Weakly nonlinear analysis of dispersive waves in mixtures of liquid and gas bubbles based on a two-fluid model

Abstract

One-dimensional nonlinear dispersive waves in liquids containing a number of microbubbles are theoretically studied based on two-fluid averaged equations derived by the present authors. The set of equations consists of the conservation laws of mass and momentum for gas and liquid phases, and the equation of motion of the bubble wall. The compressibility of liquid is taken into account, and this leads to the wave attenuation due to bubble oscillations. By using the method of multiple scales, two types of equations for nonlinear wave propagation in long ranges are derived. In a moderately low frequency band, the behavior of weakly nonlinear waves is described by the Korteweg de Vries Burgers equation. On the other hand, in a moderately high frequency band, the nonlinear modulation of quasimonochromatic wave train is described by the nonlinear Schr¨odinger equation with an attenuation term.