Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-16T11:06:40.285Z Has data issue: false hasContentIssue false
  • English
  • Français

The fluid trampoline: droplets bouncing on a soap film

Published online by Cambridge University Press:  14 April 2009

TRISTAN GILET  and
JOHN W. M. BUSH
TRISTAN GILET
Affiliation:
Group for Research and Applications in Statistical Physics, Department of Physics, University of Liege, 4000 Liege, Belgium
JOHN W. M. BUSH*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: bush@math.mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the results of a combined experimental and theoretical investigation of droplets falling onto a horizontal soap film. Both static and vertically vibrated soap films are considered. In the static case, a variety of behaviours were observed, including bouncing, crossing and partial coalescence. A quasi-static description of the soap film shape yields a force–displacement relation that provides excellent agreement with experiment, and allows us to model the film as a nonlinear spring. This approach yields an accurate criterion for the transition between droplet bouncing and crossing. Moreover, it allows us to rationalize the observed constancy of the contact time and scaling for the coefficient of restitution in the bouncing states. On the vibrating film, a variety of bouncing behaviours were observed, including simple and complex periodic states, multi-periodicity and chaos. A simple theoretical model is developed that captures the essential physics of the bouncing process, reproducing all observed bouncing states. The model enables us to rationalize the observed coexistence of multiple periodic bouncing states by considering the dependence of the energy transferred to the droplet on the phase of impact. Quantitative agreement between model and experiment is deduced for simple periodic modes, and qualitative agreement for more complex periodic and chaotic bouncing states. Analytical solutions are deduced in the limit of weak forcing and dissipation, yielding insight into the contact time and periodicity of the bouncing states.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 625 , 25 April 2009 , pp. 167 - 203
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ambravaneswaran, B., Subramani, H. J., Phillips, S. D. & Basaran, O. A. 2004 Dripping–jetting transitions in a dripping faucet. Phys. Rev. Lett. 93, 034501.CrossRefGoogle Scholar
Biance, A. L., Chevy, F., Clanet, C., Lagubeau, G. & Quéré, D. 2006 On the elasticity of an inertial liquid shock. J. Fluid Mech. 554, 47.CrossRefGoogle Scholar
Blanchette, F. & Bigioni, T. 2006 Partial coalescence of drops at liquid interfaces. Nature Physics 2, 254.CrossRefGoogle Scholar
Boudaoud, A., Couder, Y. & Ben Amar, M. 1999 Self-adaptation in vibrating soap films. Phys. Rev. Lett. 82 (19), 3847.CrossRefGoogle Scholar
Charles, G. E. & Mason, S. G. 1960 a The coalescence of liquid drops with flat liquid/liquid interfaces. J. Colloid Sci. 15, 236.CrossRefGoogle Scholar
Charles, G. E. & Mason, S. G. 1960 b The mechanism of partial coalescence of liquid drops at liquid/liquid interfaces. J. Colloid Sci. 15, 105.CrossRefGoogle Scholar
Chen, X., Mandre, S. & Feng, J. J. 2006 Partial coalescence between a drop and a liquid–liquid interface. Phys. Fluids 18, 051705.CrossRefGoogle Scholar
Clanet, C., Béguin, C., Richard, D. & Quéré, D. 2004 Maximal deformation of an impacting drop. J. Fluid Mech. 517, 199.CrossRefGoogle Scholar
Couder, Y. & Fort, E. 2006 Single-particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97, 154101.CrossRefGoogle Scholar
Couder, Y., Fort, E., Gautier, C. H. & Boudaoud, A. 2005 a From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94, 177801.CrossRefGoogle ScholarPubMed
Couder, Y., Protière, S., Fort, E. & Boudaoud, A. 2005 b Walking and orbiting bouncing droplets. Nature 437, 208.CrossRefGoogle Scholar
Coullet, P., Mahadevan, L. & Riera, C. S. 2005 Hydrodynamical models for the chaotic dripping faucet. J. Fluid Mech. 526, 1.CrossRefGoogle Scholar
Courbin, L., Marchand, A., Vaziri, A., Ajdari, A. & Stone, H. 2006 Impact dynamics for elastic membranes. Phys. Rev. Lett. 97, 244301.CrossRefGoogle ScholarPubMed
Courbin, L. & Stone, H. 2006 Impact, puncturing and the self-healing of soap films. Phys. Fluids 18, 091105.CrossRefGoogle Scholar
Dell'Aversana, P., Banavar, J. R. & Koplik, J. 1996 Suppression of coalescence by shear and temperature gradients. Phys. Fluids 8, 15.CrossRefGoogle Scholar
Dorbolo, S., Terwagne, D., Vandewalle, N. & Gilet, T. 2008 Resonant and rolling droplets. New J. Phys. 10, 113021.CrossRefGoogle Scholar
Gilet, T. & Bush, J. W. M. Chaotic bouncing of a droplet on a soap film. Phys. Rev. Lett. 102, 014501.CrossRefGoogle Scholar
Gilet, T., Mulleners, K., Lecomte, J.-P., Vandewalle, N. & Dorbolo, S. 2007 a Critical parameters for the partial coalescence of a droplet. Phys. Review E 75, 036303.CrossRefGoogle ScholarPubMed
Gilet, T., Terwagne, D., Vandewalle, N. & Dorbolo, S. 2008 Dynamics of a bouncing droplet onto a vertically vibrated interface. Phys. Rev. Lett. 100, 167802.CrossRefGoogle ScholarPubMed
Gilet, T., Vandewalle, N. & Dorbolo, S. 2007 b Controlling the partial coalescence of a droplet on a vertically vibrated bath. Phys. Rev. E 76, 035302.CrossRefGoogle ScholarPubMed
Graff, K. F. 1975 Wave Motion in Elastic Solids. Oxford University Press.Google Scholar
Honey, E. M. & Kavehpour, H. P. 2006 Astonishing life of a coalescing drop on a free surface. Phys. Rev. E 73, 027301.CrossRefGoogle ScholarPubMed
Jayaratne, O. W. & Mason, B. J. 1964 The coalescence and bouncing of water drops at an air–water interface. Proc. R. Soc. London, Ser. A 280, 545.Google Scholar
Kowalik, Z. J., Franaszek, M. & Pieranski, P. 1988 Self-reanimating chaos in the bouncing-ball system. Phys. Rev. A 37 (10).CrossRefGoogle ScholarPubMed
Landau, L. & Lifchitz, E. 1959 Fluid mechanics. In Course on Theoretical Physics, vol. 6. Addison Wesley.Google Scholar
Legendre, D., Daniel, C. & Guiraud, P. 2005 Experimental study of a drop bouncing on a wall in a liquid. Phys. Fluids 17, 097105.CrossRefGoogle Scholar
LeGoff, A., Courbin, L., Stone, H. A. & Quéré, D. 2008 Energy absorption in a bamboo foam. Europhys. Lett. 84, 36001.CrossRefGoogle Scholar
LeGrand-Piteira, N., Brunet, P., Lebon, L. & Limat, L. 2006 Propagative wave pattern on a falling liquid curtain. Phys. Rev. E 74, 026305.CrossRefGoogle Scholar
Lieber, S., Hendershott, M., Pattanaporkratana, A. & Maclennan, J. 2007 Self-organization of bouncing oil drops: two-dimensional lattices and spinning clusters. Phys. Rev. E 75, 056308.CrossRefGoogle ScholarPubMed
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130.2.0.CO;2>CrossRefGoogle Scholar
Mahajan, L. 1930 The effect of the surrounding medium on the life of liquid drops floating on the same liquid surface. Phil. Mag 10, 383.CrossRefGoogle Scholar
McLaughlin, J. B. 1981 Period-doubling bifurcations and chaotic motion for a parametrically forced pendulum. J. Stat. Physics 24 (2), 375.CrossRefGoogle Scholar
Mehta, A. & Luck, J. M. 1990 Novel temporal behavior of a nonlinear dynamical system: the completely inelastic bouncing ball. Phys. Rev. Lett. 65 (4), 393.CrossRefGoogle ScholarPubMed
Neitzel, G. P. & Dell'Aversana, P. 2002 Noncoalescence and nonwetting behavior of liquids. Annu. Rev. Fluid Mech. 34, 267.CrossRefGoogle Scholar
Okumura, K., Chevy, F., Richard, D., Quéré, D. & Clanet, C. 2003 Water spring: a model for bouncing drops. Europhys. Lett. 62 (2), 237–243.CrossRefGoogle Scholar
Pan, K. L. & Law, C. K. 2007 Dynamics of droplet–film collision. J. Fluid Mech. 587, 1.CrossRefGoogle Scholar
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85.CrossRefGoogle Scholar
Protière, S., Couder, Y., Fort, E. & Boudaoud, A. 2005 The self-organisation of surface waves sources. J. Phys. Cond. Mat. 17, S3529.CrossRefGoogle Scholar
Richard, D., Clanet, C. & Quéré, D. 2002 Contact time of a bouncing drop. Nature 417, 811.CrossRefGoogle ScholarPubMed
Richard, D. & Quéré, D. 2000 Bouncing water drops. Europhys. Lett. 50 (6), 769.CrossRefGoogle Scholar
Shaw, R. 1984 The Dripping Faucet as a Model Chaotic System. Aerial Press.Google Scholar
Sparrow, C. 1982 The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer.CrossRefGoogle Scholar
Squires, T. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics towards a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381.CrossRefGoogle Scholar
Strogatz, S. 1994 Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engineering. Perseus Books.Google Scholar
Taylor, G. I. & Howarth, L. 1959 The dynamics of thin sheets of fluid. I. Water bells. Proc. Roy. Soc. A 253, 289.Google Scholar
Taylor, G. I. & Michael, D. H. 1973 On making holes in a sheet of fluid. J. Fluid Mech. 58, 625.CrossRefGoogle Scholar
Terwagne, D., Vandewalle, N. & Dorbolo, S. 2007 Lifetime of a bouncing droplet. Phys. Rev. E 76, 056311.CrossRefGoogle ScholarPubMed
Thoroddsen, S. T. & Takehara, K. 2000 The coalescence cascade of a drop. Phys. Fluids 12 (6), 1265.CrossRefGoogle Scholar