Quasi-monochromatic weakly nonlinear waves of high frequency exceeding eigenfrequency of bubble oscillations in compressible liquid containing microbubbles

This study performs the derivation of a nonlinear wave equation for plane progressive quasi-monochromatic waves in a compressible liquid containing many spherical microbubbles that oscillate rapidly due to the pressure wave approaching the bubbles. 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, and (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 appropriate choices of scaling relations of some physical parameters such as wavelength, we can derive the nonlinear Schroedinger (NLS) equation with an attenuation term and som...