Derivation of a Nonlinear Wave Equation for High Speed and High Frequency Pressure Waves in Bubbly Liquids

Abstract

This paper derives a nonlinear wave equation for plane progressive quasi-monochromatic waves in an initially quiescent 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 phase velocity is larger than the speed of sound in a pure liquid; (iv) 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 governing equations for bubbly flows are composed of the conservation equations of mass and momentum in a two-fluid model, equation of bubble dynamics, equations of state for gas and liquid phases, and so on. By using the method of multiple scales with an appropriate choice of set of scaling relations of nondimensional parameters with respect to speed, length, and frequency in terms of nondimensional wave amplitude, the nonlinear Schrödinger (NLS) equation with a dissipation term and some correction terms can be derived from the governing equations, which describes a long-range wave propagation with dissipation and dispersion effects. The decrease of the group velocity in a far field is clarified.