Sedimentation of particlesHow can such a simple problem be so difficult?

Elisabeth Guazzelli

elisabeth.guazzelli@polytech.univ-mrs.fr

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