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
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.
One-quadrant problem statement:
The process of the flow reversal, Re = 0.1, Wi = 4, mesh is 675 nodes, t=1.7 ÷2.5
The process of the flow reversal, Re = 0.1, Wi = 4, mesh is 675 nodes, t=1.7 ÷2.5
• 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.
Vicinity of the central point:Vicinity of the central point:symmetric casesymmetric case
( , ) ( , ) ( , ) ( , )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
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
Substituting this to momentum, continuity, and UCM state equations will give…
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
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
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
Comparison with Comparison with symmetric numerical solutionsymmetric numerical solution
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
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 :
Normal stress distribution in numeric one-quadrant solution (stabilized regime), Re=0.1, Wi=4, the mesh is 2600 nodes
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
Same for the pressure
Vicinity of the central point:Vicinity of the central point:asymmetric caseasymmetric case
UCM model, Re = 0.01, Wi = 100, t = 3.55, mesh is 6400 nodes
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
Whole domain solution
UCM model, Re = 0.1, Wi = 4, t=2.7, mesh has 2090 nodes
Pressure distribution in the flow with Re = 3 and Wi = 4 at t = 3.5, mesh is 1200 nodes, Δt = 5·10-5
ConclusionsConclusions
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.
( )1inlet
tp tt
Tried lows of the pressure increase:
2
2( )1inlet
tp tt
( ) 1 tinletp t e
( ) 1inletp t
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
Convergence and quality of numerical procedure
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
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
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
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
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
Normal stress distribution in the flow with Re = 0.1 and Wi =4 at t = 3; mesh is 450 nodes, Δt= 5·10-5
( )1inlet
tp tt
Used lows of inlet pressure increase:
2
2( )1inlet
tp tt
( ) 1 exp( )inletp t t
( ) 1inletp t
The process of the flow reversal, Re = 0.1, Wi = 4, mesh is 675 nodes, t=1.7 ÷2.5
Flow picture for Re = 0.1, Wi = 4, the mesh is 4800 nodes
Flow picture for Re = 0.1, Wi = 4, the mesh is 4800 nodes
Flow picture for Re = 0.1, Wi = 4, the mesh is 4800 nodes
Extremely high Weissenberg numbers
Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
Flow picture for Re = 0.01, Wi = 100, the mesh is 4800 nodes
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