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On bifurcation in counter-flows of viscoelastic fluid

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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.

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One-quadrant problem statement:

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

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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.

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Vicinity of the central point: symmetric case

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Symmetry relative to x, y gives

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

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

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

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Pressure: from momentum equation where

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Comparison with symmetric numerical solution

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

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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 :

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

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

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Same for the pressure

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Vicinity of the central point: asymmetric case

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

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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

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Whole domain solution

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

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

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Conclusions

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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.

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Tried lows of the pressure increase:

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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

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Convergence and quality of numerical procedure

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

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

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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

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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

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

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Used lows of inlet pressure increase:

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

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

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Extremely high Weissenberg numbers

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

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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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