bush, J. W. M. 2013 Wavelike statistics from pilot-wave dynamics in a circular corral. Phys. Rev. E 88, 011001.
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. A 452, 11131126.
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143165.
Moláček, J. & Bush, J. W. M. 2013a Drops bouncing on a vibrating bath. J. Fluid Mech. 727, 582611.
Moláček, J. & Bush, J. W. M. 2013b Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory. J. Fluid Mech. 727, 612647.
Müller, H. W., Friedrich, R. & Papathanassiou, D. 1998 Theoretical and experimental investigations of the Faraday instability. In Evolution of Spontaneous Structures in Dissipative Continuous Systems (ed. Busse, F. & Müller, S. C.), Lecture Notes in Physics, vol. 55, pp. 231265. Springer.
Oza, A. U., Harris, D. M., Rosales, R. R. & Bush, J. W. M. 2013 Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. J. Fluid Mech. (submitted).
Perrard, S., Labousse, M., Miskin, M., Fort, E. & Couder, Y. 2013 Macroscopic wave–particle eigenstates. Under review.
Prosperetti, A. 1976 Viscous effects on small-amplitude surface waves. Phys. Fluids 19 (2), 195203.
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85108.
Shirokoff, D. 2013 Bouncing droplets on a billiard table. Chaos 23, 013115.
Stuart, J. T. 1958 On the nonlinear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 121.
Walker, J. 1978 Drops of liquid can be made to float on the liquid. What enables them to do so? Sci. Am. 238-6, 151–158.
Wind-Willassen, Ø., Moláček, J., Harris, D. M. & Bush, J. W. M. 2013 Exotic states of bouncing and walking droplets. Phys. Fluids 25, 082002.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed