Stefan problems characterized by moving or free boundary often due to change of phase do appear in many important applications of diverse fields. Among these, we review some recent developments in geophysical situations such as the plate techtonics and magma solitons. The various analytical approaches developed for coping with the moving boundary (Stefan) problems are reviewed and outlined. We then introduce the reader to the theory of a new series expansion method based on the Lagrange-Biirmann expansions which can be applied to the Stefan problems. Unlike classical series solutions, the new solutions demonstrate a markedly improved convergence having validity often over the entire physical time domain.

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The methods to use Lagrangian type equation of particle motion for the study of low concentration solid-gas two-phase jet are discussed. For a high concentration solid-gas two-phase jet, the measured averaged velocities and turbulent intensities of air and particles are presented and the interactions between air and particles and between particles and particles are described.

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This paper reviews a generalized lubrication theory for ultra-thin gas film derived through a linearized Boltzmann equation, which is valid for both an arbitrary Knudsen number and an arbitrary accommodation coefficient. Numerical analyses of the equation which reveal differences of lubrication characteristics between the equation and the conventional approximation equation are shown. Experimental verification and application to flying head sliders are also briefly shown.

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The randomization process of fluctuations is examined experimentally for the case of the ribbon-induced transition initially governed by a periodic T-S wave in plane Poiseuille flow with weak residual turbulence. The effects of the residual turbulence become appreciable as the T-S wave undergoes three-dimensional distortions leading to the formation of the high-shear layer. In fact it causes the fundamental and harmonic waves to modulate in both amplitude and phase at low frequencies, in particular at the position of the high-shear layer. This in turn changes the low-frequency spectrum from the original. These modulations manifest themselves as side-band spectral components around each harmonics. The randomization process is characterized by the growth of these side-hand components. At the stage of the high-frequency secondary instability, the side-band components around the higher harmonics are highly amplified and thus contribute to the breakdown of the high-shear layer into small scale eddies.

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The auto-Bäcklund transformation of KdV equation can be constructed by the solution of the Differential Perturbation Equation (abrivated by DPE), which is an extension of Lie's equation. Therefore, it is considered that the auto-Bäcklund transformations of the various integrable equations will be obtained in terms of DPE.
Furthermore, the multi-parameter DPE is introduced. One of its applications is the recursion operator, which is initiated by P. J. Olver. And the multi-parameter DPEs of Burgers equation are derived.
In particular, the concept “multi-parameter DPE” corresponds to the generalization of invariant surfaces which are suggested by S. Kumei.

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Sound emission by interaction of a vortex ring and a circular cylinder is investigated experimentally. In order to make a high-speed vortex ring, we adopted a new facility, using an electromagnetic valve. The reproducibility of the vortex ring made in our apparatus seems to be as good as that made by using the shock tube. A vortex ring moves along a horizontal path and collides against a vertical circular cylinder symmetrically. Acoustic measurements were carried out in a horizontal plane in the acoustic far field region. The sound signal recorded by a microphone there is found to consist of three major signals, which we found to be, in the order of their appearance, (i) a direct pulse signal from the nozzle exit, caused by the sudden flow-flux inflation there, (ii) a pulse of the same origin as the former but via scattering by the cylinder, and (iii) a pulse which is so-called the 'vortex sound', radiated by the vortex-cylinder interaction. We investigated the third 'vortex sound' in detail on the respect to its intensity distribution in the horizontal plane and its multi-pole character. It is found that there exist substantial components of monopole, dipole, and quadrupole, although main component is the dipole. In addition, we separated these three components from this pulse to investigate the time dependence of their intensities.

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