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Dynamics of microbubble oscillators with delay coupling

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Abstract

Two vibrating bubbles submerged in a fluid influence each others’ dynamics via sound waves in the fluid. Due to finite sound speed, there is a delay between one bubble’s oscillation and the other’s. This scenario is treated in the context of coupled nonlinear oscillators with a delay coupling term. It has previously been shown that with sufficient time delay, a supercritical Hopf bifurcation may occur for motions in which the two bubbles are in phase. In this work, we further examine the bifurcation structure of the coupled microbubble equations, including analyzing the sequence of Hopf bifurcations that occur as the time delay increases, as well as the stability of this motion for initial conditions which lie off the in-phase manifold. We show that in fact the synchronized, oscillating state resulting from a supercritical Hopf is attracting for such general initial conditions.

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References

  1. Dijkmans, P.A., et al.: Microbubbles and ultrasound: from diagnosis to therapy. Eur. J. Echocardiogr. 5, 245–256 (2004)

    Article  Google Scholar 

  2. Doinikov, A.A., Manasseh, R., Ooi, A.: Time delays in coupled multibubble systems. J. Acoust. Soc. Am. 117, 47–50 (2005)

    Article  Google Scholar 

  3. Ferrara, K., Pollard, R., Borden, M.: Ultrasound microbubble contrast agents: fundamentals and application to gene and drug delivery. Annu. Rev. Biomed. Eng. 9, 415–447 (2007)

    Article  Google Scholar 

  4. Gilmore, F.R.: “The growth or collapse of a spherical bubble in a viscous compressible liquid“. Report no. 26-4, Hydrodynamics Laboratory, California Institute of Technology, Pasadena, California (1952)

  5. Harkin, A., Kaper, T., Nadim, A.: Coupled pulsation and translation of two gas bubbles in a liquid. J. Fluid Mech. 445, 377–411 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Keller, J.B., Kolodner, I.I.: Damping of underwater explosion bubble oscillations. J. Appl. Phys. 27, 1152–1161 (1956)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Lauterborn, W., Parlitz, U.: Methods of chaos physics and their application to acousticse. J. Acoust. Soc. Am. 84(6), 1975–1993 (1988)

    Article  MathSciNet  Google Scholar 

  9. Leighton, T.G.: From sea to surgeries, from babbling brooks to baby scans: bubble acoustics at ISVR. Proc. Inst. Acoust. 26, 357–381 (2004)

    Google Scholar 

  10. Manasseh, R., Nikolovska, A., Ooi, A., Yoshida, S.: Anisotropy in the sound field generated by a bubble chain. J. Sound Vib. 278, 807–882 (2004)

    Article  Google Scholar 

  11. Mettin, R.: Dynamics of delay-coupled spherical bubbles. In: Lauterborn, W., Kurz, T. (eds.) Proceedings of the 15th International Symposium on Nonlinear Acoustics Göttingen, Germany 1–4 Sept. (1999)

    Google Scholar 

  12. Heckman, C.R., Sah, S.M., Rand, R.H.: Dynamics of microbubble oscillators with delay coupling. Commun. Nonlinear Sci. Numer. Simul. 15, 2735–2743 (2010)

    Article  Google Scholar 

  13. Plesset, M.S.: The dynamics of cavitation bubbles. J. Appl. Mech. 16, 277–282 (1949)

    Google Scholar 

  14. Plesset, M.S., Prosperettti, A.: Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145–185 (1977)

    Article  Google Scholar 

  15. Rand, R.H.: Lecture Notes in Nonlinear Vibrations, version 53. Published on-line by the Internet-First University Press (2012) http://ecommons.library.cornell.edu/handle/1813/28989

  16. Rand, R.H., Armbruster, D.: Perturbation Methods, Bifurcation Theory, and Computer Algebra. Springer, New York (1987)

    Book  MATH  Google Scholar 

  17. Rand, R.H.: Topics in Nonlinear Dynamics with Computer Algebra. Gordon and Breach Science, Langhorne (1994)

    MATH  Google Scholar 

  18. Rand, R.H., Heckman, C.R.: Dynamics of coupled bubble oscillators with delay. In: Proceedings of ASME 2009 IDETC/CIE 2009, August 30–September 2, 2009, San Diego, California

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

    Article  MATH  Google Scholar 

  20. Reddy, A.J., Szeri, A.J.: Coupled dynamics of translation and collapse of acoustically driven microbubbles. J. Acoust. Soc. Am. 112, 1346–1352 (2002)

    Article  Google Scholar 

  21. Toilliez, J.O., Szeri, A.J.: Optimized translation of microbubbles driven by acoustic fields. J. Acoust. Soc. Am. 123, 1916–1930 (2008)

    Article  Google Scholar 

  22. Wirkus, S., Rand, R.: Dynamics of two coupled Van Der Pol oscillators with delay coupling. Nonlinear Dyn. 30, 205–221 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yamakoshi, Y., Ozawa, Y., Ida, M., Masuda, N.: Effects of Bjerknes forces on Gas-Filled microbubble trapping by ultrasonic waves. Jpn. J. Appl. Phys. 40, 3852–3855 (2001)

    Article  Google Scholar 

  24. Heckman, C.R., Rand, R.H.: Asymptotic analysis of the Hopf-Hopf bifurcation in a time-delay system. J. Appl. Nonlinear Dyn. 1(2), 159–171 (2012)

    Google Scholar 

  25. Suchorsky, M.K., Sah, S.M., Rand, R.H.: Using delay to quench undesirable vibrations. Nonlinear Dyn. 62, 107–116 (2010)

    Article  MathSciNet  Google Scholar 

  26. Heckman, C.R., Rand, R.H.: Dynamics of coupled microbubbles with large fluid compressibility delays. In: Proc. EUROMECH 2011 Euro. Nonlin. Osc. Conf

  27. Rand, R.H.: Differential-Delay equations. In: Luo, A.C.J., Sun, J.-Q. (eds.) Complex Systems: Fractionality, Time-delay and Synchronization, pp. 83–117. Springer, Berlin (2011)

    Google Scholar 

  28. Engelborghs, K., Luzyanina, T., Samaey, G., Roose, D., Verheyden, K.: DDE-BIFTOOL. Available from http://twr.cs.kuleuven.be/research/software/delay/ddebiftool.shtml

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Authors and Affiliations

  1. Field of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14850, USA

    C. R. Heckman

  2. Department of Mechanical and Aerospace Engineering and Department of Mathematics, Cornell University, Ithaca, NY, 14850, USA

    R. H. Rand

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  1. C. R. Heckman
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  2. R. H. Rand
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Correspondence to R. H. Rand.

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CRH acknowledges the support of the National Science Foundation through the Graduate Research Fellowship Program. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not reflect those of the National Science Foundation.

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Heckman, C.R., Rand, R.H. Dynamics of microbubble oscillators with delay coupling. Nonlinear Dyn 71, 121–132 (2013). https://doi.org/10.1007/s11071-012-0645-2

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  • DOI: https://doi.org/10.1007/s11071-012-0645-2

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