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 Liquid crystals nematic isotropic Wormlike surfactants aligned isotropic 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 Position across 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: Micellar length n decreases in shear: Shear stress Dynamics of micellar contribution plateau solvent micelles High shear branch unstable! with

interacting pulses oscillating bands interacting defects largest Lyapunov exponent time, t single pulse interacting defects oscillating bandsinteracting pulses y t  flow curve stress evolution greyscale of Chaotic bands at applied shear rate: global constraint [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 with 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 Then find small waves along interface to be unstable… [SMF, PRL 05] Linear instability of the interface y x

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 [SMF + Olmsted, PRL 06]

Further inside unstable region: rippling wave Number of linearly unstable modes Force at wall: periodic Greyscale of

Multiple interfaces Then see erratic (chaotic??) dynamics

Vorticity banding

Vorticity banding: classical (1D) explanation Recall gradient bandingAnalogue 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% + NaNO M in water increasing with

Linear instability of flat interface to small amplitude waves z Positive growth rate  linearly unstable [SMF, submitted]

Nonlinear steady state Greyscale of“Taylor-like” velocity rolls z y z 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