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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