Numerical study on formation of an acoustic soliton in bubbly liquids based on weakly nonlinear wave equation

Abstract

Oscillation of microbubbles in bubbly liquids induces a dispersion effect of waves into weakly nonlinear pressure waves, and its propagation process is described by a KdV-Burgers (KdVB) equation for a long wave case. We numerically predict an evolution of waveform by solving the KdVB equation derived by Kanagawa et al. (2010, 2011) via a finite difference method. As a result, (i) an initial Gaussian waveform is distorted due to a nonlinear effect, and a steepening of waveform is observed on about 20 000 period; (ii) the dispersion effect is observed at a steep part of waveform, and its part becomes sharp as the initial void fraction increases; (iii) the initial waveform change into an attenuated soliton on about 100 000 period due to a dissipation effect, and the attenuation of soliton is noteworthy as the initial void fraction decreases; (iv) balance of the nonlinear, dispersion, and dissipation effects thereby forms an attenuated soliton, and its amplitude strongly depends on the value of the initial void fraction. Furthermore, the period of soliton formation depends on the type of initial waveform. Although the nonlinear, dissipation, and dispersion effects were qualitatively balanced in the KdVB equation, we conclude that these effects independently appear in the sound field.