Model for the dynamics of a spherical bubble undergoing small shape oscillations between parallel soft elastic layers

Abstract

A model is developed for a pulsating and translating gas bubble immersed in liquid in a channel formed by two soft, thin elastic parallel layers having densities equal to that of the surrounding liquid and small, but finite, shear moduli. The bubble is nominally spherical but free to undergo small shape deformations. Shear strain in the elastic layers is estimated in a way which is valid for short, transient excitations of the system. Coupled nonlinear second-order differential equations are obtained for the shape and position of the bubble, and numerical integration of an expression for the liquid velocity at the layer interfaces yields an estimate of the elastic layer displacement. Numerical integration of the dynamical equations reveals behavior consistent with laboratory observations of acoustically excited bubbles in ex vivo vessels reported by Chen et al. [Phys. Rev. Lett. 106, 034301 (2011) and Ultrasound Med. Biol. 37, 2139–2148 (2011)].