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Drops bouncing on a vibrating bath

Published online by Cambridge University Press:  28 June 2013

Jan Moláček  and
John W. M. Bush
Jan Moláček
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
John W. M. Bush*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
*
†Email address for correspondence: bush@math.mit.edu
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Abstract

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.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 727 , 25 July 2013 , pp. 582 - 611
Copyright
©2013 Cambridge University Press 

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References

Bach, G. A., Koch, D. L. & Gopinath, A. 2004 Coalescence and bouncing of small aerosol droplets. J. Fluid Mech. 518, 157–185.CrossRefGoogle Scholar
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, 505–515.Google Scholar
Bush, J. W. M. 2010 Quantum mechanics writ large. Proc. Natl. Acad. Sci. 107, 17 455–17 456.Google Scholar
Cai, Y. K. 1989 Phenomena of a liquid drop falling to a liquid surface. Exp. Fluids 7, 388–394.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Ching, B., Golay, M. W. & Johnson, T. J. 1984 Droplet impacts upon liquid surfaces. Science 226, 535–537.Google Scholar
Couder, Y. & Fort, E. 2006 Single-particle diffraction and interference at macroscopic scale. Phys. Rev. Lett. 97, 154101.Google Scholar
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.Google Scholar
Couder, Y., Protière, S., Fort, E. & Boudaoud, A. 2005b Dynamical phenomena: walking and orbiting droplets. Nature 437, 208.Google Scholar
Davis, R. B. & Virgin, L. N. 2007 Non-linear behaviour in a discretely forced oscillator. Intl J. Non-Linear Mech. 42, 744–753.Google Scholar
Eddi, A., Boudaoud, A. & Couder, Y. 2011a Oscillating instability in bouncing droplet crystals. Europhys. Lett. 94, 20004.Google Scholar
Eddi, A., Decelle, A., Fort, E. & Couder, Y. 2009a Archimedean lattices in the bound states of wave interacting particles. Europhys. Lett. 87, 56002.Google Scholar
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009b Unpredictable tunneling of a classical wave–particle association. Phys. Rev. Lett. 102, 240401.Google Scholar
Eddi, A., Moukhtar, J., Perrard, S., Fort, E. & Couder, Y. 2012 Level splitting at macroscopic scale. Phys. Rev. Lett. 108, 264503.Google Scholar
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, 433–463.CrossRefGoogle Scholar
Eddi, A., Terwagne, D., Fort, E. & Couder, Y. 2008 Wave propelled ratchets and drifting rafts. Europhys. Lett. 82, 44001.Google Scholar
Eichwald, B., Argentina, M., Noblin, X. & Celestini, F. 2010 Dynamics of a ball bouncing on a vibrated elastic membrane. Phys. Rev. E 82, 016203.Google Scholar
Everson, R. M. 1986 Chaotic dynamics of a bouncing ball. Physica D 19, 355–383.CrossRefGoogle Scholar
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, 299–340.Google Scholar
Fermi, E. 1949 On the origin of the cosmic radiation. Phys. Rev. 75, 1169–1174.CrossRefGoogle Scholar
Flemmer, R. L. C. & Banks, C. L. 1986 On the drag coefficient of a sphere. Powder Technol. 48, 217–221.CrossRefGoogle Scholar
Foote, G. B. 1975 The water drop rebound problem: dynamics of collision. J. Atmos. Sci. 32, 390–402.Google Scholar
Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. 2010 Path-memory induced quantization of classical orbits. Proc. Natl. Acad. Sci. 107, 17 515–17 520.Google Scholar
Gilet, T. & Bush, J. W. M. 2009a Chaotic bouncing of a droplet on a soap film. Phys. Rev. Lett. 102, 014501.Google Scholar
Gilet, T. & Bush, J. W. M. 2009b The fluid trampoline: droplets bouncing on a soap film. J. Fluid Mech. 625, 167–203.Google Scholar
Gopinath, A. & Koch, D. L. 2001 Dynamics of droplet rebound from a weakly deformable gas–liquid interface. Phys. Fluids 13, 3526–3532.CrossRefGoogle Scholar
Hallett, J. & Christensen, L. 1984 Splash and penetration of drops in water. J. Rech. Atmos. 18, 225–242.Google Scholar
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).Google Scholar
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, 611–613.Google Scholar
Hartland, S. 1970 The profile of the draining film between a fluid drop and a deformable fluid–liquid interface. Chem. Engng J. 1, 67–75.Google Scholar
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, 545–565.Google Scholar
Jones, A. F. & Wilson, S. D. R. 1978 The film drainage problem in droplet coalescence. J. Fluid Mech. 87, 263–288.Google Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. Math. Phys. Engng Sci. 452, 1113–1126.Google Scholar
Luck, J. M. & Mehta, A. 1993 Bouncing ball with a finite restitution: chattering, locking, and chaos. Phys. Rev. E 48, 3988–3997.Google Scholar
Luna-Acosta, G. A. 1990 Regular and chaotic dynamics of the damped Fermi accelerator. Phys. Rev. A 42, 7155–7162.CrossRefGoogle ScholarPubMed
Miller, C. A. & Scriven, L. E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417–435.Google Scholar
Moláček, J. & Bush, J. W. M. 2012 A quasi-static model of drop impact. Phys. Fluids 24, 127103.Google Scholar
Moláček, J. & Bush, J. W. M. 2013 Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory. J. Fluid Mech. 727, 612–647.Google Scholar
Okumura, K., Chevy, F., Richard, D., Quéré, D. & Clanet, C. 2003 Water spring: a model for bouncing drops. Europhys. Lett. 62, 237–243.CrossRefGoogle Scholar
Pieranski, P. 1983 Jumping particle model. Period doubling cascade in an experimental system. J. Phys. (Paris) 44, 573–578.CrossRefGoogle Scholar
Pieranski, P. & Bartolino, R. 1985 Jumping particle model. Modulation modes and resonant response to a periodic perturbation. J. Phys. (Paris) 46, 687–690.Google Scholar
Prosperetti, A. 1980 Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100, 333–347.Google Scholar
Prosperetti, A. & Oguz, H. N. 1993 The impact of drops on liquid surfaces and the underwater noise of rain. Annu. Rev. Fluid Mech. 25, 577–602.Google Scholar
Protière, S., Bohn, S. & Couder, Y. 2008 Exotic orbits of two interacting wave sources. Phys. Rev. E 78, 036204.Google Scholar
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85–108.CrossRefGoogle Scholar
Protière, S. & Couder, Y. 2006 Orbital motion of bouncing drops. Phys. Fluids 18, 091114.Google Scholar
Protière, S., Couder, Y., Fort, E. & Boudaoud, A. 2005 The self-organization of capillary wave sources. J. Phys.: Condens. Matter 17, S3529–S3535.Google Scholar
Rayleigh, Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. A 29, 71.Google Scholar
Richard, D., Clanet, C. & Quéré, D. 2002 Surface phenomena: contact time of a bouncing drop. Nature 417, 811.Google Scholar
Richard, D. & Quéré, D. 2000 Bouncing water drops. Europhys. Lett. 50, 769–775.CrossRefGoogle Scholar
Schotland, R. M. 1960 Experimental results relating to the coalescence of water drops with water surfaces. Discuss. Faraday Soc. 30, 72–77.Google Scholar
Terwagne, D. 2011 Bouncing droplets, the role of deformations. PhD thesis, Université de Liège.Google Scholar
Terwagne, D., Gilet, T., Vandewalle, N. & Dorbolo, S. 2008 From bouncing to boxing. Chaos 18, 041104.Google Scholar
Terwagne, D., Ludewig, F., Vandewalle, N. & Dorbolo, S. 2013 The role of deformations in the bouncing droplet dynamics. Phys. Fluids (submitted) arXiv:1301.7463.Google Scholar
Torby, B. J. 1984 Advanced Dynamics for Engineers. Holt, Rinehart and Winston.Google Scholar
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, 151–158.Google Scholar
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.Google Scholar