Contribution of initial bubble radius distribution to weakly nonlinear waves with a long wavelength in bubbly liquids

Abstract

In this study, the weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid was theoretically investigated. This liquid contains several small uniformly distributed spherical polydisperse gas bubbles. The polydispersity considered here represents various types of initial bubble radii, and the liquid contains multiple bubbles, each with an initial radius. Using the method of multiple scales, we first derived the Korteweg–de Vries–Burgers (KdVB) equation with a correction term as a nonlinear wave equation. This equation describes the long-range wave propagation with weak nonlinearity, low frequency, and long wavelength in the polydisperse bubbly liquid using the basic equations in a two-fluid model. The utilization of the two-fluid model incorporates the dependence of an initial void fraction on each coefficient in the nonlinear, dissipation, and dispersion terms in the KdVB equation. Furthermore, unlike previous studies on waves in polydisperse bubbly liquids, we achieved the formulation without assuming an explicit form of the polydispersity function. Consequently, we discovered the contribution of polydispersity to the various effects of wave propagation, that is, the nonlinear, dissipation, and dispersion effects. In particular, the dispersion effect of the waves was found to be strongly influenced by polydispersity.