Complex dynamics of shear banded flows Suzanne Fielding School of Mathematics, University of...
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Transcript of Complex dynamics of shear banded flows Suzanne Fielding School of Mathematics, University of...
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
Liquid crystals
nematic
isotropic
Wormlike surfactants
aligned
isotropic
high
low
Onion surfactants
ordered
disordered
[Lerouge, PhD, Metz 2000]
steady state flow curve
Cappelare et al PRE 97 Britton et al PRL 97
UNSTABLE
[Spenley, Cates, McLeish PRL 93]
Triggered by non-monotonic constitutive curve
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
increasing
shear
rate
Applied shear rate: stress fluctuates
• Type II intermittency route to chaos
time
[Sood et al, PRL 2006]
Surfactant onion phases
Schematic flow curve for disordered-to-layered transition
Shear rate density plot: bands fluctuate
[Manneville et al, EPJE 04]
SDS (6.5 wt %),octanol (7.8 wt %), brine
[Salmon et al, PRE 2003]
Time
Posi
tion
acr
oss
gap
Shear thickening wormlike micelles [Boltenhagen et al PRL 1997] TTAA/NaSal (7.5/7.5 mM) in water
Time-averaged flow curve Applied stress: shear rate fluctuates...
… along with band of shear-induced phase
Vorticity bands
[Fischer Rheol. Acta 2000]
CPyCl/NaSal (40mM/40mM) in water
Semidilute polymer solution:
fluctuations in shear rate and
birefringence at applied stress
Shear thickening wormlike micelles:
oscillations in shear & normal stress
at applied shear rate
[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
Relaxation time increases with micellar length: 00 nnn
Micellar length n decreases in shear: 0 / 1n t nn n n
tytyt ,, Shear stress
Dynamics of micellar contribution
tn g n 22yl
plateau
low high
solvent micelles
High shear branch unstable!
with 2/ 1g x x x
interacting
pulsesoscillating
bands
interactingdefects
largest Lyapunov exponent
time, t
single pulse interacting defectsoscillating bandsinteracting pulsesy
t
flow curve
stress evolution
greyscale
of ty,
Chaotic bands at applied shear rate: global constraint tydy ,
[SMF + Olmsted, PRL 04]
Theory approach 2: interfacial dynamics
• Return to stable high shear branch
• Now in a model (Johnson-Segalman) that has normal micellar stresses
2 2,t ij nm n m ijD F v l with 0xx yy
• Consider initial banded state that is 1 dimensional (flat interface)
y
x
interface width l
Linear instability of the interface
• Return to stable high shear branch
• Now in a model (Johnson-Segalman) that has normal micellar stresses
with 0xx yy
• Then find small waves along interface to be unstable…
exp xy iq x t
[SMF, PRL 05]
Linear instability of the interface
y
x
2 2,t ij nm n m ijD F v l
• Positive growth rate linearly unstable. Fastest growth: wavelength 2 x gap
[Analysis Wilson + Fielding, JNNFM 06]
Linear instability of the interface
Nonlinear interfacial dynamics
• Number of linearly unstable modes • Just beyond threshold: travelling wave
ij ij x ct
[SMF + Olmsted, PRL 06]
xL
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
Models of
shear thinning
solns of rigid
rods
Shear
thickening
Seen in worms [Fischer]; viral suspensions [Dhont]; polymers [Vlassopoulos]; onions [Wilkins]; colloidal suspensions [Zukowski]
Wormlike micelles [Wheeler et al JNNFM 98]
Already seen… Now what about…
z
Vorticity banding: possible 2D scenario
Recently observed in wormlike micelles
Lerouge et al PRL 06
CTAB wt 11% + NaNO3 0.405M in water
R100t O
L
increasing
with
O L
Linear instability of flat interface to small amplitude waves
z
Positive growth rate linearly unstable
1R100O [SMF, submitted]
exp zy iq z t
Nonlinear steady state
Greyscale of xx “Taylor-like” velocity rolls
z
y
z O L
increasing
with
[SMF, submitted]
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