Multiple-Scales Theory on High-Frequency and High-Speed Ultrasound in Compressible Liquids Containing Microbubbles

Abstract

This paper theoretically examines an weakly nonlinear propagation of plane progressive waves in an initially quiescent compressible liquid uniformly containing many spherical microbubbles. We focus on a wave propagating with a large phase velocity exceeding the speed of sound in a pure water (i.e., 1,500 m/s), which is accomplished by the consideration of compressibility of the liquid phase. For simplicity, the attenuation of wave owing to the 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 compressibility. The set of governing equations for bubbly flows is composed of the conservation laws of mass and momentum for gas and liquid phases, equation of motion for radial oscillations of representative bubble, and the equation of state for both phases. By using the method of multiple scales and an appropriate 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 speed of sound in a pure liquid in terms of small but finite wave amplitude (i.e., perturbation), lead to derive a nonlinear wave equation describing the wave behavior at a far field.