Nonlinear Analysis on Fast Propagation of Quasi-Monochromatic Waves in Compressible Bubbly Liquids

Abstract

This study performs the derivation of a nonlinear wave equation for plane progressive quasi-monochromatic waves in a compressible liquid containing many spherical microbubbles. Main assumptions are as follows: (i) the wave frequency is larger than an eigenfrequency of single bubble oscillations. (ii) The compressibility of the liquid phase is incorporated. (iii) The effect of viscosity in the gas phase, heat conduction in the gas and liquid phases, phase change across the bubble wall, and thermal conductivities of the gas and liquid, are neglected. The basic equations for bubbly flows are composed of a set of conservation equations of mass and momentum in a two-fluid model, the equation of bubble dynamics, and so on. From the method of multiple scales with an appropriate choice of scaling relations of some physical parameters, i.e., wavelength, wave frequency, propagation speed, and amplitude of waves concerned, we can derive the nonlinear Schrödinger (NLS) equation with an attenuation term and some correction terms, which describes the long-range wave propagation, where the phase velocity is larger than the speed of sound in a pure liquid. We clarified a decrease of the group velocity in a far field.