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Complex dynamics of shear banded flows

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. Liquid crystals. Onion surfactants.

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Complex dynamics of shear banded flows

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  1. Complex dynamics of shear banded flows Suzanne FieldingSchool of Mathematics, University of ManchesterPeter OlmstedSchool of Physics and Astronomy, University of LeedsHelen WilsonDepartment of Mathematics, University College London Funding: UK’s EPSRC

  2. Liquid crystals Onion surfactants nematic ordered isotropic disordered Shear banding Wormlike surfactants aligned isotropic

  3. Cappelare et al PRE 97 Britton et al PRL 97 Triggered by non-monotonic constitutive curve steady state flow curve [Lerouge, PhD, Metz 2000] UNSTABLE [Spenley, Cates, McLeish PRL 93]

  4. Experiments showing oscillating/chaotic bands

  5. Shear thinning wormlike micelles [Holmes et al, EPL 2003, Lopez-Gonzalez, PRL 2004] 10% w/v CpCl/NaSal in brine Applied shear rate: stress fluctuates Time-averaged flow curve Velocity greyscale: bands fluctuate radial displacement

  6. 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

  7. Surfactant onion phases Schematicflow curve for disordered-to-layered transition [Salmon et al, PRE 2003] Shear rate density plot: bands fluctuate [Manneville et al, EPJE 04] SDS (6.5 wt %), octanol (7.8 wt %), brine Position across gap Time

  8. 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

  9. 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

  10. Theory approach 1: flat interface

  11. The basic idea… bulk instability of high shear band • Existing model predictsstable, time-independentshear 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

  12. Micellar length n decreases in shear: plateau Simple model: couple flow to micellar length Shear stress micelles solvent Dynamics of micellar contribution with Relaxation time increases with micellar length: High shear branch unstable!

  13. stress evolution Chaotic bands at applied shear rate: global constraint flow curve largest Lyapunov exponent S interacting pulses interacting defects oscillating bands S time, t single pulse interacting pulses oscillating bands interacting defects y greyscale of t [SMF + Olmsted,PRL 04]

  14. Theory approach 2: interfacial dynamics

  15. 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 with • Consider initial banded state that is 1 dimensional (flat interface)

  16. Linear instability of the interface • Return to stable high shear branch y x • Now in a model (Johnson-Segalman) that has normal micellar stresses with • Then find small waves along interface to be unstable… [SMF,PRL 05]

  17. Linear instability of the interface • Positive growth rate  linearly unstable. Fastest growth: wavelength 2 x gap [Analysis Wilson + Fielding, JNNFM 06]

  18. Nonlinear interfacial dynamics • Number of linearly unstable modes • Just beyond threshold: travelling wave [SMF + Olmsted, PRL 06]

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

  20. Multiple interfaces Then see erratic (chaotic??) dynamics

  21. Vorticity banding

  22. 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]

  23. Wormlike micelles [Wheeler et al JNNFM 98]

  24. Vorticity banding: possible 2D scenario Already seen… Now what about… z

  25. Recently observed in wormlike micelles Lerouge et al PRL 06 increasing with CTAB wt 11% + NaNO3 0.405M in water

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

  27. Nonlinear steady state Greyscale of “Taylor-like” velocity rolls z increasing with z [SMF, submitted] y

  28. 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

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