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Chaotic mode competition in the shape oscillations of pulsating bubbles

Published online by Cambridge University Press:  26 April 2006

D. Zardi  and
G. Seminara
D. Zardi
Affiliation:
Istituto di Idraulica, Facoltà di Ingegneria, Università di Genova, Via Montallegro 1, 16145 Genova, Italy
G. Seminara
Affiliation:
Istituto di Idraulica, Facoltà di Ingegneria, Università di Genova, Via Montallegro 1, 16145 Genova, Italy
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Abstract

A possible mechanism for the occurrence of the phenomenon of erratic drift of bubbles in liquids subjected to acoustic waves was proposed by Benjamin & Ellis (1990) who showed that nonlinear interactions between adjacent perturbation modes expressed in terms of spherical harmonics of any order may lead to the excitation of mode 1 which is equivalent to a displacement of the bubble centroid. We show that indeed such a mechanism can give rise to a chaotic process at least under the conditions experimentally investigated by Benjamin & Ellis (1990). In fact we examine the case in which the angular frequency ω of the incident wave is sufficiently close to both the natural frequency of mode n + 1 (ωn + 1) and twice the natural frequency of mode n (2ωn) thus exciting simultaneously a subharmonic mode n and a synchronous mode n + 1. The value of n is set equal to 3 in accordance with Benjamin & Ellis' (1990) observation. A classical multiple scale analysis allows us to follow the development of these perturbations in the weakly nonlinear regime to find an autonomous system of quadratically coupled nonlinear differential equations governing the evolution of the amplitudes of the perturbations on a slow time scale. As obtained by Gu & Sethna (1987) for the Faraday resonance problem, we find both regular and chaotic solutions of the above system. Chaos is found to develop for large enough values of the amplitude of the acoustic excitation within some region in the parameter space and is reached through a period-doubling sequence displaying the typical characteristics of Feigenbaum scenario.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 286 , 10 March 1995 , pp. 257 - 276
Copyright
© 1995 Cambridge University Press

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References

Benjamin, T. B. 1987 Hamiltonian theory for motions of bubbles in an infinite liquid. J. Fluid Mech. 181, 349–379.Google Scholar
Benjamin, T. B. & Ellis, A. T. 1990 Self-propulsion of asymmetrically vibrating bubbles. J. Fluid Mech. 212, 65–80.Google Scholar
Benjamin, T. B. & Strasberg, M. 1958 Excitation of oscillations in the shape of pulsating gas bubbles; Theoretical work. (Abstract) J. Acoust. Soc. Am. 30, 697.Google Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Elder, S. A. 1959 Cavitation microstreaming J. Acoust. Soc. Am. 31, 54–64.Google Scholar
Eller, A. I. & Crum, L. A. 1970 Instability of the motion of a pulsating bubble in a sound field. J. Acoust. Soc. Am. 47, 762–767.Google Scholar
Feng, Z. C. & Leal, L. G. 1994 Bifurcation and chaos in shape and volume oscillations of a periodically driven bubble with two-to-one internal resonance. J. Fluid Mech. 266, 209–242.Google Scholar
Feng, Z. C. & Sethna, P. R. 1989 Symmetry-breaking bifurcation in resonant surface waves. J. Fluid Mech. 199, 495–518.Google Scholar
Gaines, N. 1932 Magnetostriction oscillator producing intense audible sound and some effects obtained. Physics 3, 209–229.Google Scholar
Gould, R. K. 1966 Heat transfers across a solid–liquid interface in the presence of acoustic streaming. J. Acoust. Soc. Am. 40, 219–225.Google Scholar
Gu, X. M. & Sethna, P. R. 1987 Resonant surface waves and chaotic phenomena. J. Fluid Mech. 183, 543–565.Google Scholar
Hall, P. & Seminara, G. 1980 Nonlinear oscillations of non-spherical cavitation bubbles in acoustic fields. J. Fluid Mech. 101, 423–444.Google Scholar
Holmes, P. 1986 Chaotic motions in a weakly nonlinear model for surface waves. J. Fluid Mech. 162, 365–388.Google Scholar
Hsieh, D. Y. & Plesset, M. S. 1961 Theory of rectified diffusion of mass into gas bubbles. J. Acoust. Soc. Am. 33, 206–215.Google Scholar
Kambe, T. & Umeki, M. 1990 Nonlinear dynamics of two-mode interactions in parametric excitation of surface waves. J. Fluid Mech. 212, 373–393.Google Scholar
Kornfeld, M. & Suvorov, L. 1944 On the destructive action of cavitation. J. Appl. Phys. 15, 495–506.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th Edn. Cambridge University Press.
Longuet-Higgins, M. S. 1989a Monopole emission of sound by asymmetric bubble oscillations. Part 1. Normal modes J. Fluid Mech. 201, 525–541.Google Scholar
Longuet-Higgins, M. S. 1989b Monopole emission of sound by asymmetric bubble oscillations. Part 2. An initial-value problem J. Fluid Mech. 201, 543–565.Google Scholar
Mei, C. C. & Zhou, X. 1991 Parametric resonance of a spherical bubble J. Fluid Mech. 229, 29–50.Google Scholar
Miles, J. W. 1984 Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Moon, F. C. 1987 Chaotic Vibrations. John Wiley & Sons.
Norris, J. W. 1991 Abstract presented at First European Fluid Mechanics Conference, Cambridge.
Parker, T. S. & Chua, L. O. 1989 Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag.
Pearson, C. E. 1983 Handbook of Applied Mathematics. Van Nostrand Reinhold Company.
Plesset, M. S. & Mitchell, T. P. 1956 On the stability of the spherical shape of a vapor cavity in a liquid Q. Appl. Maths 13, 419–430.Google Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Ann. Rev. Fluid Mech. 9, 145–185.Google Scholar
Prosperetti, A. 1974 Nonlinear oscillation of gas bubbles in liquids: steady state solutions. J. Acoust. Soc. Am. 56, 878–885.Google Scholar
Ralston, A. & Wilf, H. S. 1960 Mathematical methods for digital computers, vol. 1. John Wiley & Sons.
Rayleigh, Lord 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94–98.Google Scholar
Saffman, P. G. 1967 The self-propulsion of a deformable body in a perfect fluid. J. Fluid Mech. 28, 385–389.Google Scholar
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471–494.Google Scholar
Strasberg, M. & Benjamin, T. B. 1958 Excitation of oscillations in the shape of pulsating gas bubbles; Experimental work. (Abstract) J. Acoust. Soc. Am. 30, 697.Google Scholar
Thompson, J. M. T. & Stewart, H. B. 1986 Nonlinear Dynamics and Chaos. John Wiley & Sons.