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Parametric resonance of a spherical bubble

Published online by Cambridge University Press:  26 April 2006

Chiang C. Mei  and
Xianchu Zhou
Chiang C. Mei
Affiliation:
Civil Engineering Department, Massachusetts Institute of Technology. Cambridge MA 02139, USA
Xianchu Zhou
Affiliation:
Institute of Mechanics, Academia Sinica, Beijing, China
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Abstract

We modify a recent theory of Longuet-Higgins (1989a, b) to study the resonant interaction between an isotropic mode and one or two distort ional modes of an oscillating bubble in water when the isotropic mode is forced by ambient sound. Gravity and buoyant rise are ignored. The energy exchange between modes is strong enough so that both (or all three) can attain comparable amplitudes after a long time. We show that for two-mode interactions the mode-coupling equations are similar to those studied in other physical contexts such as nonlinear optics, coupled oscillators and standing waves in a basin. Instability around fixed points is examined for various bubble radii, phase mismatch, and detuning of the external forcing. Numerical evidences of chaotic bubble oscillations and sound radiation are discussed. It is found that in a certain parameter domain, Hopf bifurcations are possible, and chaos is reached via a period-doubling sequence. However, when there are three interacting modes, each of the two distortion modes interacts with the breathing mode directly and the route to chaos is via a quasi-periodic 2-torus. Possible relevance of this theory to the observed erratic drifting of a bubble is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 29 - 50
Copyright
© 1991 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Erdélyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1953 Higher Transcendental Functions, vol. 2. McGraw-Hill.
Evans, D. V. & Linton, C. M. 1991 Trapped modes in open channels. Fluid Mech. 225, 153–175.Google Scholar
Evans, D. V. & McIver, P. 1991 Trapped waves over symmetric thin bodies. J. Fluid Mech. 223, 509–519.Google Scholar
Jones, D. S. 1953 The eigenvalues of 2u + LDu = 0 when the boundary conditions are given in semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668–684.Google Scholar
Leblond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.
Mciver, P. 1991 Trapping of surface water waves by fixed bodies in channels. Q. J. Mech. Appl. Maths (to appear.)Google Scholar
Mciver, P. & Evans, D. V. 1985 The trapping of surface waves above a submerged horizontal cylinder. J. Fluid Mech. 151, 243–255.Google Scholar
Ursell, F. 1950 Surface waves on deep water in the presence of a submerged circular cylinder I. Proc. Camb. Phil. Soc. 46, 141–152.Google Scholar
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 346–358.Google Scholar
Ursell, F. 1952 Edge waves over a sloping beach. Proc. R. Soc. Lond. A214, 79–97.Google Scholar
Ursell, F. 1968 On the head seas travelling along a horizontal cylinder. J. Inst. Math. Appl. 4, 414–427.Google Scholar
Ursell, F. 1987 Mathematical aspects of trapping modes in the theory of surface waves. J. Fluid Mech. 183, 421–437.Google Scholar
Watson, G. N. 1966 A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.