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