Oscillation of Droplets and Bubbles

  • N. AshgrizEmail author
  • M. Movassat


A liquid droplet may go through shape oscillation if it is forced out of its equilibrium spherical shape, while gas bubbles undergo both shape and volume oscillations because they are compressible. This can happen when droplets and bubbles are exposed to an external flow or an external force. Liquid droplet oscillation is observed during the atomization process when a liquid ligament is first separated from a larger mass or when two droplets are collided. Droplet oscillations may change the rate of heat and mass transport. Bubble oscillations are important in cavitation problems, effervescent atomizers and flash atomization where large number of bubbles oscillate and interact with each other. This chapter provides the basic theory for the oscillation of liquid droplet and gas bubbles.


Bjerknes force Bubble breakup Bubble interaction Bubble oscillation Chaotic oscillation Damping rate Droplet oscillation Nonlinear oscillation Oscillation frequency RPNNP equation Shape modes Spherical harmonics Volume oscillation 


  1. 1.
    J. W. S. Rayleigh, On the capillary phenomena of jets, Proc. R. Soc. Lond. 29, 71, 1879.CrossRefGoogle Scholar
  2. 2.
    F.H. Busse, J. Fluid Mech. 142, 1, 1984.zbMATHCrossRefGoogle Scholar
  3. 3.
    H. Lamb, Hydrodynamics, 6th ed. Cambridge University Press, Cambridge, 1932.zbMATHGoogle Scholar
  4. 4.
    S. Chandrasekhar, The oscillations of a viscous liquid globe, Proc. Lond. Math. Soc. 9, 141 1959.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    W. H. Reid, The oscillations of a viscous liquid drop, Q. Appl. Math. 18, 86, 1960.Google Scholar
  6. 6.
    A. Prosperetti, Free oscillations of drops and bubbles: the initial-value problem, J. Fluid Mech. 100, 333, 1980.zbMATHCrossRefGoogle Scholar
  7. 7.
    J. A. Tsamopoulos and R. A. Brown, Nonlinear oscillations of inviscid drops and bubbles, J. Fluid Mech. 127, 519, 1983.zbMATHCrossRefGoogle Scholar
  8. 8.
    G. B. Foote, A numerical method for studying simple drop behavior: simple oscillation, J. Comput. Phys. 11, 507, 1973.CrossRefGoogle Scholar
  9. 9.
    C. T. Alonso, The Dynamics of Colliding and Oscillating Drops, in Proceedings of the International Colloquium on Drops and Bubbles, edited by D. J. Collins, M. S. Plesset, and M. M. Saffren, Jet Propulsion Laboratory, 1974.Google Scholar
  10. 10.
    T. S. Lundgren and N. N. Mansour, Oscillations of drops in zero gravity with weak viscous effects, J. Fluid Mech. 194, 479, 1991.CrossRefMathSciNetGoogle Scholar
  11. 11.
    T. W. Patzek, R. E. Benner, Jr., O. A. Basaran, and L. E. Scriven, Nonlinear oscillations of inviscid free drops, J. Comput. Phys. 97, 489 1991.zbMATHCrossRefGoogle Scholar
  12. 12.
    O. A. Basaran, Nonlinear oscillations of viscous liquid drops, J. Fluid Mech. 241, 169, 1992.zbMATHCrossRefGoogle Scholar
  13. 13.
    E. Trinh and T. G. Wang, Large-amplitude free and driven drop shape oscillations: experimental observations, J. Fluid Mech. 122, 315, 1982.CrossRefGoogle Scholar
  14. 14.
    E. Becker, W. J. Hiller, and T. A. Kowalewski, Experimental and theoretical investigation of large amplitude oscillations of liquid droplets, J. Fluid Mech. 231, 180, 1991.CrossRefGoogle Scholar
  15. 15.
    E. Becker, W. J. Hiller, and T. A. Kowalewski, Nonlinear dynamics of viscous droplets, J. Fluid Mech. 258, 191, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    F. Mashayek and N. Ashgriz, Nonlinear oscillation of liquid drops With internal circulation, Phys. Fluids 10(5), 1071–1082, May 1998.CrossRefGoogle Scholar
  17. 17.
    C. A. Miller and L. E. Scriven, The oscillations of a fluid droplet immersed in another fluid, J. Fluid Mech. 32, 417, 1968.CrossRefGoogle Scholar
  18. 18.
    P. L. Marston, Shape oscillation and static deformation of drops and bubbles driven by modulated radiation stresses: theory, J. Acoust. Soc. Am. 67, 15, 1980.zbMATHCrossRefGoogle Scholar
  19. 19.
    A. Prosperetti, Normal-mode analysis for the oscillations of a viscous liquid drop immersed in another liquid, J. Me´c. 19, 142, 1980.MathSciNetGoogle Scholar
  20. 20.
    O. A. Basaran, T. C. Scott, and C. H. Byers, Drop oscillations in liquid-liquid systems, AIChE. J. 35, 1263, 1989.CrossRefGoogle Scholar
  21. 21.
    E. Trinh, A. Zwern, and T. G. Wang, An experimental study of small amplitude drop oscillations in immiscible liquid systems, J. Fluid Mech. 115, 453, 1982.CrossRefGoogle Scholar
  22. 22.
    O. A. Basaran and D. W. Depaoli, Nonlinear oscillations of pendant drops, Phys. Fluids 6, 2923, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    D. W. DePaoli, J. Q. Feng, O. A. Basaran, and T. C. Scott, Hysteresis in forced oscillations of pendant drops, Phys. Fluids 7, 1181, 1995.CrossRefGoogle Scholar
  24. 24.
    J. A. Tsamopoulos and R. A. Brown, Resonant oscillations of inviscid charged drops, J. Fluid Mech. 147, 373, 1984.zbMATHCrossRefGoogle Scholar
  25. 25.
    E. H. Trinh, R. G. Holt, and D. B. Thiessen, The dynamics of ultrasonically levitated drops in an electric field, Phys. Fluids 8, 43, 1996.CrossRefGoogle Scholar
  26. 26.
    T. Shi and R. E. Apfel, Oscillatiaons of a deformed liquid drop in an acoustic field, Phys. Fluids 7(7), 1545, 1995.zbMATHCrossRefGoogle Scholar
  27. 27.
    E. H. Trinh, D. B. Thiessen, and R. G. Holt, Driven and freely decaying nonlinear shape oscillations of drops and bubbles immersed in a liquid, experimental results, J. Fluid Mech. 364, 253, 1998.zbMATHCrossRefGoogle Scholar
  28. 28.
    N.J. Holter and W.R. Glasscock, Vibrations of evaporating liquid drops, J. Acoust. Soc. Am. 24, 682, 1952.CrossRefGoogle Scholar
  29. 29.
    K. Adachi and R. Takaki, Vibration of a flattened drop. I. Observation, J. Phys. Soc. Jpn. 53, 4184, 1984.CrossRefGoogle Scholar
  30. 30.
    K. Adachi and R. Takaki, Vibration of a flattened drop. II. Normal mode analysis, J. Phys. Soc. Jpn. 54, 2462, 1985.CrossRefGoogle Scholar
  31. 31.
    N. Yoshiyasu, K. Matsuda, and R. Takaki, Self-induced vibration of a water drop placed on an oscillating plate, J. Phys. Soc. Jpn. 65, 2068, 1996.CrossRefGoogle Scholar
  32. 32.
    A. J. James, B. Vukasinovic, M. K. Smith, and A. Glezer, Vibration–induced drop atomization and bursting, J. Fluid Mech. 476, 1, 2003.zbMATHGoogle Scholar
  33. 33.
    L. Rayleigh, The pressure developed in a liquid during the collapse of a spherical cavity, Phil. Mag. 34, 94, 1917.Google Scholar
  34. 34.
    M. S. Plesset and T. P. Mitchell, On the stability of the spherical shape of a vapor cavity in a liquid, Q. Appl. Math. 13, 419, 1955.MathSciNetGoogle Scholar
  35. 35.
    A. I. Eller, Force on a bubble in a standing acoustic wave, J. Acoust. Soc. Am. 43, 170, 1968.CrossRefGoogle Scholar
  36. 36.
    V. F. K. Bjerknes, Fields of Force. Columbia University Press, New York, 1906.Google Scholar
  37. 37.
    A. Prosperetti, Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids, J. Acoust. Soc. Am. 61, 17, 1977.CrossRefGoogle Scholar
  38. 38.
    A. Prosperetti, Bubble phenomena in sound fields: part one, Ultrasonics 22, 69, 1984.CrossRefGoogle Scholar
  39. 39.
    A. Prosperetti, Bubble phenomena in sound fields: part two, Ultrasonics 22, 115, 1984.CrossRefGoogle Scholar
  40. 40.
    T. Barbat, N. Ashgriz, and C. Liu, Dynamics of two interacting bubbles in an acoustic field, J. Fluid Mech. 389, 137, 1999.zbMATHCrossRefGoogle Scholar
  41. 41.
    T. Barbat and N. Ashgriz, Planner dynamics of two interacting bubbles in an acoustic field, Appl. Math. Comput. 157, 775, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    A. I. Eller and L. A. Crum, Instability of the motion of a pulsating bubble in a sound field, J. Acoust. Soc. Am. 47, 762, 1970.CrossRefGoogle Scholar
  43. 43.
    C. C. Mei and X. Zhou, Parametric resonance of a spherical bubble, J. Fluid Mech. 229, 26, 1991.CrossRefGoogle Scholar
  44. 44.
    A. A. Doinikov, Translational motion of a bubble undergoing shape oscillations, J. Fluid Mech. 501, 1, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    T. B. Benjamin and A. T. Ellis, Self-propulsion of asymmetrically vibrating bubbles, J. Fluid Mech. 212, 65, 1990.zbMATHCrossRefGoogle Scholar
  46. 46.
    I. Sh. Akhatov and S. I. Konovalova, Regular and chaotic dynamics of a spherical bubble, J. Appl. Math. Mech., 69, 575, 2005.CrossRefMathSciNetGoogle Scholar
  47. 47.
    T. Watanabe and Y. Kukita, Translational and radial motions of a bubble in an acoustic standing wave field, Phys. Fluids A. 5, 2682, 1993.CrossRefGoogle Scholar
  48. 48.
    M. Movassat, N. Ashgriz, and M. Bussmann, Bubble dynamics under forced oscillation in microgravity environment, in Proceedings of ASME International Mechanical Engineering Congress and Exposition, November 2009, Lake Buena Vista, FL.Google Scholar
  49. 49.
    H. N. Yoshikawa, F. Zoueshtigh, H. Caps, P. Kurowski, and P. Petitjeans, Bubble splitting in oscillatory flows on ground and in reduced gravity, Eur. Phys. J. E. 31, 191, 2010.CrossRefGoogle Scholar
  50. 50.
    M. Movassat, N. Ashgriz, and M. Bussmann, Three-dimensional numerical simulation of bubble interaction under forced vibration, Presented in COSPAR Meeting, July 2010, Bremen, Germany.Google Scholar

Copyright information

© Springer US 2011

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCA

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