Rome 2015-03-24 Clustering of chiral particles in flows with broken parity invariance K. Gustavsson1), L. Biferale1) 1) Department of Physics, University of Tor Vergata, Italy Motion of an ‘isotropic helicoid’ Equations for velocity helicoid: and angular velocity for small isotropic Happel & Brenner, Low Reynolds number hydrodynamics (1963) Stokes’ law translation – rotation coupling (scalar) Fluid velocity Half fluid vorticity Particle relaxation time Particle ‘size’ (defined by mass and moment of inertia ) Helicoidality Equations break spatial reflection symmetry ( and pseudovectors) 1 Dimensionless parameters Stokes number with Size and Helicoidality smallest time- and length scales of flow. Constraint on : Using , The equations of motion becomes with Solution blows up unless and constrained by particle density higher than that of the fluid and geometrical size must be smaller than . 2 Example of an isotropic helicoid Recipe from Lord Kelvin: “An isotropic helicoid can be made by attaching projecting vanes to the surface of a globe in proper positions; for instance cutting at 45º each, at the middles of the twelve quadrants of any three great circles dividing the globe into eight quadrantal triangles.“ Kelvin, Phil. Mag. 42 (1871) 3 Example of an isotropic helicoid Recipe from Lord Kelvin (1884) Start with a sphere 4 Example of an isotropic helicoid Recipe from Lord Kelvin (1884) Start with a sphere Draw 3 great circles 5 Example of an isotropic helicoid Recipe from Lord Kelvin (1884) Start with a sphere Draw 3 great circles Identify 12 vane positions at midpoints of quarter-arcs 6 Example of an isotropic helicoid Recipe from Lord Kelvin (1884) Start with a sphere Draw 3 great circles Identify 12 vane positions at midpoints of quarter-arcs Put a vane on each vane position (45º to arc line) 7 Chirality In a constant flow , the isotropic helicoid starts spinning around the flow direction with angular velocity . The spinning direction depends on the chirality of the vanes. Left handed Right handed 8 Clustering at small Expand compressibility of particle-velocity field Centrifuge effect with modified amplitude to first order in Term due to parity breaking of system Maxey, J. Fluid Mech. 174 (1987) Reflection-invariant systems have Isotropic helicoids violate that relation Same for parity-breaking flows Helicity parameter Right-handed structures ( Left-handed structures ( ) more common ) more common 9 Clustering at small in random flow Small- theory Gustavsson & Mehlig EPL 96 (2011) Small- Spherical particle ( Right-handed particle ( Right-handed particle ( Right-handed particle ( limit ) in neutral flow ) in left-handed flow ) in neutral flow ) in left-handed flow 10 Clustering at small in random flow 11

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