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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, i.e., the nonlinear, dissipation, and dispersion effects. In particular, the dispersion effect of the waves was found to be strongly influenced by polydispersity.

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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.