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On bifurcation in counter-flows of viscoelastic fluid

On bifurcation in counter-flows of viscoelastic fluid. Preliminary work. Mackarov I. Numerical observation of transient phase of viscoelastic fluid counterflows // Rheol. Acta. 2012, Vol. 51, Issue 3, Pp. 279-287 DOI 10.1007/s00397-011-0601-y.

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On bifurcation in counter-flows of viscoelastic fluid

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  1. On bifurcation in counter-flowsof viscoelastic fluid

  2. Preliminary work • Mackarov I. Numerical observation of transient phase ofviscoelasticfluid counterflows // Rheol. Acta. 2012, Vol. 51, Issue 3,Pp. 279-287DOI 10.1007/s00397-011-0601-y. • Mackarov I. Dynamic features of viscoelastic fluid counter flows // Annual Transactions of the Nordic Rheology Society. 2011. Vol. 19. Pp. 71-79.

  3. One-quadrant problem statement:

  4. The process of the flow reversal, Re = 0.1, Wi= 4, mesh is675nodes,t=1.7 ÷2.5

  5. The process of the flow reversal, Re= 0.1, Wi = 4, mesh is675nodes,t=1.7 ÷2.5

  6. G.N. Rocha and P.J. Oliveira.Inertial instability in Newtonian cross-slot flow– A comparison againstthe viscoelastic bifurcation.Flow Instabilities and Turbulence in Viscoelastic Fluids, Lorentz Center, July 19-23, 2010, Leiden, Netherlands • R. J. Poole, M. A. Alves, and P. J. Oliveira.Purely Elastic Flow Asymmetries. Phys. Rev. Lett., 99,164503, 2007.

  7. Vicinity of the central point: symmetric case

  8. Symmetry relativeto x, y gives

  9. Symmetry relativeto x, y definesthe most general asymptotic form of velocities: … and stresses:

  10. Substituting this to momentum, continuity, and UCM state equations will give…

  11. Symmetry on x, y involves (21) Therefore, for the rest of the coefficients in solution ,

  12. Pressure: from momentum equation where

  13. Comparison with symmetric numerical solution

  14. Via finite-difference expressions of coefficients in velocities expansions, we get from the numeric solution: A ≈ -0.006B ≈ 0.0032

  15. STRESS: Via finite-difference determination of coefficients in velocities expansions get : Σx= -0.0573α = 0.0286 β = 0.026≈ α “Numerical” stress in the central point : σxx= -0.0518

  16. Normal stress distribution in numericone-quadrantsolution (stabilized regime),Re=0.1, Wi=4, the mesh is 2600 nodes

  17. PRESSURE: Via finite-difference values of coefficients in velocities expansions, we get : Px=0.0642 Py=-0.0641 ≈-Px

  18. Same for the pressure

  19. Vicinity of the central point: asymmetric case

  20. UCM model, Re = 0.01, Wi = 100, t=3.55, mesh is6400nodes

  21. Looking into nature of the flow reversal: analogy with simpler flows • Couette flow • Poiseuille flow

  22. Whole domain solution

  23. UCM model, Re = 0.1, Wi = 4, t=2.7, mesh has2090nodes

  24. Pressure distributionin the flow with Re = 3andWi= 4at t =3.5, meshis1200nodes, Δt= 5·10-5

  25. Conclusions

  26. Both some features reported before and new details were observed in simulation of counter flows within cross-slots (acceleration phase). • Among the new ones: the pressure and stresses singularities both at the stagnation point and at the walls corner, flow reversal with vortex-like structures. • The flow reverse is shown to result fromthe wave nature of a viscoelastic fluid flow.

  27. Tried lows of the pressure increase:

  28. Flow picture (UCM model, Re = 0.05, Wi = 4, t=6.2), with exponential lowof the pressure increase (α = 1) the meshis432nodes

  29. Convergence and quality of numerical procedure

  30. Picture of vortices typical fortypical for small Re.UCM model, Re = 0.1, Wi = 4, t=2.6, mesh is1200nodes

  31. The same flow snapshot(UCM model, Re = 0.1, Wi = 4, t=2.6), obtained on a non-elastic meshwith 1200 nodes

  32. Normal stress distribution in the flow with Re = 0.01 andWi=100 at t = 3; UCM model, meshis2700nodes, Δt= 5·10-5

  33. Sequence of normal stress abs. values at the stagnation point. Smaller markers correspond to time step 0.0001, bigger ones are for time step 0.00005

  34. High Weissenberg numbers: Sequence of normal stress abs. values at the stagnation point. Smaller markers correspond to time step 0.0001, bigger ones are for time step 0.00005

  35. Normal stress distribution in the flow with Re = 0.1 andWi =4at t = 3; meshis 450 nodes, Δt= 5·10-5

  36. Used lows of inlet pressure increase:

  37. The process of the flow reversal, Re = 0.1, Wi= 4, mesh is675nodes,t=1.7 ÷2.5

  38. Flow picture for Re = 0.1, Wi = 4, the mesh is4800nodes

  39. Flow picture for Re = 0.1, Wi = 4, the mesh is4800nodes

  40. Flow picture for Re = 0.1, Wi = 4, the mesh is4800nodes

  41. Extremely high Weissenberg numbers

  42. Flow picture for Re = 0.01, Wi = 100, the mesh is4800nodes

  43. Flow picture for Re = 0.01, Wi = 100, the mesh is4800nodes

  44. Flow picture for Re = 0.01, Wi = 100, the mesh is4800nodes

  45. Flow picture for Re = 0.01, Wi = 100, the mesh is4800nodes

  46. Flow picture for Re = 0.01, Wi = 100, the mesh is4800nodes

  47. Flow picture for Re = 0.01, Wi = 100, the mesh is4800nodes

  48. Flow picture for Re = 0.01, Wi = 100, the mesh is4800nodes

  49. A flow snapshot from S. J. Haward et. al., The rheology of polymer solution elastic strands in extensional flow, Rheol Acta (2010) 49:781-788

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