Shape oscillation and stability of an encapsulated microbubble translating in an acoustic wave

Encapsulated microbubbles (EMBs) are associated with a wide variety of important medical applications, including sonography, drug delivery, and sonoporation. The nonspherical oscillations, or shape modes, of EMBs strongly affect their stability and acoustic signature, and thus are an important factor to consider in the design and utilization of EMBs. Under acoustic forcing, EMBs often translate with significant velocity, which can excite shape modes, yet few studies have addressed the effect of translation on the shape stability of EMBs. In this work, the shape stability of an EMB subject to translation is investigated through development of an axisymmetric model for the case of small deformations. The potential flow in the bulk volume of the external flow is modeled using an asymptotic analysis. Viscous effects within the thin boundary layer at the interface are included, owing to the no-slip boundary condition, using Prosperetti's theory [Q. Appl. Math. 34, 339 (1977)]. In-plane stress and bending moment due to the encapsulation are incorporated into the model through the dynamic boundary condition at the interface. The evolution equations for radial oscillation, translation, and shape oscillation of an EMB are derived, which can be reduced to model an uncoated gas bubble by neglecting the encapsulation properties. These equations are solved numerically to analyze the shape mode stability of an EMB and a gas bubble subject to an acoustic, traveling plane wave. The findings demonstrate the counterintuitive result that translation has a more destabilizing effect on an EMB than on a gas bubble. The no-slip condition at the encapsulating membrane is the main factor responsible for mediating this interfacial instability due to translation.