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