Skip to main content

Drops bouncing on a vibrating bath

  • Jan Moláček (a1) and John W. M. Bush (a1)

We present the results of a combined experimental and theoretical investigation of millimetric droplets bouncing on a vertically vibrating fluid bath. We first characterize the system experimentally, deducing the dependence of the droplet dynamics on the system parameters, specifically the drop size, driving acceleration and driving frequency. As the driving acceleration is increased, depending on drop size, we observe the transition from coalescing to vibrating or bouncing states, then period-doubling events that may culminate in either walking drops or chaotic bouncing states. The drop’s vertical dynamics depends critically on the ratio of the forcing frequency to the drop’s natural oscillation frequency. For example, when the data describing the coalescence–bouncing threshold and period-doubling thresholds are described in terms of this ratio, they collapse onto a single curve. We observe and rationalize the coexistence of two non-coalescing states, bouncing and vibrating, for identical system parameters. In the former state, the contact time is prescribed by the drop dynamics; in the latter, by the driving frequency. The bouncing states are described by theoretical models of increasing complexity whose predictions are tested against experimental data. We first model the drop–bath interaction in terms of a linear spring, then develop a logarithmic spring model that better captures the drop dynamics over a wider range of parameter space. While the linear spring model provides a faster, less accurate option, the logarithmic spring model is found to be more accurate and consistent with all existing data.

Corresponding author
Email address for correspondence:
Hide All
Bach G. A., Koch D. L. & Gopinath A. 2004 Coalescence and bouncing of small aerosol droplets. J. Fluid Mech. 518, 157185.
Benjamin T. & Ursell F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505515.
Bush J. W. M. 2010 Quantum mechanics writ large. Proc. Natl. Acad. Sci. 107, 17 45517 456.
Cai Y. K. 1989 Phenomena of a liquid drop falling to a liquid surface. Exp. Fluids 7, 388394.
Chandrasekhar S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Ching B., Golay M. W. & Johnson T. J. 1984 Droplet impacts upon liquid surfaces. Science 226, 535537.
Couder Y. & Fort E. 2006 Single-particle diffraction and interference at macroscopic scale. Phys. Rev. Lett. 97, 154101.
Couder Y., Fort E., Gautier C. H. & Boudaoud A. 2005a From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94, 177801.
Couder Y., Protière S., Fort E. & Boudaoud A. 2005b Dynamical phenomena: walking and orbiting droplets. Nature 437, 208.
Davis R. B. & Virgin L. N. 2007 Non-linear behaviour in a discretely forced oscillator. Intl J. Non-Linear Mech. 42, 744753.
Eddi A., Boudaoud A. & Couder Y. 2011a Oscillating instability in bouncing droplet crystals. Europhys. Lett. 94, 20004.
Eddi A., Decelle A., Fort E. & Couder Y. 2009a Archimedean lattices in the bound states of wave interacting particles. Europhys. Lett. 87, 56002.
Eddi A., Fort E., Moisy F. & Couder Y. 2009b 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. 2011b Information stored in Faraday waves: the origin of a path memory. J. Fluid Mech. 674, 433463.
Eddi A., Terwagne D., Fort E. & Couder Y. 2008 Wave propelled ratchets and drifting rafts. Europhys. Lett. 82, 44001.
Eichwald B., Argentina M., Noblin X. & Celestini F. 2010 Dynamics of a ball bouncing on a vibrated elastic membrane. Phys. Rev. E 82, 016203.
Everson R. M. 1986 Chaotic dynamics of a bouncing ball. Physica D 19, 355383.
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.
Fermi E. 1949 On the origin of the cosmic radiation. Phys. Rev. 75, 11691174.
Flemmer R. L. C. & Banks C. L. 1986 On the drag coefficient of a sphere. Powder Technol. 48, 217221.
Foote G. B. 1975 The water drop rebound problem: dynamics of collision. J. Atmos. Sci. 32, 390402.
Fort E., Eddi A., Boudaoud A., Moukhtar J. & Couder Y. 2010 Path-memory induced quantization of classical orbits. Proc. Natl. Acad. Sci. 107, 17 51517 520.
Gilet T. & Bush J. W. M. 2009a Chaotic bouncing of a droplet on a soap film. Phys. Rev. Lett. 102, 014501.
Gilet T. & Bush J. W. M. 2009b The fluid trampoline: droplets bouncing on a soap film. J. Fluid Mech. 625, 167203.
Gopinath A. & Koch D. L. 2001 Dynamics of droplet rebound from a weakly deformable gas–liquid interface. Phys. Fluids 13, 35263532.
Hallett J. & Christensen L. 1984 Splash and penetration of drops in water. J. Rech. Atmos. 18, 225242.
Harris D. M., Moukhtar J., Fort E., Couder Y. & Bush J. W. M. 2013 Wave-like statistics from pilot-wave dynamics in a circular corral. Phys. Rev. E (submitted).
Hartland S. 1969 The effect of circulation patterns on the drainage of the film between a liquid drop and a deformable liquid–liquid interface. Chem. Engng Sci. 24, 611613.
Hartland S. 1970 The profile of the draining film between a fluid drop and a deformable fluid–liquid interface. Chem. Engng J. 1, 6775.
Jayaratne O. W. & Mason B. J. 1964 The coalescence and bouncing of water drops at an air/water interface. Proc. R. Soc. Lond. A 280, 545565.
Jones A. F. & Wilson S. D. R. 1978 The film drainage problem in droplet coalescence. J. Fluid Mech. 87, 263288.
Kumar K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. Math. Phys. Engng Sci. 452, 11131126.
Luck J. M. & Mehta A. 1993 Bouncing ball with a finite restitution: chattering, locking, and chaos. Phys. Rev. E 48, 39883997.
Luna-Acosta G. A. 1990 Regular and chaotic dynamics of the damped Fermi accelerator. Phys. Rev. A 42, 71557162.
Miller C. A. & Scriven L. E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417435.
Moláček J. & Bush J. W. M. 2012 A quasi-static model of drop impact. Phys. Fluids 24, 127103.
Moláček J. & Bush J. W. M. 2013 Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory. J. Fluid Mech. 727, 612647.
Okumura K., Chevy F., Richard D., Quéré D. & Clanet C. 2003 Water spring: a model for bouncing drops. Europhys. Lett. 62, 237243.
Pieranski P. 1983 Jumping particle model. Period doubling cascade in an experimental system. J. Phys. (Paris) 44, 573578.
Pieranski P. & Bartolino R. 1985 Jumping particle model. Modulation modes and resonant response to a periodic perturbation. J. Phys. (Paris) 46, 687690.
Prosperetti A. 1980 Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100, 333347.
Prosperetti A. & Oguz H. N. 1993 The impact of drops on liquid surfaces and the underwater noise of rain. Annu. Rev. Fluid Mech. 25, 577602.
Protière S., Bohn S. & Couder Y. 2008 Exotic orbits of two interacting wave sources. Phys. Rev. E 78, 036204.
Protière S., Boudaoud A. & Couder Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85108.
Protière S. & Couder Y. 2006 Orbital motion of bouncing drops. Phys. Fluids 18, 091114.
Protière S., Couder Y., Fort E. & Boudaoud A. 2005 The self-organization of capillary wave sources. J. Phys.: Condens. Matter 17, S3529S3535.
Rayleigh Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. A 29, 71.
Richard D., Clanet C. & Quéré D. 2002 Surface phenomena: contact time of a bouncing drop. Nature 417, 811.
Richard D. & Quéré D. 2000 Bouncing water drops. Europhys. Lett. 50, 769775.
Schotland R. M. 1960 Experimental results relating to the coalescence of water drops with water surfaces. Discuss. Faraday Soc. 30, 7277.
Terwagne D. 2011 Bouncing droplets, the role of deformations. PhD thesis, Université de Liège.
Terwagne D., Gilet T., Vandewalle N. & Dorbolo S. 2008 From bouncing to boxing. Chaos 18, 041104.
Terwagne D., Ludewig F., Vandewalle N. & Dorbolo S. 2013 The role of deformations in the bouncing droplet dynamics. Phys. Fluids (submitted) arXiv:1301.7463.
Torby B. J. 1984 Advanced Dynamics for Engineers. Holt, Rinehart and Winston.
Walker J. 1978 Drops of liquid can be made to float on the liquid. What enables them to do so? The Amateur Scientist, Sci. Am. 238, 151158.
Zou J., Wang P. F., Zhang T. R., Fu X. & Ruan X. 2011 Experimental study of a drop bouncing on a liquid surface. Phys. Fluids 23, 044101.
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? *



Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 201 *
Loading metrics...

Abstract views

Total abstract views: 484 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd November 2017. This data will be updated every 24 hours.