1. Controlling the rotation of particles in a suspension
Philippe Peyla
LSP
Université de Grenoble
Levan Jibuti (LSP)
Salima Rafaï (LSP - CNRS)
Ashok Sangani (Syracuse Univ.) Andréas Acrivos (City College of NY)
Eppur si muove !
Nancy, 2010

2. Why controlling rotation of particles in a suspension?
Audi R8
Smart fluids:
Industry:
Clutches, dampers, brakes
Before applying
a magnetic field
… After
Nature:
chloroplast .
Consequence on rheology,
Flow focusing, …

3. Controlling the rotation of particles in a suspension
Rotation in presence of an external torque (Smart fluids)

4. Rotation of a particle in a shearV0 H
wz=-g/2
g=V0/H

5. Rotation of a particle in a shearV0 ( ) ( ) ( )
V0X g x =
Voy y H ( ) ( ) ( ) ( ) x x
g/2
g/2 = + y
g/2 y y
-g/2 y y
wz=-g/2 x
g=V0/H x x
Rotation wz=-g/2
Extension/compression

6. Rotation of a particle in a shearV0
2nd Faxen Law:
Torque exerted by the fluid on the particle:
T=-8pa3 h (1/2 rot V0-w)
Torque free particule:
T=0, donc w=1/2 rot V0=- g/2 ez H a y y
wz=-g/2 x
g=V0/H x
Rotation wz=-g/2

7. Control of the particle rotation by an external field Rheology of smart fluids
Torque-free particle
External torque
External torque
wz=-g/2
wz>-g/2
wz<-g/2
heff=sxy/g
heff>h0eff
heff<h0eff

8. Control of the particle rotation by an external field Rheology of smart fluids
Dilute regime g
2nd Faxen Law : y
T=-8phR3 [1/2 rot V0-w] x z T
External torque

9. Control of the particle rotation by an external field Rheology of smart fluids
Dilute regime g
2nd Faxen Law : y
T=-8phR3 [1/2 rot V0-w] w x
Tz=8phR3 [g/2+wz] z T
External torque

10. Control of the particle rotation by an external field
Dilute regime with N particles
sxy=s0xy+sRxy g g
2nd Faxen Law : y
T=-8phR3 [1/2 rot V0-w] wz x
Tz=8phR3 [g/2+wz] z
sRxy= N Tz/2V T
External torque

11. Control of the particle rotation by an external field
Dilute regime with N particles
sxy=s0xy+sRxy g g
2nd Faxen Law : y
T=-8phR3 [1/2 rot V0-w] wz x
Tz=8phR3 [g/2+wz] z
sRxy= N Tz/2V T
Q=(wz+g/2)/(g/2)
hReff=sRxy/g=3/2 h f Q
External torque
f=N 4/3 p R3/V
heff=h0eff+hReff=(s0xy+sRxy)/g=h(1+5/2 f + 3/2 f Q)

12. Control of the particle rotation by an external field
Dilute regime with N particles
(heff-h)/h=5/2 f + 3/2 f Q
sxy=s0xy+sRxy g g y w x
(heff-h)/h z T
External torque Q
(heff-h)/h=0 if
Q=-5/31.67

13. Control of the particle rotation by an external field
More concentrated regimes
(no dipole-dipole interactions)
(heff-h)/h= (h0eff-h)/h (1+3/5Q)
sxy=s0xy+sRxy g g y w x
(heff-h)/h z T
External torque Q
h0eff(f)=h (1-f/fm)-5/2fm
Krieger & Dougherty law:

14. Control of the particle rotation by an external field
More concentrated regimes g g y
sxy w x
F(Q,f) z T
External torque Q
F(Q,f)=(heff-h)/(h0eff-h)= (1+3/5Q)

15. Control of the particle rotation by an external field
More concentrated regimes g g
A Faxen law for more concentrated regimes: y
sxy
<T>=-(12V/5N) (h0eff-h) (1/2 rot V-w) w x z T wz
External torque f f
<T>Tf  0=-8pR3h (1/2 rot V-w)

16. Controlling the rotation of particles in a suspension
External torque
Torque-free particle
External torque
wz<-g/2
wz=-g/2
wz>-g/2
heff<h0eff
heff=sxy/g
heff>h0eff

17. Controlling the rotation of particles in a suspension
Rotation in presence of walls (Microfluidic conditions)

18. Rotation of a very confined particle in a shearwz
Increases or
decreases?? wz 2H y
-V0 x
g=V0/H

19. Rotation of a very confined particle in a shear
Naive argument :
wz  V0/a V0/H = g 2H 2a
wz/(g/2) y 2
-V0 x 1
g=V0/H
H/a 1

20. Rotation of a very confined particle in a shear
wz/(g/2) y
-V0 x
Our numerical simulations
g=V0/H

21. Rotation of a very confined particle in a shear
wz/(g/2) y
-V0 x
Our numerical simulations
And
Reflection method (A. Sangani)
g=V0/H

22. Rotation of a very confined particle in a shear
VT(r)
VT(r) - V0(r) y y x z x x

23. Rotation of a very confined particle in a shear
Pure shear flow =
Rotation g/2 + ext./compr. flow
VT(r)
VT(r) - V0(r)
Rotation
g/2 y x z

24. Rotation of a very confined particle in a shear
VT(r)
VT(r)-V0(r)

25. Rotation of a very confined particle in a shear
VT(r)
VT(r)-V0(r)
Also obtained by B. Kaoui
et al, on circular vesicles
(To be published)

26. Control of the particle rotation
Rotation is modified both by
- confinement
- external field
Dipole-dipole interaction should be added
(changes the rheology at small shear rate)