Transcript Slide 1

Complex dynamics of shear banded flows
Suzanne Fielding
School of Mathematics, University of Manchester
Peter Olmsted
School of Physics and Astronomy, University of Leeds
Helen Wilson
Department of Mathematics, University College London
Funding: UK’s EPSRC
Shear banding
   yv
Wormlike surfactants
high
low
Liquid crystals
aligned
nematic
isotropic
isotropic
Onion surfactants
ordered
disordered
Triggered by non-monotonic constitutive curve
steady state flow curve


[Lerouge, PhD, Metz 2000]
Cappelare et al PRE 97 Britton et al PRL 97
[Spenley, Cates, McLeish PRL 93]
Experiments showing oscillating/chaotic bands
Shear thinning wormlike micelles
[Holmes et al, EPL 2003, Lopez-Gonzalez, PRL 2004] 10% w/v CpCl/NaSal in brine
Time-averaged flow curve
Applied shear rate: stress fluctuates
Velocity greyscale: bands fluctuate
radial displacement
Shear thinning wormlike micelles
[Sood et al, PRL 2000] CTAT (1.35 wt %) in water
Time averaged flow curve
Applied shear rate: stress fluctuates
increasing
shear
rate
• Type II intermittency route to chaos
[Sood et al, PRL 2006]
time
Surfactant onion phases
Schematic flow curve for disordered-to-layered transition
[Salmon et al, PRE 2003]
Position across gap
Shear rate density plot: bands fluctuate
[Manneville et al, EPJE 04]
SDS (6.5 wt %),
octanol (7.8 wt %),
brine
Time
Shear thickening wormlike micelles
[Boltenhagen et al PRL 1997] TTAA/NaSal (7.5/7.5 mM) in water
Applied stress: shear rate fluctuates...
Time-averaged flow curve
… along with band of shear-induced phase
Vorticity bands
Shear thickening wormlike micelles:
oscillations in shear & normal stress
at applied shear rate
[Fischer Rheol. Acta 2000]
CPyCl/NaSal (40mM/40mM) in water
Semidilute polymer solution:
fluctuations in shear rate and
birefringence at applied stress
[Hilliou et al Ind. Eng. Chem. Res. 02]
Polystyrene in DOP
Theory approach 1: flat interface
The basic idea… bulk instability of high shear band
• Existing model predicts stable, time-independent shear bands
• What if instead we have an unstable high shear constitutive branch…
• See also (i) Aradian + Cates EPL 05, PRE 06 (ii) Chakrabarti, Das et al PRE 05, PRL 04
Simple model: couple flow to micellar length
t     y, t     y, t 
Shear stress
solvent
micelles
Dynamics of micellar contribution
  n   t    g   n    l 2  2y
Relaxation time increases with micellar length:
Micellar length n decreases in shear:
plateau

low


with g  x   x / 1  x 2

 n   0 n n0 

 n t n  n  n0 / 1    n  
High shear branch unstable!
high
Chaotic bands at applied shear rate: global constraint    dy   y, t 
flow curve
largest Lyapunov exponent


interacting oscillating interacting
pulses
defects
bands


stress evolution

time, t
single pulse
y
interacting pulses
oscillating bands
interacting defects
greyscale
of  y, t 
t
[SMF + Olmsted, PRL 04]
Theory approach 2: interfacial dynamics
Linear instability of the interface
• Return to stable high shear branch
interface width l
y
x
• Now in a model (Johnson-Segalman) that has normal micellar stresses
Dt ij  F  nm , nvm   l   ij
2
2
with
 xx   yy  0
• Consider initial banded state that is 1 dimensional (flat interface)
Linear instability of the interface
• Return to stable high shear branch
  y  exp  iqx x  t 
y
x
• Now in a model (Johnson-Segalman) that has normal micellar stresses
Dt ij  F  nm , nvm   l   ij
2
2
with
 xx   yy  0
• Then find small waves along interface to be unstable…
[SMF, PRL 05]
Linear instability of the interface
• Positive growth rate  linearly unstable. Fastest growth: wavelength 2 x gap
[Analysis Wilson + Fielding, JNNFM 06]
Nonlinear interfacial dynamics
• Number of linearly unstable modes
• Just beyond threshold: travelling wave
Lx
 ij   ij  x  ct 
[SMF + Olmsted, PRL 06]
Further inside unstable region: rippling wave
• Number of linearly unstable modes
• Force at wall: periodic
• Greyscale of
 xx
Multiple interfaces
Then see erratic (chaotic??) dynamics
Vorticity banding
Vorticity banding: classical (1D) explanation
Recall gradient banding
Analogue for vorticity banding
Shear
thickening
Models of
shear thinning
solns of rigid
rods
Seen in worms [Fischer]; viral suspensions [Dhont];
polymers [Vlassopoulos]; onions [Wilkins];
colloidal suspensions [Zukowski]
Wormlike micelles
[Wheeler et al JNNFM 98]
Vorticity banding: possible 2D scenario
Already seen…
Now what about…
z
Recently observed in wormlike micelles
Lerouge et al PRL 06
L
  O  L
increasing
with

t  O 100 R 
CTAB wt 11% + NaNO3 0.405M in water
Linear instability of flat interface to small amplitude waves
z
  y  exp  iqz z  t 
Positive growth rate  linearly unstable
[SMF, submitted]
  O 100 R 
1
Nonlinear steady state
Greyscale of
 xx
“Taylor-like” velocity rolls
  O  L
z
increasing
with
z
[SMF, submitted]
y

Summary / outlook
• Two approaches
a) Bulk instability of (one of) bands – (microscopic) mechanism ?
b) Interfacial instability – mechanism ?
(Combine these?)
• Wall slip – in most (all?) experiments
• 1D vs 2D: gradient banding can trigger vorticity banding