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Surface instability of an encapsulated bubble induced by an ultrasonic pressure wave

Published online by Cambridge University Press:  06 December 2011

Yunqiao Liu ,
Kazuyasu Sugiyama ,
Shu Takagi  and
Yoichiro Matsumoto
Yunqiao Liu*
Affiliation:
Department of Mechanical Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Kazuyasu Sugiyama
Affiliation:
Department of Mechanical Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Shu Takagi
Affiliation:
Department of Mechanical Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Organ and Body Scale Team, CSRP, Riken, 2-1, Hirosawa, Wako-shi, Saitama 351-0198, Japan
Yoichiro Matsumoto
Affiliation:
Department of Mechanical Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
*
†Email address for correspondence: yqliu@fel.t.u-tokyo.ac.jp
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Abstract

In this paper, we investigate the shape stability of a nearly spherical bubble encapsulated by a viscoelastic membrane in an ultrasound field. To describe the dynamic balance on the bubble surface, the in-plane stress and the bending moment are incorporated into the governing equations for the perturbed radial flow of viscous incompressible fluid (Prosperetti, Q. Appl. Math., vol. 34, 1977, p. 339). The radial motion of the bubble is obtained by solving the Rayleigh–Plesset equation with elastic stress. The deflection therefrom is linearized and expanded with respect to the Legendre polynomial of order . Two amplitudes for each shape mode are introduced because the membrane moves not only in the radial direction but also in the tangential direction. The system with a boundary layer approximation is reduced to Mathieu’s equation. A simple expression for the natural frequency of the shape mode is derived, which is validated by direct numerical simulation. Stability diagrams for the higher-order shape mode are mapped out in the phase space of driving amplitude and frequency over a range of values of the elastic modulus of the membrane. The most unstable driving frequency is found to satisfy an integer multiple relationship of the form , due to the structure of Mathieu’s equation in the system. In addition to the resonance interaction, liquid viscosity plays an important role in the stability of the encapsulated bubble.

JFM classification

Drops and Bubbles: Bubble dynamics Biological Fluid Dynamics: Membranes Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 691 , 25 January 2012 , pp. 315 - 340
Copyright
Copyright © Cambridge University Press 2011

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