Kang, I. S. and Leal, L. G. (1988) Smallamplitude perturbations of shape for a nearly spherical bubble in an inviscid straining flow (steady shapes and oscillatory motion). Journal of Fluid Mechanics, 187 . pp. 231266. ISSN 00221120. https://resolver.caltech.edu/CaltechAUTHORS:20120604111537526

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Abstract
The method of domain perturbations is used to study the problem of a nearly spherical bubble in an inviscid, axisymmetric straining flow. Steadystate shapes and axisymmetric oscillatory motions are considered. The steadystate solutions suggest the existence of a limit point at a critical Weber number, beyond which no solution exists on the steadystate solution branch which includes the spherical equilibrium state in the absence of flow (e.g. the critical value of 1.73 is estimated from the thirdorder solution). In addition, the firstorder steadystate shape exhibits a maximum radius at θ = 1/6π which clearly indicates the barrellike shape that was found earlier via numerical finitedeformation theories for higher Weber numbers. The oscillatory motion of a nearly spherical bubble is considered in two different ways. First, a small perturbation to a spherical base state is studied with the ad hoc assumption that the steadystate shape is spherical for the complete Webernumber range of interest. This analysis shows that the frequency of oscillation decreases as Weber number increases, and that a spherical bubble shape is unstable if Weber number is larger than 4.62. Secondly, the correct steadystate shape up to O(W) is included to obtain a rigorous asymptotic formula for the frequency change at small Weber number. This asymptotic analysis also shows that the frequency decreases as Weber number increases; for example, in the case of the principal mode (n = 2), ω^2 = ω_0^0(1−0.31W), where ω_0 is the oscillation frequency of a bubble in a quiescent fluid.
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Additional Information:  © 1988 Cambridge University Press. Received 10 April 1987 and in revised form 29 July 1987. Published Online April 21 2006. This work was supported by a grant from the Fluid Mechanics Program of the National Science Foundation. The authors wish to thank Professor R. A. Brown for his insightful comments on an earlier version of this paper.  
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Record Number:  CaltechAUTHORS:20120604111537526  
Persistent URL:  https://resolver.caltech.edu/CaltechAUTHORS:20120604111537526  
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Deposited By:  Tony Diaz  
Deposited On:  05 Jun 2012 21:35  
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