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