on bifurcation in counter-flows of viscoelastic fluid
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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. - PowerPoint PPT PresentationTRANSCRIPT
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On bifurcation in counter-flowsOn bifurcation in counter-flowsof viscoelastic fluid of viscoelastic fluid
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Preliminary workPreliminary 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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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:Vicinity of the central point:symmetric casesymmetric case
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( , ) ( , ) ( , ) ( , )u x y u x y u x y u x y
( , ) ( , ) ( , ) ( , )v x y v x y v x y v x y
( , ) ( , ) ( , ) ( , )p x y p x y p x y p x y
Symmetry relative to x, y gives
( , ) ( , ) ( , ) ( , )xx xx xx xxx y x y x y x y
( , ) ( , ) ( , ) ( , )xy xy xy xyx y x y x y x y
( , ) ( , ) ( , ) ( , )yy yy yy yyx y x y x y x y
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32 3 3o, (13
)u x y A x B x y x yB x
2 3 331,3
o( )v x y A y B x y x yB y
Symmetry relative to x, y defines the most general asymptotic form of velocities:
… and stresses:
32 32, = o( )xx x xx y x y y
32 32, = o( )yy y xx y x y y
3 3=σ , o( )xy x yx y x y
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Substituting this to momentum, continuity, and UCM state equations will give…
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2 1x
A
Re A Wi
2 1y
A
Re A Wi
2 24
4 1
A B Wi
Re A Wi
2 1B
Re AWi
2 1 4 1
BRe AWi AWi
2 1
B
Re AWi
2 1 4 1
B
Re AWi AWi
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2 1 4 1
B
Re AWi AWi
(21)
( , ) ( , )u x y v y x ( , ) ( , )v x y u y x
,
Symmetry on x, y involves
Therefore, for the rest of the coefficientsin solution
2 1
B
Re AWi
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Pressure:
from momentum equation
2 20
3 3(, )( ) x yp x y P P x P y x y
where
2
2 2( , 2)
1 4xBAB
Re WiP A
A
2
2 2( , ) ,
4) 2(
1y xP A B P A B BARe A Wi
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Comparison with Comparison with symmetric numerical solutionsymmetric numerical solution
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2 31,3
u x y A x B x y B x
2 31,3
v x y A y B x y B y
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:
Σx= -0.0573 α = 0.0286 β = 0.026 ≈ α
σxx= -0.0518
Via finite-difference determination of coefficientsin velocities expansions get :
2 2, =xx xx y x y
“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 :
2 20( , ) x yp x y P P x P y
Px=0.0642 Py=-0.0641 ≈ -Px
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Same for the pressure
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Vicinity of the central point:Vicinity of the central point:asymmetric caseasymmetric 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 Looking into nature of the flow reversalreversal: analogy with simpler flowsanalogy with simpler flows
• Couette flowCouette flow
• Poiseuille flowPoiseuille 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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ConclusionsConclusions
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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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( )1inlet
tp tt
Tried lows of the pressure increase:
2
2( )1inlet
tp tt
( ) 1 tinletp t e
( ) 1inletp t
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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
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
X
Y
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0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
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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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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( )1inlet
tp tt
Used lows of inlet pressure increase:
2
2( )1inlet
tp tt
( ) 1 exp( )inletp t t
( ) 1inletp t
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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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Flow picture for Re = 0.1, Wi = 4, the mesh is 4800 nodes
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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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Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
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Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
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Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
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Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
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Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
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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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