Multiple-scales analysis on high speed and high frequency pressure waves induced by liquid compressibility in bubbly liquids

This paper theoretically examines weakly nonlinear propagation of plane progressive waves in an initially quiescent compressible liquid containing many spherical microbubbles. Waves propagate with a large phase velocity exceeding the speed of sound in a pure water, which is induced by the incorporation of compressibility of the liquid phase. For simplicity, the wave dissipation owing to viscosity in the gas phase and heat conduction in the gas and liquid phases are ignored, and wave dissipation is thereby owing to the liquid viscosity and liquid compressibility. The set of governing equations for bubbly flows is composed of conservation equations of mass and momentum for gas and liquid phases, the equations of motion describing radial oscillations of a representative bubble, and the equation of state for both phases. By using the method of multiple scales and the determination of size of three nondimensional parameters, i.e., the bubble radius versus wavelength, wave frequency versus eigenfrequency of single bubble oscillations, and wave propagation speed versus sound speed in pure liquid in terms of small but finite wave amplitude (i.e., perturbation), we can derive a nonlinear wave equation describing the wave behavior at a far field.This paper theoretically examines weakly nonlinear propagation of plane progressive waves in an initially quiescent compressible liquid containing many spherical microbubbles. Waves propagate with a large phase velocity exceeding the speed of sound in a pure water, which is induced by the incorporation of compressibility of the liquid phase. For simplicity, the wave dissipation owing to viscosity in the gas phase and heat conduction in the gas and liquid phases are ignored, and wave dissipation is thereby owing to the liquid viscosity and liquid compressibility. The set of governing equations for bubbly flows is composed of conservation equations of mass and momentum for gas and liquid phases, the equations of motion describing radial oscillations of a representative bubble, and the equation of state for both phases. By using the method of multiple scales and the determination of size of three nondimensional parameters, i.e., the bubble radius versus wavelength, wave frequency versus eigenfrequency of sin...