Unified Theory Based on Parameter Scaling for Derivation of Nonlinear Wave Equations in Bubbly Liquids

Abstract

We propose a systematic derivation method of the Korteweg-de Vries-Burgers (KdVB) equation and nonlinear Schrödinger (NLS) equation for nonlinear waves in bubbly liquids on the basis of appropriate choices of scaling relations of physical parameters. The basic equations are composed of a set of conservation equations for mass and momentum and the equation of bubble dynamics in a two-fluid model. The scaling of parameters is related to the wavelength, frequency, propagation speed, and amplitude of waves concerned. With the help of the method of multiple scales, appropriate choices of the parameter scaling allow us to derive various nonlinear wave equations systematically from a set of basic equations. The result shows that the one-dimensional nonlinear propagation of a long wave with a low frequency is described by the KdVB equation, and that of an envelope of a carrier wave with a high frequency by the NLS equation. Thus, we establish a unified theory of derivation of nonlinear wave equations in bubbly liquids.

Two types of nonlinear wave equations for diffractive beams in bubbly liquids with nonuniform bubble number density

“…1, the linear dispersion relation for waves in bubbly liquids neglecting the dissipation and liquid compressibility can be depicted conceptually (van Wijngaarden, 1968(van Wijngaarden, , 1972. Then, Kanagawa et al (2010) clarified that the NLS equation including the dissipation effect for envelopes of high-frequency short-wavelength carrier wave and the KdVB equation for waves of low-frequency long-wavelength ( Fig. 1) can be derived by applying the following scaling relations:…”

“…Electronic mail: kanagawa@kz.tsukuba.ac.jp are based on the basic equations for gas-liquid two-phase flows, and which incorporate the diffraction effect as quasiplane waves and the nonuniformity of the bubble distribution, will progress the predictions for various problems on acoustics in bubbly liquids. Kanagawa et al (2010) (or Yano et al, 2013 proposed a unified method to derive nonlinear wave equations for plane wave propagation in uniform bubbly liquids. This method is based on a set of scaling relations among the physical parameters that are appropriate to a specific wave phenomenon, and parameter scaling is the measurement of the nondimensional magnitudes of physical quantities in terms of a typical nondimensional amplitude of the wave under consideration.…”

“…The real numbers a, b, and c in the right-hand side of Eq. (1) should be chosen appropriate to a wave concerned, for instance, Kanagawa et al (2010) chosen so as to derive the NLS and KdVB equations, as follows:…”

“…(1) Kanagawa et al (2010) were restricted to plane waves, i.e., one-dimensional waves in which the averaged physical quantities are functions of the time and spatial coordinate in the direction of wave propagation. Furthermore, the initial void fraction was assumed to be constant, which implies that initial number density of the bubbles in the liquid is spatially uniform.…”

“…The spatial distribution of the number density of the bubbles, initially in a quiescent state, is assumed to be a slowly varying function of the spatial coordinates; the amplitude of variation is assumed to be small compared to the mean number density. A previous derivation method of nonlinear wave equations for plane progressive waves in uniform bubbly liquids [Kanagawa, Yano, Watanabe, and Fujikawa (2010). J. Fluid Sci.…”