Surface instability of an encapsulated bubble induced by an ultrasonic pressure wave
Surface instability of an encapsulated bubble induced by an ultrasonic pressure wave
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DOI：
10.1017/jfm.2011.477
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发表时间：
2011-12
影响因子：
3.7
通讯作者：
Yunqiao Liu;K. Sugiyama;S. Takagi;Y. Matsumoto
中科院分区：
文献类型：
--
作者：
Yunqiao Liu;K. Sugiyama;S. Takagi;Y. Matsumoto
摘要本文研究了超声场中粘弹性膜包裹的近球形气泡的形状稳定性。为了描述气泡表面上的动态平衡，将面内应力和弯矩并入粘性不可压缩流体的扰动径向流的控制方程（Prosperetti，Q.应用数学：第34卷，1977，第339页）。通过求解含弹性应力的Rayleigh-Plesset方程，得到了气泡的径向运动。从偏转线性化和扩展相对于勒让德多项式的顺序$k\geq 2$。每个形状模式的两个振幅被引入，因为膜不仅在径向方向上，而且在切向方向上移动。在边界层近似下，系统简化为Mathieu方程.推导了形状模态固有频率的简单表达式，并通过直接数值模拟进行了验证。高阶形状模式的稳定性图绘制在相空间中的驱动振幅和频率的范围内的膜的弹性模量的值。最不稳定的驱动频率被发现满足形式为2 {\omega }_{k} / {\omega }_{d} = n$的整数倍关系，由于在系统中的Mathieu方程的结构。除了共振相互作用外，液体粘度对气泡的稳定性也起着重要的作用。
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 $k\geq 2$. 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 $2{\omega }_{k} / {\omega }_{d} = n$, 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.