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Nonlinear effects in the dynamics of shape and volume oscillations for a gas bubble in an external flow

Published online by Cambridge University Press:  26 April 2006

S. M. Yang ,
Z. C. Feng  and
L. G. Leal
S. M. Yang
Affiliation:
Department of Chemical and Nuclear Engineering, University of California – Santa Barbara, Santa Barbara, California, USA Permanent address: Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea.
Z. C. Feng
Affiliation:
Department of Chemical and Nuclear Engineering, University of California – Santa Barbara, Santa Barbara, California, USA
L. G. Leal
Affiliation:
Department of Chemical and Nuclear Engineering, University of California – Santa Barbara, Santa Barbara, California, USA
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Abstract

This paper considers the dynamics of a gas bubble in response to either a pressure pulse or a pressure step at t = 0, both in the presence and absence of a mean flow. Our work utilizes small-deformation, domain perturbation analysis carried to second and higher order in the amplitude of deformation, ε. In the absence of a mean flow, our analysis of the small deformation problem for an initial impulsive perturbation of the bubble volume and shape is closely related to recently published work by Longuet-Higgins on the time-dependent oscillations of an initially deformed bubble in a quiescent fluid. However, in the presence of a mean flow which deforms the bubble, the bubble response to pressure changes is more complex. Specifically, the present analysis identifies a number of different mechanisms for resonant interaction between shape deformation modes and the volume or radial breathing mode of oscillation. This includes not only a fundamental change in the resonant interactions at 0(ε2) - where resonant interaction is also found in the absence of mean flow – but resonant interactions also at the level of 0(ε3/2;) which are not present without the mean flow. On the other hand, the bubble dynamics in response to a step change in the pressure distribution in a quiescent fluid exhibits similar resonant interactions at 0(ε2) to those obtained for a pressure pulse in the presence of mean flow because the bubble oscillates around a non-spherical steady-state shape owing to the non-uniform pressure distribution on the bubble surface in both the cases.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 247 , February 1993 , pp. 417 - 454
Copyright
© 1993 Cambridge University Press

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