On the interaction of two encapsulated bubbles in an ultrasound field

Abstract

We establish a theoretical model for the radial oscillations, translational motions and deformations of two interacting encapsulated bubbles. The flow field outside the bubbles is approximated by a potential flow with a viscous correction. The in-plane stresses and bending moments of the viscoelastic membranes are balanced by the hydrodynamic tractions at the interfaces of the bubbles. Since the material points move along the membranes accompanied by their movements in the radial direction when the encapsulated bubbles undergo deformations, stress balance in both the tangential and normal directions and the no-velocity-jump condition at the bubble surface are applied. The derived expression for the viscous drag includes the quasisteady drag force and the history force, which is validated by the solution of the unsteady Stokes equation. With an appropriate choice of the interface parameters, the present model is suitable for bubbles with free-slip, viscoelastic or no-slip interfaces. The viscous correction and the potential part of our solution are validated, respectively, by comparing them with previous experimental observations. The encapsulated bubble shows more stability in resisting shape oscillation. The attractive or repulsive movements of the two bubbles subjected to a driving frequency are consistent with the prediction by Bjerknes’ theory. For gas bubbles, the drag is mainly from the quasisteady component of the flow. For encapsulated bubbles, the no-velocity-jump condition enhances viscous dissipation, and thus contributes significantly to the history force in the viscous drag, generating more damping in the translational motion.