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## Clustering and Mixing of Floaters by Waves

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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.**Effect of surface tension**Capillarity breaks Archimedes’ law • Hydrophilic particles are lighter • Hydrophobic particles are heavier than displaced fluid Two bodies of the same weight displace different amount of water depending on their material (wetting conditions)**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**Standing wave**Small amplitude wave:**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**Experimentalsetup**PW Laser CW Laser**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**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 > 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**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**Standing wave : Particle concentration and Wave amplitude**are characterized by the dispersion of the light intensity F=100.9 Hz, l=8 mm, s=5 mm, A=0.983 g T1**Wave Amplitude vs AccelerationF= 100.9 Hz Cell: 58.3 x 9.6**mm Ac=0.965 0.01**2D k-spectrum of the parametric waves in a turbulent mode**averaged over 100 measurements**∑λ<0 → singular (fractal) distribution –**Sinai-Ruelle-Bowen measure multi-fractal measure Balkovsky, Fouxon, Falkovich, Gawedzki, Bec, Horvai**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.**Random particle distribution**n=2000 in the AOI, std(n)=39**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.**How waves move small particles?**• Stokes drift (1847): • Kundt’s tube stiration in a sound waves (King, 1935): E – the mean energy density,