Bubble–bubble interaction governs nonlinear acoustic waves in polydisperse bubbly liquids

In the weakly nonlinear regime, pressure wave propagation in bubbly liquids converts either into weak shock waves or into (acoustic) solitons by the balance of nonlinearity, dissipation, and dispersion. To predict whether such waves change to weak shock waves or solitons, it is essential to quantitatively evaluate these effects. The Korteweg–de Vries–Burgers (KdVB) equation provides a useful mathematical model for this purpose and has previously been derived under various assumptions. In this study, we derived the KdVB equation for polydisperse bubbly liquids considering bubble–bubble interactions. Although our previous work derived KdVB equation considering bubble–bubble interactions has been limited to the monodisperse case [Hemmi & Kanagawa, Results Eng., 25, 103752 (2025)], the present work successfully extend the KdVB equation to include polydispersity. Therefore, the result can be applied to systems with multiple initial bubble sizes. We incorporate the effect of bubble–bubble interactions into the Keller equation and derive the KdVB equation as the nonlinear wave equation for polydisperse bubbly liquids using the singular perturbation method. We investigated how bubble–bubble interactions and polydispersity affect nonlinear wave propagation in bubbly liquids. Our analysis, based on two discrete bubble radii and a bubble radius distribution, shows that the dispersion effect becomes markedly large when the difference in bubble size is significant, and that increasing polydispersity leads to higher values of all coefficients in the KdVB equation. Compared with the monodisperse case, the enhancement due to interactions is more pronounced in polydisperse systems. These results demonstrate that interactions and polydispersity synergistically strengthen the nonlinear, dissipation, and dispersion characteristics of pressure waves.