Skip to main content

Bubble expansion rates in viscous compressible liquid


A new equation has been derived for radial expansion of a bubble in viscous liquid, taking into account the compressibility of liquid. This model is important in understanding the time-evolution of the bubble growth in a cavitating flow. We solved the equation of motion of viscous liquid and obtained the velocity field and the pressure distribution in liquid. It is found that the expansion rate of a bubble in viscous compressible liquid is governed by the friction force due to viscosity as well as the surface tension, the inertial force, and the gas pressure. Further, the bubble expansion rate is brought by the same expression to that in an incompressible liquid only in the case that the liquid is in a hydrostatic state before the bubble grows.


  • Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, New York, 1967.

    Google Scholar 

  • John, F., Partial Differential Equations, 4th edition, Springer-Verlag, New York, 1982.

    Book  Google Scholar 

  • Keller, B. J. and I. I. Kolodner, Damping of underwater explosion bubble oscillations, J. Appl. Phys., 27, 1152–1161, 1956.

    Article  Google Scholar 

  • Keller, B. J. and M. Miksis, Bubble oscillations of large amplitude, J. Acoust. Soc. Am., 68(2), 628–633, 1980.

    Article  Google Scholar 

  • Landau, L. D. and E. M. Lifshitz, Fluid Mechanics, 2nd edition, Butterworth-Heinemann, Oxford, 1987.

    Google Scholar 

  • Lauterborn, W., Numerical investigation of nonlinear oscillations of gas bubbles in liquids, J. Acoust. Soc. Am., 59(2), 283–293, 1976.

    Article  Google Scholar 

  • Papale, P., Volcanic conduit dynamics, FROM MAGMA TO TEPHRA, Modelling Physical Processes of Explosive Volcanic Eruptions, Elsevier, Amsterdam, chapter 3, 2001.

    Google Scholar 

  • Plesset, M. S., The dynamics of cavitation bubbles, J. Appl. Mech., 16, 277–282, 1949.

    Google Scholar 

  • Plesset, M. S. and A. Prosperetti, Bubble dynamics and cavitation, Ann. Rev. Fluid Mech., 9, 145–185, 1977.

    Article  Google Scholar 

  • Prosperetti, A. and A. Lezzi, Bubble dynamics in a compressible liquid. Part 1. First-order theory, J. Fluid Mech., 168, 457–478, 1986.

    Article  Google Scholar 

  • Prosperetti, A., L. A. Crum, and K. W. Commander, Nonlinear bubble dynamics, J. Acoust. Soc. Am., 83(2), 502–514, 1988.

    Article  Google Scholar 

  • Proussevitch, A. A. and D. L. Sahagian, Dynamics and energetics of bubble growth in magmas: analytical formulation and numerical modeling, J. Geophys. Res., 103(B8), 18,223–18,251, 1998.

    Article  Google Scholar 

  • Rayleigh, L., On the pressure developed in a liquid during the collapse of a spherical cavity, Philos. Mag., 34, 94–98, 1917.

    Article  Google Scholar 

  • Scriven, L. E., On the dynamics of phase growth, Chem. Eng. Sci., 10, 1–13, 1959.

    Article  Google Scholar 

  • Sparks, R. S. J., The dynamics of bubble formation and growth in magmas: a review and analysis, J. Volcanol. Geotherm. Res., 3, 1–37, 1978.

    Article  Google Scholar 

  • Toramaru, A., Numerical study of nucleation and growth of bubbles in viscous magmas, J. Geophys. Res., 100(B2), 1913–1931, 1995.

    Article  Google Scholar 

  • Trilling, L., The collapse and rebound of a gas bubble, J. Appl. Phys., 23(1), 14–17, 1952.

    Article  Google Scholar 

  • Yamada, K., H. Tanaka, K. Nakazawa, and H. Emori, A new theory of bubble formation in magma, J. Geophys. Res., 110, B02203, doi:10.1029/2004JB003113, 2005.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Kou Yamada.

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit

Reprints and Permissions

About this article

Cite this article

Yamada, K., Emori, H. & Nakazawa, K. Bubble expansion rates in viscous compressible liquid. Earth Planet Sp 58, 865–872 (2006).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Key words

  • Bubble
  • expansion
  • compressible
  • viscous
  • cavitation