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

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

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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

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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 ii YY Duct Core fluid Imposed flow rate Q W(r) Plug ii YY

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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 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)

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

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

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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

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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

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

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

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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

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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

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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

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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

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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

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