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Nonlinear Wave Propagation in Bubbly Liquids

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Bubble Dynamics and Shock Waves

Part of the book series: Shock Wave Science and Technology Reference Library ((SHOCKWAVES,volume 8))

Abstract

Weakly nonlinear wave equations for pressure waves in bubbly liquids are derived in a general and systematic way based on the asymptotic expansion method of multiple scales. The derivation procedure is explained in detail with a special attention to scaling relations between physical parameters characterizing the wave motions concerned. In the framework of the present theory, one can systematically deal with various weakly nonlinear wave motions for various systems of governing equations of bubbly liquids, thereby deriving such as the Korteweg–de Vries–Burgers equation, the nonlinear Schrödinger equation, and the Khokhlov–Zabolotskaya–Kuznetsov equation. In this sense, the method may be called a unified theory of weakly nonlinear waves in bubbly liquids.

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References

  1. Carstensen, E.L., Foldy, L.L.: Propagation of sound through a liquid containing bubbles. The Journal of the Acoustical Society of America 19(3), 481–501 (1947)

    Article  Google Scholar 

  2. Fox, F.E., Curley, S.R., Larson, G.S.: Phase velocity and absorption measurements in water containing air bubbles. The Journal of the Acoustical Society of America 27(3), 534–539 (1955)

    Article  Google Scholar 

  3. van Wijngaarden, L.: On the equations of motion for mixtures of liquid and gas bubbles. Journal of Fluid Mechanics 33, 465–474 (1968)

    Article  MATH  Google Scholar 

  4. van Wijngaarden, L.: One-dimensional flow of liquids containing small gas bubbles. Annual Review of Fluid Mechanics 4, 369–394 (1972)

    Article  Google Scholar 

  5. Noordzij, L., van Wijngaarden, L.: Relaxation effects, caused by relative motion, on shock waves in gas-bubble/liquid mixtures. Journal of Fluid Mechanics 66, 115–143 (1974)

    Article  MATH  Google Scholar 

  6. Kuznetsov, V.V., Nakoryakov, V.E., Pokusaev, B.G., Shreiber, I.R.: Propagation of perturbations in a gas-liquid mixture. Journal of Fluid Mechanics 85, 85–96 (1978)

    Article  Google Scholar 

  7. Caflisch, R.E., Miksis, M.J., Papanicolaou, G.C., Ting, L.: Effective equations for wave propagation in bubbly liquids. Journal of Fluid Mechanics 153, 259–273 (1985)

    Article  MATH  Google Scholar 

  8. Nigmatulin, R.I.: Dynamics of Multiphase Media, vol. 1 & 2. Hemisphere, New York (1991)

    Google Scholar 

  9. Gumerov, N.A.: Quasi-monochromatic weakly non-linear waves in a low-dispersion bubble medium. Journal of Applied Mathematics and Mechanics 56(1), 50–59 (1992)

    Article  MathSciNet  Google Scholar 

  10. Watanabe, M., Prosperetti, A.: Shock waves in dilute bubbly liquids. Journal of Fluid Mechanics 274, 349–381 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Akhatov, I., Parlitz, U., Lauterborn, W.: Towards a theory of self-organization phenomena in bubble-liquid mixtures. Physical Review E 54(5), 4990–5003 (1996)

    Article  Google Scholar 

  12. Khismatullin, D.B., Akhatov, I.S.: Sound-ultrasound interaction in bubbly fluids: theory and possible applications. Physics of Fluids 13(12), 3582–3598 (2001)

    Article  Google Scholar 

  13. Smereka, P.: A Vlasov equation for pressure wave propagation in bubbly fluids. Journal of Fluid Mechanics 454, 287–325 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gasenko, V.G., Nakoryakov, V.E.: Nonlinear three-wave equation for a polydisperse gas-liquid mixture. Journal of Engineering Thermophysics 17(3), 158–165 (2008)

    Article  Google Scholar 

  15. Kanagawa, T., Yano, T., Watanabe, M., Fujikawa, S.: Unified theory based on parameter scaling for derivation of nonlinear wave equations in bubbly liquids. Journal of Fluid Science and Technology 5(3), 351–369 (2010)

    Article  Google Scholar 

  16. Kanagawa, T., Yano, T., Watanabe, M., Fujikawa, S.: Nonlinear wave equation for ultrasound beam in nonuniform bubbly liquids. Journal of Fluid Science and Technology 6(2), 279–290 (2011)

    Article  Google Scholar 

  17. Kanagawa, T., Watanabe, M., Yano, T., Fujikawa, S.: Nonlinear wave equations for pressure wave propagation in liquids containing gas bubbles (comparison between two-fluid model and mixture model). Journal of Fluid Science and Technology 6(6), 838–850 (2011)

    Article  Google Scholar 

  18. Zabolotskaya, E.A., Khokhlov, R.V.: Quasi-plane waves in the nonlinear acoustics of confined beams. Soviet Physics Acoustics 15, 35–40 (1969)

    Google Scholar 

  19. Kuznetsov, V.P.: Equation of nonlinear acoustics. Soviet Physics Acoustics 16, 467–470 (1971)

    Google Scholar 

  20. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Soviet Physics Doklady 15, 539–541 (1970)

    MATH  Google Scholar 

  21. Drew, D.A., Passman, S.L.: Theory of Multicomponent Fluids. Springer, New York (1999)

    Google Scholar 

  22. Elder, S.A.: Cavitation microstreaming. The Journal of the Acoustical Society of America 31(1), 54–64 (1959)

    Article  Google Scholar 

  23. Bjerknes, V.F.K.: Fields of Force. Columbia University Press, New York (1906)

    Google Scholar 

  24. Crum, L.A.: Bjerknes forces on bubbles in a stationary sound field. The Journal of the Acoustical Society of America 57(6), 1363–1370 (1975)

    Article  Google Scholar 

  25. Keller, J.B., Kolodner, I.I.: Damping of underwater explosion bubble oscillations. Journal of Applied Physics 27(10), 1152–1161 (1956)

    Article  Google Scholar 

  26. Egashira, R., Yano, T., Fujikawa, S.: Linear wave propagation of fast and slow modes in mixtures of liquid and gas bubbles. Fluid Dynamics Research 34, 317–334 (2004)

    Article  MATH  Google Scholar 

  27. Yano, T., Egashira, R., Fujikawa, S.: Linear analysis of dispersive waves in bubbly flows based on averaged equations. Journal of the Physical Society of Japan 75(10), 104401 (2006)

    Article  Google Scholar 

  28. Jones, A.V., Prosperetti, A.: On the suitability of first-order differential models for two-phase flow prediction. International Journal of Multiphase Flow 11(2), 133–148 (1985)

    Article  MATH  Google Scholar 

  29. Zhang, D.Z., Prosperetti, A.: Averaged equations for inviscid disperse two-phase flow. Journal of Fluid Mechanics 267, 185–219 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Eames, I., Hunt, J.C.R.: Forces on bodies moving unsteadily in rapidly compressed flows. Journal of Fluid Mechanics 505, 349–364 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)

    MATH  Google Scholar 

  32. Jeffrey, A., Kawahara, T.: Asymptotic Methods in Nonlinear Wave Theory. Pitman, London (1982)

    MATH  Google Scholar 

  33. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  34. Bogoliubov, N.N., Mitropolsky, Y.A.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon and Breach, New York (1961)

    Google Scholar 

  35. Inoue, Y., Matsumoto, Y.: Nonlinear wave modulation in dispersive media. Journal of the Physical Society of Japan 36(5), 1446–1455 (1974)

    Article  Google Scholar 

  36. Kakutani, T., Inoue, Y., Kan, T.: Nonlinear capillary waves on the surface of liquid column. Journal of the Physical Society of Japan 37(2), 529–538 (1974)

    Article  Google Scholar 

  37. Kakutani, T., Sugimoto, N.: Krylov–Bogoliubov–Mitropolsky method for nonlinear wave modulation. Physics of Fluids 17(8), 1617–1625 (1974)

    Article  MathSciNet  Google Scholar 

  38. Zemanek, J.: Beam behavior within the nearfield of a vibrating piston. The Journal of the Acoustical Society of America 49, 181–191 (1971)

    Article  Google Scholar 

  39. Blackstock, D.T.: Fundamental of Physical Acoustics, ch. 13. Wiley, New York (2000)

    Google Scholar 

  40. Hamilton, M.F., Blackstock, D.T. (eds.): Nonlinear Acoustics, ch. 8. Academic Press, New York (1998)

    Google Scholar 

  41. Fujikawa, S., Yano, T., Watanabe, M.: Vapor-Liquid Interface, Bubbles and Droplets: Fundamentals and Applications. Springer, Berlin (2011)

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. Osaka University, Osaka, Japan

    T. Yano & S. Fujikawa

  2. University of Tokyo, Tokyo, Japan

    T. Kanagawa

  3. Hokkaido University, Hokkaido, Japan

    M. Watanabe

Authors
  1. T. Yano
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  2. T. Kanagawa
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  3. M. Watanabe
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  4. S. Fujikawa
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Corresponding author

Correspondence to T. Yano .

Editor information

Editors and Affiliations

  1. , Department of Mechanical Engineering, Isik University, Mesrutiyet Koyu, Universite Sokak 2, Sile, 34980, Turkey

    Can F. Delale

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Yano, T., Kanagawa, T., Watanabe, M., Fujikawa, S. (2013). Nonlinear Wave Propagation in Bubbly Liquids. In: Delale, C. (eds) Bubble Dynamics and Shock Waves. Shock Wave Science and Technology Reference Library, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34297-4_4

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