1. University of British Columbia Sarah Hormozi, Kerstin Wielage-Burchard, Ian Frigaard & Mark Martinez Exotic Flows in visco-plastic lubrication

2. 2 Outline  Motivation  Visco-plastic lubrication  Strory so far  Start-up and entry length effect  Stability of the established flow  Conclusion

3. 3 Introduction Multi-layer flow applications  Co-extrusion: product is made of >1 layers simultaneously  Film coating: layer applied to fluid substrate  Lubricated transport: lubricating fluid lies in a layer between wall and transported fluid  Whenever fluid-fluid interfaces are present, rate of throughput/production is limited by interfacial instability

4. 4 Visco-plastic Lubrication Lubricating fluid  Outer fluid has yield stress  Inner fluid unimportant  Duct cross-section also unimportant  Flow rates controlled to have plug at the interface Lubricating fluid Duct Core fluid Imposed flow rate Q W(r) Plug ii YY Duct Core fluid Imposed flow rate Q W(r) Plug ii YY

5. 5 Results so far:  Linear stability: Visco-plastically lubricated multi-layer flows can be more stable than equivalent single fluid flows  Frigaard, JNNFM 100, (2001)  Nonlinear stability: (Newtonian core) Conditional stability for Re in 10-100 range Stability conditional on amplitude, but not weakly nonlinear Energy method  Moyers-Gonzalez, Frigaard & Nouar, JFM 506, (2004)  Experimental demonstration: (Xanthan+Carbopol) Stable flows, where predicted, for inner fluid Re~103  Huen, Frigaard & Martinez, JNNFM 142, (2007)

6. 6 240 seconds >300s 2.2m Huen, Frigaard & Martinez, JNNFM 142, (2007)

7. 7 Equations of Motion r z r=R i r=r i Fluid 1 Fluid 2 W(r) r=1

8. 8 Computational Solution  Implemented within PELICA`NS Open source code (IRSN, France) C++ Numerical PDE Solution Package Meshing capabilities & parallel comp. 10 years of internal development  Mixed FE/FV scheme  VOF method to handle 2 fluids  Yield stress fluid rheology handled either by viscosity regularisation or augmented Lagrangian method  PELICANS has various standard benchmarks computed Developers have also used for yield stress fluid flows (Vola & Latche) C=0 C=1 r z r=R i r=r i Fluid 1 Fluid 2 W(r) r=1

9. 9 Can code produce experimentally observed multi-fluid flow structures? Pearl and mushroom instability, D’Olce et al., Phys. Fluids 20 (2008) Pearl and mushroom instability, Produced by the code

10. 10 Start-up flow t=4t=8t=12|u||u| . Re=20, m=1, ri=0.4

11. 11 Established flow m = 1, B=10, r i = 0.4 Re=5 Re=20 Re=40

12. 12 Stability computations: Method  Fix (m,B,ri) to have suitable base flow  Periodic cell in flow direction, run to steady state from analytic base solution  discrete steady flow  Superimpose perturbation to base flow  Divergence free, initial perturbations that break plug (A), or leave intact (B)  Normalise initial perturbation & scale with amplitude u=A(vr,vz)  Run transient computation + Case(A) + Case(B) U u U+u

13. 13 Initial stage: plug reforms quickly Pipe flow: Re=1,B=20, m=10, r i =0.4, r y =0.71,Initial perturbation amplitude = 40% Colourmaps of strain rate + axial velocity superimposed CASE A; t=0, 0.001, 0.002, 0.005, 1 CASE B; t=0, 0.001, 0.002, 0.005, 1 Decay of velocity perturbation for ri=0.4,m=10,B=20,Re=1,case(A). Different curves denote initial perturbation amplitudes: A=0.01,0.1,0.4,0.6,1,3 A

14. 14 Pearl Instability Pipe flow: Re=100,B=20, m=10, r i =0.4, r y =0.71,Initial perturbation amplitude = 300%, Case(B), Colourmaps of strain rate + axial velocity superimposed

15. 15 Mushroom Instability Pipe flow: Re=100,B=20, m=10, r i =0.6, r y =0.72,Initial perturbation amplitude = 80%, Case(B), Colourmaps of strain rate + axial velocity superimposed

16. 16 Conclusion  If (m,B,ri) have “case 1” solution, (plug at interface) transients converge to flow that is approximately the base parallel flow Displacement fronts eventually advected from tube,Smaller m more problematic (large velocity gradients),Moderate expansions (ri > Ri) and contractions (ri < Ri) are OK  Main discrepancy from diffusion/dispersion at interface No flow instabilities observed for Re 10^4  Established steady flows: Shortest development lengths when Ri = ri,Development lengths longer with expansion than contraction,3 different development length definitions possible,Development lengths increase with Re, but not linear relationship  Perturbed flows stable at serious Re & amplitudes Not weakly nonlinear,Incomplete decay of ||u|| due to mixing/dispersion: New secondary flows,What if immiscible fluids?  Caution: pipe results are axisymmetric