Sergei Lukaschuk, Petr Denissenko

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Clustering and Mixing of Floaters by Waves

• Sergei Lukaschuk, Petr Denissenko
• Grisha Falkovich
• The University of Hull, UK
• The Weizmann Institute of Science, Israel

Warwick Turbulent Symposium. December 8, 2005.
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Effect of surface tension
Capillarity breaks Archimedes law
Two bodies of the same weight displace different
amount of water depending on their material
(wetting conditions)

• Hydrophilic particles are lighter
• Hydrophobic particles are heavier than displaced
  fluid

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Small hydrophilic particles climb up, and
hydrophobic particles slide down along inclined
surface. Similar particles attract each other
and form clusters. A repulsion may exist in the
case of non-identical particles Cheerious effect

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Standing wave
Small amplitude wave
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Van Dyke, An Album of Fluid Motion
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Equation for the depth of the submerged part, ?
M p. mass, md mass of displaced fluid, Fc
capillary force, ?v - friction coefficient ( )
Equation of motion for horizontal projection
For the long gravity waves when
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Experimental setup
PW Laser
CW Laser
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Working liquid water surface tension
71.6 mN/m refraction index
1.33 Particles glass hollow spheres average
size 60 ? m density 0.6 g/cm3
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Measurement System

• Cell geometry 9.6 x 58.3 x 10 mm, 50 x 50 x 10
  mm
• Boundary conditions pinned meniscus flat
  surface
• Acceleration measurements Single Axis
  Accelerometer, ADXL150 (Resolution 1 mg / Hz1/2 ,
  Range ? 25 g, 16-bit A-to-D, averaging 10 s,
  Relative error 0.1)
• Temperature control 0.2ºC
• Vibrations Electromagnetic shaker controlled by
  digital waveform generator. Resonant frequency gt
  1 kHz
• Illumination expanded beam
• CW Laser to characterise particles concentration,
  wave configuration and the amplitude
• PIV pulsed (10 nsec) Yag laser for the particle
  motion
• Imaging
• 3 PIV cameras synchronized with shaker
  oscillation

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Measurement methods

• Particle Concentration
• off-axis imaging synchronized with zero-phase of
  the surface wave
• measuring characteristic light intensity, its
  dispersion and moments averaged over area of
  different size
• Wave configuration
• shadowgraph technique
• 2D Fourier transform in space to measure averaged
  k-vector
• Wave amplitude measurement
• refraction angle of the light beam of 0.2 mm
  diam.
• dispersion of the light intensity

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Standing wave Particle concentration and Wave
amplitude are characterized by the dispersion of
the light intensity F100.9 Hz, ?l8 mm, ?s5
mm, A0.983 g
T1
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Wave Amplitude vs Acceleration F 100.9 Hz
Cell 58.3 x 9.6 mm
Ac0.965 ? 0.01
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2D k-spectrum of the parametric waves in a
turbulent mode averaged over 100 measurements
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Distribution in random flow (wave turbulence)
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??lt0 ? singular (fractal) distribution
Sinai-Ruelle-Bowen measure
multi-fractal measure
Balkovsky, Fouxon, Falkovich, Gawedzki, Bec,
Horvai
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Moments of concentrations 2,3,4,5 and 6th versus
the scale of coarse graining. Inset scaling
exponent of the moments of particle number versus
moment number.
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Random particle distribution
n2000 in the AOI, std(n)39
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PDF of the number of particles in a bin 128x128
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PDF of the number of particles in a bin 256 x 256
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Conclusion
Small floaters are inertial ? they
drift and form clusters in a standing wave
wetted particles form clusters in the
nodes unwetted - in the antinodes clustering
time is proportional to A2 they create
multi-fractal distribution in random waves.
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How waves move small particles?

• Stokes drift (1847)
• Kundts tube stiration in a sound waves (King,
  1935)

E the mean energy density,