Part IIb. A seminar on fluctuations in sedimentation · 2014. 5. 13. · Part IIb. A seminar on uctuations in sedimentation May 13, 2014. Fluctuations in the velocities of sedimenting

Part IIb. A seminar on fluctuations insedimentation

May 13, 2014

Fluctuations in the velocities of sedimentingparticles

John Hinch

DAMTP, Cambridge

May 13, 2014

In collaboration with Élisabeth Guazzelli & Laurence Bergougnoux

and their students

Guazzelli & Hinch (2011) Ann. Rev. Fluid Mech. 43, 97–116

Fluctuating velocitiesParticles do no fall at a constant speed in a suspension

Trajectories of two spheres at φ = 0.3

Nicolai, Herzhaft, Hinch, Oger & Guazzelli. (1995) Phys. Fluids 7, 12–23.

The divergence paradox

I Theory: depend on size L of box w ′ = VS

√φLa

I Experiments: no such dependence

w′/VS

0

0.5

1

1.5

2

2.5

0 500 1000 1500 2000

20 x 20 x 40 cm3

10 x 10 x 40 cm3

4 x 10 x 40 cm3

4 x 4 x 40 cm3

W

t

t/tS

Nicolai & Guazzelli. (1995) Phys. Fluids 7, 3–5.

Theory of scaling

I Dilute: pair separated by r have w ′ ∼ VS ar ,

so averaging with p ∼ n (const)∫

w ′2p dV diverges like V 2Sφ

L

a

Caflisch & Luke (1985) Phys. Fluids 28, 759-60.

I Explanation Hinch (1988) Disorder and Mixing 153–60

N2 +√N N2 −

√N

...................

.................

........

........

........

.......

.............

.................................. ............ ............. ............... ............... .............

..............................................

.............

...............

................

.................

..................I

w ′

w ′ =

√Nmg

6πµL= VS

√φL

a

‘Poisson’ value

Theory of scaling

I Dilute: pair separated by r have w ′ ∼ VS ar ,so averaging with p ∼ n (const)

∫w ′

2p dV diverges like V 2Sφ

L

a



N2 +√N N2 −

√N

...................

.................

........

........

........

.......

.............

.................................. ............ ............. ............... ............... .............

..............................................

.............

...............

................

.................

..................I

w ′

w ′ =

√Nmg

6πµL= VS

√φL

a

‘Poisson’ value

Theory of scaling


∫w ′


L

a



N2 +√N N2 −

√N

...................

.................

........

........

........

.......

.............

.................................. ............ ............. ............... ............... .............

..............................................

.............

...............

................

.................

..................I

w ′

w ′ =

√Nmg

6πµL

= VS

√φL

a

‘Poisson’ value

Theory of scaling


∫w ′


L

a



N2 +√N N2 −

√N

...................

.................

........

........

........

.......

.............

.................................. ............ ............. ............... ............... .............

..............................................

.............

...............

................

.................

..................I

w ′

w ′ =

√Nmg

6πµL= VS

√φL

a

‘Poisson’ value

Big effect of a little stratification

B lawdziewicz c1995, private communication - ignored.

Luke (2000) Phys. Fluids 12, 1619–21.

I If vertical change in density exceeds statistical fluctuation,then heavy side sinks only to level of neutral buoyancy.

I Blobs smaller than ` unaffected, with ` given by

`∂n`3

∂z=√n`3 so ` = n1/5

(−∂n∂z

)−2/5

Hence

w ′ = Vs

√φ`

a= VSφ

3/5

(−a∂φ

∂z

)−1/5

Tee, Mucha, Cipelletti, Manley & Brenner (2002) PRL 89:054501


B lawdziewicz c1995, private communication - ignored. Luke (2000) Phys. Fluids 12, 1619–21.



`∂n`3

∂z=√n`3 so ` = n1/5

(−∂n∂z

)−2/5

Hence

w ′ = Vs

√φ`

a= VSφ

3/5

(−a∂φ

∂z

)−1/5




I If vertical change in density exceeds statistical fluctuation,

then heavy side sinks only to level of neutral buoyancy.


`∂n`3

∂z=√n`3 so ` = n1/5

(−∂n∂z

)−2/5

Hence

w ′ = Vs

√φ`

a= VSφ

3/5

(−a∂φ

∂z

)−1/5






`∂n`3

∂z=√n`3 so ` = n1/5

(−∂n∂z

)−2/5

Hence

w ′ = Vs

√φ`

a= VSφ

3/5

(−a∂φ

∂z

)−1/5






`∂n`3

∂z=√n`3

so ` = n1/5(−∂n∂z

)−2/5

Hence

w ′ = Vs

√φ`

a= VSφ

3/5

(−a∂φ

∂z

)−1/5






`∂n`3

∂z=√n`3 so ` = n1/5

(−∂n∂z

)−2/5

Hence

w ′ = Vs

√φ`

a= VSφ

3/5

(−a∂φ

∂z

)−1/5






`∂n`3

∂z=√n`3 so ` = n1/5

(−∂n∂z

)−2/5

Hence

w ′ = Vs

√φ`

a

= VSφ3/5

(−a∂φ

∂z

)−1/5






`∂n`3

∂z=√n`3 so ` = n1/5

(−∂n∂z

)−2/5

Hence

w ′ = Vs

√φ`

a= VSφ

3/5

(−a∂φ

∂z

)−1/5


Computer simulations to test effect of stratificationInitially stratified

z

0

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1 1.2 1.4

φ

Concentration profileat different times

∆φ/φ = 0.4, 2500 particles,

average over 40 realisations

w′2n

0

0.005

0.01

0.015

0.02

0 0.1 0.2 0.3 0.4 0.5

t

Velocity fluctuations for∆φ/φ = 0,. . . ,0.4

104 particles, h = 10

Decay to a plateau value w ′∞

Chehata Gómez, Bergougnoux, Guazzelli & Hinch (2009) Phys. Fluids 21: 093304

Computer simulations to test effect of stratificationInitially stratified

Plateau value w ′∞2 plotted against stratification

w ′∞2

Poisson

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

n1/5(−∂n/∂z)−2/5

Different n & δx

I Hence

w ′∞ = 0.94VSφ3/5

(−a∂φ

∂z

)−1/5


Computer simulations to test effect of stratificationInitially uniform - stratified in descending front

Viewed in windows at different heights: top, bottom

w ′2

Poisson

∂φ∂z

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5

t

Velocity fluctuations reduced when front arrives in window

Computer simulations to test effect of stratificationInitially uniform - stratified in descending front

w ′2 in front plotted against stratification

w ′2

Poisson

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2

aLφ

1/5(−a∂φ∂z

)−2/5

Fair agreement only, but recall time delay for initial value to decay

ExperimentsInitially uniform - stratified in descending front

I Four experiments at φ = 0.3%, with different box size anddifferent particle sizes and densities. View in fixed window.

I Open symbols w ′/VS . Filled symbols −a∂φ/∂z (difficult).

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

a ) b )

d )c )

w′VS

w′VS

t/tS t/tS

t/tSt/tS

− adφdz

− adφdz

− adφdz

I II III I II III

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

I II III I II III

-1

0

1

2

3

4

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

w′VS

w′VS

− adφdz

I – Decay of initial state, II – plateau, III – in front




-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

a ) b )

d )c )

w′VS

w′VS

t/tS t/tS

t/tSt/tS

− adφdz

− adφdz

− adφdz

I II III I II III

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

I II III I II III

-1

0

1

2

3

4

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

w′VS

w′VS

− adφdz

I – Decay of initial state,

II – plateau, III – in front




-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

a ) b )

d )c )

w′VS

w′VS

t/tS t/tS

t/tSt/tS

− adφdz

− adφdz

− adφdz

I II III I II III

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

I II III I II III

-1

0

1

2

3

4

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

w′VS

w′VS

− adφdz

I – Decay of initial state, II – plateau,

III – in front




-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

a ) b )

d )c )

w′VS

w′VS

t/tS t/tS

t/tSt/tS

− adφdz

− adφdz

− adφdz

I II III I II III

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

-0.5

0

0.5

1

1.5

2

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

I II III I II III

-1

0

1

2

3

4

-5 10-7

0

5 10-7

1 10-6

1.5 10-6

2 10-6

0 200 400 600 800 1000 1200

w′VS

w′VS

− adφdz

I – Decay of initial state, II – plateau, III – in front


I Velocity fluctuations inhibited by stratification

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1 1.2

w′VPoisson

(βc/β)1/5

Filled symbols on plateau (II),open in front (I).


Experiments

And initial values are the old divergent scaling

w ′0 = VS

√φL

a

√

φ0L/a

v ′

VS

w′

VS

0

1

2

3

4

5

0 1 2 3 4 5

0

0


Diffusing front

I Does front between top of suspension and clear fluid diffuse?

I Self-diffusivity D = w ′` = 2.75Vsaφ4/5 (−a∂φ/∂z)−3/5

I Nonlinear diffusion equation

∂φ

∂t− ∂(Vsφ)

∂z=

∂

∂z

(2.75Vsa

2/5φ4/5(−∂φ∂z

)2/5)

Mucha & Brenner (2003) Phys. Fluids 15: 1305-13

I Numerical value 2.75 of diffusivity from similarity solution . . .

Diffusing front


I Self-diffusivity D = w ′`

= 2.75Vsaφ4/5 (−a∂φ/∂z)−3/5


∂φ

∂t− ∂(Vsφ)

∂z=

∂

∂z

(2.75Vsa

2/5φ4/5(−∂φ∂z

)2/5)



Diffusing front


I Self-diffusivity D = w ′` = 2.75Vsaφ4/5 (−a∂φ/∂z)−3/5


∂φ

∂t− ∂(Vsφ)

∂z=

∂

∂z

(2.75Vsa

2/5φ4/5(−∂φ∂z

)2/5)



Diffusing front

I Similarity thickness of front

δ = 3.07aφ1/7(Vst/a)5/7

δ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3

10000, 10, 103072, 6, 83072, 6, 5

3072, 6, 12750, 6, 5

10368, 6, 12

t5/7(N/h)1/7

Diffusing front

Similarity plot of concentration profile

c

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-2 -1 0 1 2 3 4

t=0.5t=1.0t=1.5t=2.0

thy

[z − z(c = 0.5)]/δ(t)

I Nonlinear diffusion equation predicts concentration profile indiffusing front at top of suspension

Open question: effect of small inertia

The Poisson estimate was for blobs at low Reynolds numbers.

With inertia, estimate:

w ′ = Vsφ1/3Re

−1/3p

Recent preliminary experiments(Bergougnoux 2011)

Open question: effect of small inertia

The Poisson estimate was for blobs at low Reynolds numbers.With inertia, estimate:

w ′ = Vsφ1/3Re

−1/3p

Recent preliminary experiments(Bergougnoux 2011)

Documents

Part IIb. A seminar on fluctuations in sedimentation · 2014. 5. 13. · Part IIb. A seminar on uctuations in sedimentation May 13, 2014. Fluctuations in the velocities of sedimenting