Bubble Motions in Bubble Rafts under Steady Shear

Michael Dennin

Department of Physics

U. C. Irvine

Supported by: Department of Energy grant DE-FG02-03ED46071, Sloan Foundation, Petroleum Research Fund, and UCI UROP

General Outline

• Questions raised/addressed in this talk

• Overview of the system

• Initial results

Two Questions

• What is the average flow behavior of slowly sheared bubble raft? (How does this relate to flow of foams?)

• What is the connection between average flow behavior and individual bubble motions?

General properties

• Fluctuations in stress/energy.

• “Particle” rearrangements (T1 events, non-affine motions)

• Non-uniform shear

• Diffusive motion of “particles”.

Two “types” of non-uniform shear

Non-uniform shear: region of non-zero and zero shear rate coexist

1) strain rate is continuous (usually exponential velocity).

2) strain rate is discontinuous.

“Two-dimensional” foam

Debregeas, Tabuteau, Di Meglio, PRL 87 (2001)

Three dimensional suspension

Coussot, Raynaud, et al., PRL 88, 218301 (2002)

Definition of T1 Event

T1 event:Neighbor switching

Apparatus

Schematic of Apparatus

Inner radius ri: 3.84 cmOuter radius ro: 7.43 cmArea fraction: 0.95Boundary conditions: no slip at both walls, but inner cylinder is free to move.

Definition of Terms

Outer barrier moves with V

Strain: x/r

Strain Rate: d/dt = v/r

Viscosity: = stress/(strain rate)

r

strain

elastic

flowingstress

Shear stress: xy = F/L (two-dimensions)

( )/

d v rd dt r

dr r

Bubble Motions

Reminder of Geometry Consequences

• Couette Geometry: average stress, , proportional to 1/r2

• Yield stress, y:

=> critical radius beyond which “rigid” body or elastic behavior, strain rate is a continuous function of r.

( ) ny

4.0 4.5 5.0 5.5 6.0 6.5 7.00.0

0.5

1.0

1.5

2.0

stre

ss (

mN

/m)

radial position (cm)

"flowing"

zero shearrate: "rigid body"

Effective Viscosity: stress/(strain rate)

-3 -2 -1 01

2

3

4

log

(vis

cosi

ty)

log (strain rate)

1/3 1/3 1/3(0.8 mN/m)( / ) (1.8 mNs /m)( / ) y a d dt d dt

Stress versus strain

0 500 1000 15000.0

0.5

1.0

1.5

2.0

2.5

DC

BA

stre

ss (

mN

/m)

time (s)

(1)

(2)

(1) strain rate = 3 x 10-2 s-1 (2) strain rate = 4 x 10-3 s-1

y= 0.8 mN/m

rc=6.3 cm

rc=6.7 cm

Average Velocity Profile

5 6 70.0

0.5

1.0

6 7

0.8

1.0

radial position (cm)

v(r

)/r

v(r

)/r

radial position (cm)

V(r)/r = 1 => rigid body rotation.

Fit is to vel. profile for a power law viscosity.

Some Questions

• What sets the “critical” radius?• Why is strain rate discontinuous?

Consider “flow” during individual events and T1 events.

• What is the role of stress chains, if they exist?

T1 events and stress

3 4 5 6

4

5

6

7

1.4

1.6

1.8

2.0

2.2

2.4

ra

dia

l po

sitio

n (

cm)

strain

str

ess

(d

yne

/cm

)

T1 events and bubble motions

3.2 3.3 3.4 3.5 3.6 3.74

5

6

7

1.5

1.6

1.7

1.8

1.9

6.0 6.1 6.2 6.3 6.44

5

6

7

1.8

2.0

2.2

2.4

A

po

sitio

n (c

m)

strain

(a)

B C D E str

ess

(mN

/m)

strain

pos

ition

(cm

)

(b)

A B C D E

str

ess

(mN

/m)

“Local” Displacements

4.5 5.0 5.5 6.0 6.5 7.0-3

-2

-1

0

1

2

t)

radial position

EB,C

A,D

T1 events and average velocity

0.0

0.2

0.4

0.6

0.8

1.0

4.5 5.0 5.5 6.0 6.5 7.0

0.02

0.04

0.06

0.08

#

T1

eve

nts

/ b

ub

ble

radial postion (cm)

v(r

)/r

Summary

• Apparent disagreement between average stress measurements and average velocity profile: strain-rate discontinuity needs to be understood.

• Connection between T1 events and short time bubble motions. Not clear the connection between T1 events and average velocity.

• Time averages rapidly converge despite very nonlinear short time motion.

Acknowledgments

• Video images of bubble raft: John Lauridsen

• Viscosity measurements: Ethan Pratt• Initial Bubble tracking software: Gregory

Chanan• Funding: Department of Energy grant DE-

FG02-03ED46071, Sloan Foundation, Petroleum Research Fund, and UCI UROP