Derivation of an amplitude equation for pressure wave propagation in polydisperse bubbly liquids

Abstract

A weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent water uniformly containing small gas bubbles is theoretically investigated. In the present study, the bubbles do not coalesce, break up, appear, and disappear. The bubbles are spherical, and these oscillations are spherically symmetric. In addition, the viscosity of gas inside the bubbles and the thermal conductivities of the both phases are neglected. Although abovementioned assumptions were used in our previous studies [e.g., Kanagawa et al., J. Fluid Sci. Technol.(2010); Kanagawa, J. Acoust. Soc. Am. (2015)] and classical studies [e.g., van Wijngaarden, J. Fluid Mech. (1968)], we here incorporate polydispersity of bubbly liquids. From the method of multiple scales, an amplitude (or a nonlinear wave) equation describing a long-range propagation of waves in bubbly liquids is derived from the basic equations in a two-fluid model. By comparing the present result with the previous results assuming monodispersity, we qualitatively and quantitatively discuss an effect of polydispersity on the wave propagation process in bubbly liquids.