Ryuji Takaki

Vibration of a Flattened Drop. I. Observation

Vibration of a single drop of liquid nitrogen, oxygen or argon was observed by putting it on a horizontal plate with the room temperature. The drop had a shape of flattened disk and floated on the plate owing to rapid evaporation. It showed a characteristic mode of vibration, namely a standing wave appeared along the periphery and the plane view showed a polygonal shape. As the radius of drop reduced through evaporation, sudden transitions to modes with smaller number of vertices occurred. The measured vibrational frequency was proportional to (wave number) 3/2 , and depended in a simple way on the surface tension and the density of the liquid.

Thermal convection of a fluid with temperature-dependent viscosity

Numerical analysis has been made for two-dimensional Benard convection of a fluid with temperature-dependent viscosity of exponential type based on the Boussinesque approximation. The stress-free surface condition and periodic structure in the horizontal direction were assumed, where the horizontal length of periodicity was chosen to be twice of the depth. Parameters characterizing the flow are the Rayleigh number Ra0, the Prandtl number Pr0, and the log viscosity ratio c, i.e., the logarithm of the ratio of viscosities at the upper and the lower boundaries. Solutions of the stationary states were obtained for Ra0 = 3000 and 1600. For both of the values Ra0 = 3000 and 1600, two stationary solutions were found for c to 8 and c to 10, respectively, namely the system has a multi-valued nature.

Vibration of a Flattened Drop.II.Normal Mode Analysis

A hydrodynamical model is developed to analyse vibration of a liquid drop on a horizontal bed. The drop has a flattened shape by the gravity and the surface tension. The fluid is assumed inviscid and to move nearly horizontally, so that the shallow water theory is applied, and a set of governing equations for the vibration is derived. It is applied to a small-amplitude oscillation and a normal mode solution is obtained, where the drop shows a standing wave along its periphery. The predicted vibration frequency agrees with the experiment by the present authors.

Statistical theory of vortex merger in the two‐dimensional mixing layer

The phenomenon of vortex merging in a two‐dimensional mixing layer is treated statistically by introducing the probability density of spacings between neighboring vortices. The rate of merging is the other statistical variable and is a function of the spacing. The mixing layer is assumed to be statistically uniform along the streamwise coordinate and not to change with time. The following further assumptions were made: no correlation between adjacent spacings, similarity of the probability density, etc., to all times, and that the rate of merging for a single pair of vortices is inversely proportional to the square of their distance. The probability density was obtained by solving the governing equation. The result agrees well with that data measured by Brown and Roshko.

Self-Induced Vibration of a Water Drop Placed on an Oscillating Plate

Vibration of a water drop placed on a vertically oscillating plate was observed, and the amplitudes and frequencies of the plate were measured with which the drop showed large amplitude vibrations. A special cloth was used so that the drop did not wet the plate. The results are compared with a simple theoretical prediction based on the Mathieu equation. The experimental data are shown to lie within the unstable growing region derived from the theory.

Numerical Analysis of Distortion of a Vortex Filament

Motion of a vortex filament with a narrow core in an inviscid and incompressible fluid otherwise at rest is investigated numerically. Two types of initial conditions, a sinusoidal form with a single Fourier component and that of a superposition of the first and the third components, are assumed. In both cases, interaction between different components occurs owing to the nonlinear property of the governing equation, but the filament does not show random behavior and after a certain time it recurs nearly to the initial state. Analytical treatment is made and the numerical results are confirmed. Next, the case where an external flow field with an axisymmetric potential flow exists is examined, and an onset of randomness is observed. This is considered as due to a computational error amplified through the interaction between the filament and the external flow.

Non-linear Stability of Liquid Flow down an Inclined Plane

The non-linear stability of the flow of a liquid layer down an inclined plane is studied, taking into account the surface tension and the Reynolds stress due to the first periodic disturbance. The wave number and the wave amplitude of the new laminar flow are obtained as function of the Reynolds number.

Optimal Time Step Control for the Numerical Solution of Ordinary Differential Equations

In solving differential equations using a finite stepsize

Mechanism of self-induced vibration of a liquid drop based on the surface tension fluctuation

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Linear stability of a two-dimensional uniform fluidized bed

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