Delay differential equations for single-bubble dynamics in a compressible liquid

Abstract

Most common methods to include the effects of liquid compressibility in models for single-bubble dynamics rely on series expansions to some order in the inverse of the sound speed in the liquid. It has been shown that the Keller-Miksis model for single-bubble dynamics can be obtained from a series expansion of a delay differential equation related to the Rayleigh-Plesset equation for a single bubble. The iterative approach used to obtain the series expansion of the delay-based model becomes unworkable for more complicated models of bubble dynamics. Therefore, to provide an alternative, simpler method to model the effects of liquid compressibility, the delay differential equation model proposed by Ilinskii and Zabolotskaya [J. Acoust. Soc. Am. 92, 2837 (1992)] is analyzed directly. The results of the delay-based formulations are compared to those produced by models based on common series expansions. Alternative formulations of the delay differential equation are also considered and compared.