1. On bifurcation in counter-flows of viscoelastic fluid

2. 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. 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 is 675 nodes, t=1.7 ÷2.5

6. G.N. Rocha and P.J. Oliveira. Inertial instability in Newtonian cross-slot flow – A comparison against the 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 relative to x, y gives

9. Symmetry relative to x, y defines the most general asymptotic form of velocities: … and stresses:

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

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

13. Pressure: from momentum equation where

14. Comparison with symmetric numerical solution

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

16. STRESS: Σ x = -0.0573 α = 0.0286 β = 0.026 α σ xx = -0.0518 Via finite-difference determination of coefficients in velocities expansions get : Numerical stress in the central point :

17. Normal stress distribution in numeric one-quadrant solution (stabilized regime), Re=0.1, Wi=4, the mesh is 2600 nodes

18. PRESSURE: Via finite-difference values of coefficients in velocities expansions, we get : P x =0.0642 P y =-0.0641 - P x

19. Same for the pressure

20. Vicinity of the central point: asymmetric case

21. UCM model, Re = 0.01, Wi = 100, t = 3.55, mesh is 6400 nodes

22. Looking into nature of the flow reversalanalogy with simpler flows Looking into nature of the flow reversal : analogy with simpler flows Couette flow Couette flow Poiseuille flow Poiseuille flow

23. Whole domain solution

24. UCM model, Re = 0.1, Wi = 4, t=2.7, mesh has 2090 nodes

25. Pressure distribution in the flow with Re = 3 and Wi = 4 at t = 3.5, mesh is 1200 nodes, Δt = 5·10 -5

26. Conclusions

27. 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 from the wave nature of a viscoelastic fluid flow.

28. Tried lows of the pressure increase:

29. Flow picture (UCM model, Re = 0.05, Wi = 4, t=6.2), with exponential low of the pressure increase (α = 1) the mesh is 432 nodes

30. Convergence and quality of numerical procedure

31. Picture of vortices typical for typical for small Re. UCM model, Re = 0.1, Wi = 4, t=2.6, mesh is 1200 nodes

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

33. Normal stress distribution in the flow with Re = 0.01 and Wi =100 at t = 3; UCM model, mesh is 2700 nodes, Δt= 5·10 -5

34. 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

36. Normal stress distribution in the flow with Re = 0.1 and Wi =4 at t = 3; mesh is 450 nodes, Δt= 5·10 -5

37. Used lows of inlet pressure increase:

38. The process of the flow reversal, Re = 0.1, Wi = 4, mesh is 675 nodes, t=1.7 ÷2.5

39. Flow picture for Re = 0.1, Wi = 4, the mesh is 4800 nodes

42. Extremely high Weissenberg numbers

43. Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes

50. 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