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Small-amplitude perturbations of shape for a nearly spherical bubble in an inviscid straining flow (steady shapes and oscillatory motion)

Kang, I. S. and Leal, L. G. (1988) Small-amplitude perturbations of shape for a nearly spherical bubble in an inviscid straining flow (steady shapes and oscillatory motion). Journal of Fluid Mechanics, 187 . pp. 231-266. ISSN 0022-1120. doi:10.1017/S0022112088000412.

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The method of domain perturbations is used to study the problem of a nearly spherical bubble in an inviscid, axisymmetric straining flow. Steady-state shapes and axisymmetric oscillatory motions are considered. The steady-state solutions suggest the existence of a limit point at a critical Weber number, beyond which no solution exists on the steady-state 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 third-order solution). In addition, the first-order steady-state shape exhibits a maximum radius at θ = 1/6π which clearly indicates the barrel-like shape that was found earlier via numerical finite-deformation 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 steady-state shape is spherical for the complete Weber-number 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 steady-state 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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ID Code:31796
Deposited By: Tony Diaz
Deposited On:05 Jun 2012 21:35
Last Modified:09 Nov 2021 20:00

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