Weakly nonlinear propagation of pressure waves in bubbly liquids with a polydispersity based on two-fluid model equations

Abstract

Herein, the weak, nonlinear propagation of pressure waves in an initially quiescent liquid containing many small spherical gas bubbles is theoretically studied. We focus on the initial, polydispersity features of the bubble radius and bubble number density. Our analysis was not based on any assumptions about explicit polydispersity forms. Using equations based on a two-fluid model and the method of multiple scales with perturbation expansions, the Korteweg–de Vries–Burgers equation for a low-frequency long wave and nonlinear Schrödinger equation for a high-frequency short wave were derived. In both cases, the polydispersity contributes to the advection and dissipation effects of waves, and every coefficient in both equations includes the initial void fraction as one of the most important parameters, owing to the use of the two-fluid model.