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Sedimentation of particlesHow can such a simple problem be so difficult?
Elisabeth Guazzelli
IUSTI – CNRS – Polytech’Marseille
Sedimentation of particles – 2005 – p.1/27
Sedimentation of particles
Re = aU/ν � 1 ⇒ Stokes’ flow
Pe = aU/D � 1 ⇒ Only hydrodynamics
Solid and monodisperse particles for simplicity!
Sedimentation of particles – 2005 – p.2/27
Sedimentation of particles
Sedimentation of a suspension of spheres
Sedimentation of a suspension of fibers
Sedimentation of a cloud of particles
Sedimentation of particles – 2005 – p.3/27
Sedimentation of a single sphere
PSfrag replacementsg
−6πµaUS
4
3πa3(ρp − ρf )g
Stokes’ velocityUS = 2(ρp − ρf )a2g/9µ
Pozrikidis 1997
Long-range interactionsu ∼ O(aUS
r )
Stokes 1851Sedimentation of particles – 2005 – p.4/27
Uniformly dispersed spheresVelocity of a pair of spheres at a separation r:
PSfrag replacementsUS + ∆U(r)
r
PSfrag replacementsUS + ∆U(r)
r Averaging over all possible separations occurring withprobability p(r):
〈U〉 = US +
∫
∆U(r) p(r) dr3 diverges!
O(aUr )O(1)O(r2) as r → ∞
Multibody long-range hydrodynamic interactions
Sedimentation of particles – 2005 – p.5/27
Sedimentation of spheres in a vessel
Sedimentation of particles – 2005 – p.6/27
Mean velocity
0
0.2
0.4
0.6
0.8
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Ham and Homsy (650)Ham and Homsy (470)Nicolai et al. (788)Richardson-ZakiBatchelor
Hindered settling
Concentration
Hindered settling:〈U〉 = USf(φ)Richardson-Zaki 1954:f(φ) = (1 − φ)5
Main effect = Back-flow
Batchelor 1972:f(φ) = 1 − 6.55φ + O(φ2)assuming uniformlydispersed spheres
Results depend on mi-crostructure in turn deter-mined by hydrodynamics
Sedimentation of particles – 2005 – p.7/27
Velocity fluctuations
Large anisotropic fluctuations
Sphere-tracking in an index-matched suspensionHam & Homsy 1988, Nicolai et al. 1995
Sedimentation of particles – 2005 – p.8/27
Divergence of velocity fluctuations?
PSfrag replacements
N/2 +√
N N/2 −
√
Nl
Randomly distributed particles
Box of size aφ−1/3 < l < L
Statistical fluctuations√
N →∆U‖ ∼
√N 4
3πa3(ρs−ρ)g6πµl ∼ US
√
φ la
Large-scale fluctuations aredominant∆U‖ ∼ US
√
φLa diverges!
Caflisch & Luke 1985, Hinch 1988
BUT no such divergence seen in experimentsNicolai & Guazzelli 1995, Segrè et al. 1997, Guazzelli 2001
Sedimentation of particles – 2005 – p.9/27
More theories . . .Koch & Shaqfeh 1991: a non-random microstructure
Tong & Ackerson 1998: turbulent convection analogy
Levine et al. 1998: stochastic model
da Cunha 1995, Ladd 2002: impenetrable bottom
Brenner 1999: wall effect
Luke 2000: stratification → fluctuation decay
Tee et al. 2002, Mucha et al. 2003-04: diffusivespreading of the front → stratification → fluctuationdecay
Nguyen & Ladd 2005: polydispersity → stratification
Hinch 1985, Asmolov 2004, Luke 2005: bottom and top= sink of large-scale disturbances
Sedimentation of particles – 2005 – p.10/27
Relaxation of large-scale fluctuations
0 20 40 60 80 1000
50
100
150
200
250
300
350
400
t/ts = 0.0
interparticule distance (aφ-1/3)
inte
rpar
ticul
e di
stan
ce (a
φ-1/
3 )
0 0.5 1
φ(z)/φ0
0 20 40 60 80 1000
50
100
150
200
250
300
350
400
t/ts = 83.8
0 0.5 1 0 20 40 60 80 100
0
50
100
150
200
250
300
350
400
t/ts = 167.5
0 0.5 1 0 20 40 60 80 1000
50
100
150
200
250
300
350
400
t/ts = 251.3
0 0.5 1
Initially, the large-scale fluctuations dominate the dynamics.But, they are transient as the heavy parts settle to the
bottom and light parts raise to the top.Sedimentation of particles – 2005 – p.11/27
Left with smaller-scale fluctuations
0 20 40 60 80 1000
20
40
60
80
100
120
140
t/ts = 418.8
interparticule distance (a φ−1/3)
inte
rpar
ticul
e di
stan
ce (a
φ−1
/3)
0 20 40 60 80 1000
20
40
60
80
100
120
140
t/ts = 628.3
interparticule distance (a φ−1/3)
inte
rpar
ticul
e di
stan
ce (a
φ−1
/3)
0 20 40 60 80 1000
20
40
60
80
100
120
140
t/ts = 837.7
interparticule distance (a φ−1/3)
inte
rpar
ticul
e di
stan
ce (a
φ−1
/3)
Then, smaller-scale fluctuations are dominant until thearrival of the upper sedimentation front.
Why fluctuation length-scale ∼ 20aφ−1/3?Chehata, Bergougnoux, Guazzelli, & Hinch 2005
Sedimentation of particles – 2005 – p.12/27
Sedimentation of particles
Sedimentation of a suspension of spheres
Sedimentation of a suspension of fibers
Sedimentation of a cloud of particles
Sedimentation of particles – 2005 – p.13/27
Sedimentation of a single fiber
PSfrag replacements Drift velocity
Coupling between orientation and velocity
Sedimentation of particles – 2005 – p.14/27
Sedimentation of fibers in a vessel
Sedimentation of particles – 2005 – p.15/27
Mean Velocity and orientation
Enhanced sedimentation and vertical orientation
Fiber-tracking in an index-matched suspensionHerzhaft & Guazzelli 1999
Sedimentation of particles – 2005 – p.16/27
Packet instability → Streamers
Fluorescing fibers within a laser sheet
Metzger, Guazzelli, & Butler 2005
Sedimentation of particles – 2005 – p.17/27
Large-scale streamers
t=0 t=33 t=99 t=132
-2Vs
3Vs
Vertical velocity versus time from PIV measurements
Metzger, Guazzelli, & Butler 2005
Sedimentation of particles – 2005 – p.18/27
Modeling the instability
Koch & Shaqfeh 1989, Mackaplow & Shaqfeh 1998, Butler& Shaqfeh 2002, Saintillan, Darve, & Shaqfeh 2005
Sedimentation of particles – 2005 – p.19/27
Simulations versus Experiments
Steady state? Wave-length selection?
Saintillan, Shaqfeh, Darve,Metzger, Guazzelli, & Butler 2005
see more on Video Entry #17 (Gallery of Fluid Motion)Sedimentation of particles – 2005 – p.20/27
Sedimentation of particles
Sedimentation of a suspension of spheres
Sedimentation of a suspension of fibers
Sedimentation of a cloud of particles
Sedimentation of particles – 2005 – p.21/27
Spherical cloud of spheres
PSfrag replacements Rg
−2πµ 2+3λλ+1
R UCloud
N 4
3πa3(ρp − ρ)g
Drag force (Hadamard,Rybczynski 1911):
Fh = −2πµ2 + 3λ
λ + 1RUCloud
with λ = µs/µ
Settling velocity:
UCloud =N 4
3πa3(ρp − ρ)g
2πµ2+3λλ+1 R
Suspension mixture = effective fluid of viscosity µs
Sedimentation of particles – 2005 – p.22/27
Stability of the cloud?
It is important to note that the drop is found to be stablewithout any surface tension to maintain the sphericalshape. Feuillebois 1984.
A sperical blob shape is especially well suited to a studyof random particle migration . . . because it maintainsessentially constant form. Nitsche & Batchelor 1997.
A single spherical drop does not deform substantially.Machu et al. 2000.
At creeping flow conditions the suspension drop retainsa compact, roughly spherical shape while settling.Bosse et al. Gallery of Fluid Motion 2005.
Sedimentation of particles – 2005 – p.23/27
But the cloud is unstable!
PSfrag replacementsSpherical cloud
TorusBreak-up
and so on
PSfrag replacements
Spherical cloud
Torus
Break-up
and so on
PSfrag replacementsSpherical cloud
TorusBreak-up
and so on
Sedimentation of particles – 2005 – p.24/27
Evolution of the cloud
Cloud composed of 3000 point-particles
Successive instabilities? Break-up?
Metzger, Ekiel-Jezewska, & Guazzelli 2005see more on talk FK.00007
Sedimentation of particles – 2005 – p.25/27
Conclusions
Long-range nature of the multi-body hydrodynamicinteractionsCoupling between hydrodynamics and suspensionmicrostructure→ Collective dynamics: swirls, streamers, instabilities
More open problemsLarger concentrationsBidisperse or polydisperse particlesAnisotropic particles (platelets)Deformable particles: Saintillan et al. 2005Non-Newtonian fluids: Mora, Talini, & Allain 2005Inertia
Sedimentation of particles – 2005 – p.26/27
Collaborations
B. Herzhaft, H. Nicolai, Y. Peysson (ESPCI Paris) andD. Chehata, B. Metzger (IUSTI Marseille)
L. Bergougnoux (IUSTI Marseille)
E. J. Hinch (University of Cambridge)
M. L. Ekiel-Jezewska (IPPT-PAN Warsaw)
J. E. Butler (University of Florida)
E. Darve, M. B. Mackaplow, D. Saintillan, andE. S. G. Shaqfeh (Stanford University)
G. M. Homsy (University of California Santa Barbara)
Sedimentation of particles – 2005 – p.27/27