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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 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

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:

Van Dyke, “An Album of Fluid Motion”

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

Experimental setup PW Laser CW Laser

Working liquid: water surface tension: 71. 6 m. N/m refraction index: 1. 33 Particles: glass hollow spheres average size 60 m density 0. 6 g/cm 3

Measurement System § Cell geometry: 9. 6 x 58. 3 x 10 mm, 50 x 10 mm § Boundary conditions: pinned meniscus = flat surface § Acceleration measurements: Single Axis Accelerometer, ADXL 150 (Resolution 1 mg / Hz 1/2 , 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 k. Hz § 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 ü off-axis imaging synchronized with zero-phase of the surface wave ü measuring characteristic – light intensity, its dispersion and moments averaged over area of different size Wave configuration: ü shadowgraph technique ü 2 D Fourier transform in space to measure averaged kvector Wave amplitude measurement ü refraction angle of the light beam of 0. 2 mm diam. ü 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 T 1

Wave Amplitude vs Acceleration F= 100. 9 Hz Cell: 58. 3 x 9. 6 mm Ac=0. 965 0. 01

2 D k-spectrum of the parametric waves in a turbulent mode averaged over 100 measurements

Distribution in random flow (wave turbulence)

∑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 6 th 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

PDF of the number of particles in a bin 128 x 128

PDF of the number of particles in a bin 256 x 256

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 A 2 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,