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.

Iron oxide thin films were prepared by chemical vapor deposition. The source material was iron (III) acetylacetonate. The Fe2O3 films were produced at a substrate temperature above 200°C. The films deposited at a substrate temperature above 300°C were polycrystalline β‐Fe2O3 . Reduction and oxidation of the amorphous films in a 0.3 M LiClO4 propylene carbonate solution caused desirable changes in optical absorption. Coulometry indicated that the coloration efficiency was 6.0 to 6.5 cm2 C−1 .

In our previous paper (Kanagawa et al., J. Fluid Sci. Tech., 5, 2010), we have proposed a systematic method for derivation of various types of nonlinear wave equations for plane waves in bubbly liquids. The method makes use of an asymptotic expansion with multiple scales in terms of a small wave amplitude as an expansion parameter and a set of scaling relations of physical parameters, based on basic equations of two-fluid model of bubbly flows. In this paper, we extend the method so as to handle a weakly diffracted ultrasound beam in a quiescent liquid containing a number of spherical gas bubbles distributed with a weak nonuniformity. Because of the high expandability of the original method, the extension can be accomplished by adding a scaling relation of the diameter of the beam to the original set of scaling relations. As a result, we derive a generalized Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation [or a generalized Kadomtsev-Petviashvili (KP) equation] for a long wave and low frequency case.

Jet breakup is an important behavior at a core disruptive accident for a sodium-cooled fast reactor. The lattice Boltzmann (LB) method is adopted to simulate the jet breakup behavior. The Multiple-Relaxation Time (MRT) scheme is introduced into the existing three-dimensional 19-velocity (D3Q19) LB model for immiscible two-phase flow to enhance the numerical stability for low kinematic viscosity. The simulation results show that the present LB model using MRT enables to simulate the jet breakup behavior, where the kinematic viscosity is of the order of 10 -3 . The velocity field and interfacial shape are compared with the experimental result using PIV and Laser-Induced Fluorescence (LIF). The interfacial instability and fragmentation behavior of the jet can be also simulated. Comparison of the LB simulation with experimental data shows that the time series of jet leading edge can be simulated within an error of around 10%.