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TECHNICAL PAPERS

# The Effects of Vibrations on Particle Motion Near a Wall in a Semi-Infinite Fluid Cell

[+] Author and Article Information
Samer Hassan

Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON M5S 3E5, Canada

Masahiro Kawaji

Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON M5S 3E5, Canadakawaji@ecf.utoronto.ca

Tatyana P. Lyubimova, Dmitry V. Lyubimov

Theoretical Physics Department, Russian Academy of Sciences, 15, Bukirev str., Perm 614600, Russia

J. Appl. Mech 73(4), 610-621 (Nov 14, 2005) (12 pages) doi:10.1115/1.2165229 History: Received October 15, 2004; Revised November 14, 2005

## Abstract

The effects of small vibrations on a particle-fluid system relevant to material processing such as crystal growth in space have been investigated experimentally and theoretically. An inviscid model for a spherical particle of radius, $R0$, suspended by a thin wire and moving normal to a cell wall in a semi-infinite liquid-filled cell subjected to external horizontal vibrations, was developed to predict the vibration-induced particle motion under normal gravity. The wall effects were studied by varying the distance between the equilibrium position of the particle and the nearest cell wall, $H$. The method of images was used to derive the equation of motion for the particle oscillating in an inviscid fluid normal to the nearest cell wall. The particle amplitude in a semi-infinite cell increased linearly with the cell vibration amplitude as expected from the results for an infinite cell, however, the particle amplitude also changed with the distance between the equilibrium position of the particle and the nearest wall. The particle amplitude was also found to increase or decrease depending on whether the cell vibration frequency was below or above the resonance frequency, respectively. The theoretical predictions of the particle amplitudes in the semi-infinite cell agreed well with the experimental data, where the effect of the wall proximity on the particle amplitude was found to be significant for $(H∕R0<2)$ especially near the resonance frequency. Experiments performed at high frequencies well above the resonance frequency showed that the particle amplitude reaches an asymptotic value independent of the wire length.

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## Figures

Figure 1

A semi-infinite cell

Figure 2

Experimental setup.

Figure 3

Semi-infinite cell model

Figure 4

Variation of the ratio of the particle amplitudes predicted by Eqs. 51,52 with the particle-wall distance to particle radius ratio, H∕R0, for (a) below-resonance frequencies and (b) above-resonance frequencies (steel particle in water, wire length=76mm)

Figure 5

Variation of the ratio of the particle amplitudes predicted by Eqs. 51,52 with the particle-wall distance to particle radius ratio, H∕R0, for (a) below-resonance frequencies and (b) above-resonance frequencies (steel particle in water, wire length=76mm)

Figure 6

Variation of dimensionless particle amplitudes with particle-wall distance to particle radius ratio predicted by Eq. 51 for (a) below-resonance frequencies and (b) above-resonance frequencies (steel particle in water, wire length=76mm)

Figure 7

Variation of the ratio of the particle amplitude for a semi-infinite cell (Eq. 51) to that for an in infinite cell with the particle-wall distance to particle radius ratio, H∕R0, for (a) below-resonance frequencies and (b) above-resonance frequencies (cell amplitude=16 and 30μm, respectively)

Figure 8

Variations of the dimensionless resonance frequencies for different particles in a semi—infinite cell predicted by Eqs. 54,55 with H∕R0 (wire length=76mm, (—) Eq. 54, (---) Eq. 55)

Figure 9

Variation of the dimensionless resonance frequency predicted by Eq. 54 with the particle-wall distance to particle radius ratio, H∕R0 (wire length=76mm)

Figure 10

Variation of particle amplitudes with particle-wall distance to particle radius ratio predicted by Eqs. 51,52 (steel particle in water, f=1.0Hz, cell amplitude=1.0mm, wire length=76mm)

Figure 11

Variation of particle amplitudes with particle-wall distance to particle radius ratio predicted by Eqs. 51,52 (steel particle in water, f=1.25Hz, cell amplitude=0.50mm, wire length=76mm)

Figure 12

Variation of particle amplitudes with particle-wall distance to particle radius ratio predicted by Eqs. 51,52 (steel particle in water, f=2.0Hz, cell amplitude=0.25mm, wire length=76mm)

Figure 13

Variation of particle amplitudes with particle-wall distance to particle radius ratio predicted by Eqs. 51,52 (steel particle in water, f=3.0Hz, cell amplitude=0.50mm, wire length=76mm)

Figure 14

Theoretical and experimental variations of particle amplitude with cell vibration frequency (steel particle in water, wire length=70mm, widest cell tested, cell width=50.8mm)

Figure 15

Comparison of measured particle amplitudes with the predictions of Eqs. 51,52 for different vibration frequencies (H∕R0=1.18, cell amplitude=0.47mm, wire length=76mm)

Figure 16

Particle amplitude variation with wire length for different cell vibration frequencies (steel particle in water, cell amplitude=1mm, H∕R0=1.18)

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