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

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≥ 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 2ω ₖ / ω 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.