Lagrangian description of the interaction of a bubble with a particle

Abstract

The Lagrangian formalism used previously to describe pulsations and translations of interacting bubbles [Ilinskii et al., Proc. 17th ISNA, Penn State, edited by Atchley and Sparrow (AIP, New York, 2006)] has been extended to model interactions of bubbles and elastic particles. The bubbles and particles are assumed to be spherical. Of eventual interest is interaction of a cavitation bubble cluster with fragments of kidney stones produced during shock-wave lithotripsy. Here, we discuss only the interaction of a single gas bubble with a single rigid particle. Numerical simulations illustrate change in separation distance in the presence of an applied sound field. Using the averaging procedure employed by Harkin et al. [J. Fluid Mech. 445, 377 (2001)] for two bubbles, we develop approximate equations for the translation under the conditions of weak excitation and weak interaction. Radiation force between the bubble and particle is fifth order in terms of the separation distance, as found previously by Coakley and Nyborg [in Ultrasound: Its Applications in Medicine and Biology, edited by F. Fry (Elsevier, Amsterdam, 1978), pp. 77–159]. Numerical and analytical results are compared with results obtained by others using different approaches. [T.A.H was supported by an NSF Graduate Research Fellowship.]