Elastic flow asymmetries in microfluidic flow-focusing devicesfpinho/pdfs/MSNO_Liverpool2008.pdf ·...

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Complex Flows of Complex Fluids, Liverpool 2008

a) Dep. de Engenharia Química, CEFT, Faculdade de Engenharia da Universidade do Porto, 4200-465 Porto, Portugal. b) CEFT, Faculdade de Engenharia da Universidade do Porto, 4200-465 Porto, Portugal. c) Universidade do Minho, 4704-553 Braga, Portugal. d) Dep. of Engineering, University of Liverpool, Liverpool, L69 3GH, United Kingdom. e) Dep. de Engenharia Electromecânica, Universidade da Beira Interior, 6201-001 Covilhã, Portugal

Mónica S. N. Oliveira a) Fernando T. Pinho b), c)

Rob J. Pooled) Paulo J. Oliveira e) Manuel A. Alves a)

Elastic flow asymmetries in microfluidicflow-focusing devices


Extensional Flows

• Inkjet printing, spray drying, precise reactant delivery

• High viscosity solvents, polymer melts

• Large range of polymer concentrations

low Rehigh De

Macro-scale

Micro-scale

• Low viscosity solvents• Low concentrations of high

MW polymer

moderate Remoderate - high De

• Microfluidics – DNA stretchingWong et al. (2003)

• Lab-on-a-Chip- Complex microfluidic flows with non-Newtonian working fluid

From Macro to Micro Scales

• Polymer extrusion processes

• Extensional viscosity of dilute polymer solutions difficult to measure

2


Converging flows in microchannels

Abrupt contraction Hyperbolic contraction

• Construct a micro-fluidic based ‘rheometer-on-a-chip’

• True Planar Elongational Flow

γ

ε

d const.d

zVz

ε = =

James and Saringer, J. Fluid Mech., 97 (1980)James, AIChE J., 37 (1991)

Ideal Axial Velocity Profiles along the centerline

- high deformation rates- absence of inertia- low viscosity complex fluids


Flow in the hyperbolic channel

εH = 2 εH = 3Streaklines

Vector Map εH = 3

• Lower Strain Shorter channel– Entrance effects extend to large portion of the

channel • Higher Strain Longer channel:

– Contractions entrance and exit effects confined to small fractions of the channel

– Region of approximately constant strain rate– Shear significant in the downstream section of

the contraction

Newtonian and PEO solutions

3


VR = U2 / U1

FR = D2 U2 / D1 U1

Flow-focusing device

WR = D2 / D1

Cross-slot geometry with threeinlets and one outlet.

Operational parameters

0.3 ≤ WR ≤ 5

1 ≤ VR ≤ 500

Values and ranges for this studyU2U2

U1

U3

D2D2

D1

D1

L=30 D1

L

Deborah number

De = λ U2 / D1


Governing equations

2λ η∇

+ =τ τ D

2μ=τ D

0∇⋅ =u Conservation of mass

Conservation of momentum

• Isothermal Incompressible Flow

UCM model

Newtonian fluid

• Constitutive Equations

Creeping flow

Re = 0

0p−∇ +∇ ⋅ =τ

Isolate viscoelastic effects

4


• Finite Volume Method, using a time-marching algorithm*: *Oliveira et al., JNNFM,79 (1998) 1-43– Governing equations discretized in time over a small time step, δt– Differential equations integrated over control volumes, CV

• Implicit first-order Euler scheme for time-derivative discretization.• Central differences for discretization of diffusive terms.• CUBISTA** high-resolution scheme for discretization of the advective terms of the momentum

equations. **Alves et al., Int. J. Num. Methods in Fluids, 122 (2003) 47-75

• Governing equations are transformed into algebraic form and solved sequentially for each dependent variable with conjugate gradient solvers:

– Dependent Variables (p, u, τ ) evaluated at the cells center

• Pressure-velocity and velocity-stress coupling ensured at the CV faces –SIMPLEC algorithm******Patankar and Spalding, Int. J. Heat and Mass Transfer, 15 (1972) 1787–806

• The ensemble of all control volumes defines the computational mesh.

Numerical Method


Mesh Characteristics & Boundary Conditions

Mesh Characteristics

• Two-dimensional• Full Domain

• Structured, orthogonal and non-uniform• Δxmin= Δymin= 0.02 D1• 5 blocks

Boundary Conditions

• INLET -Fully-developed velocity profiles-Null stress components

• OUTLET -Vanishing streamwise gradients of velocity and stress components-Constant pressure gradient

• WALLS No-slip conditionsx = 0

(0,0) x

y

13965154711647523001355514860374205

0.30.40.51.02.03.05.0

NCellsWR = D2/D1

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• Separation streamlines naturally evolve to a nearly hyperbolic shape

• Hencky strain controlled by operating parameters:

• Approximately constant strain rate at the centerline

Converging Flow – varying VR

Re = 0WR = 1 (D2 = D1)De = λ U2 /D1 = 0.2

( )1'3

3ln ln 1 22H

D VR WRD

ε⎛ ⎞ ⎡ ⎤= = + ⋅ ⋅⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

-2.0 -1.0 0.0 1.0 2.0 3.0

Dimensionless Axial Position D

imen

sion

less

Axi

al V

eloc

ity VR = 1VR = 20

1

2

3

4

5

-2 -1 0 1 2 3

VR = 1

VR = 20

D’3

D1


Converging Flow – Varying WR

Re = 0FR = 1 WR = 1

De = 0 De = 0.31

WR = 0.5De = 0 De = 0.38

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0.0

1.0

2.0

3.0

4.0

5.0

6.0

-2.0 -1.0 0.0 1.0 2.0 3.0

Axial Position

Axi

al V

eloc

ity

De = 0De = 0.01De = 0.10De = 0.20De = 0.31


Re = 0FR = 1 WR = 1

De = 0 De = 0.31

Velocity undershoot characteristic of diverging streamlines

Velocity overshoot

WR = 0.5De = 0 De = 0.38

WR = 1


0.0

1.0

2.0

3.0

4.0

5.0

6.0

-2.0 -1.0 0.0 1.0 2.0 3.0

Axial Position

Axi

al V

eloc

ity

De = 0De = 0.01De = 0.10De = 0.20De = 0.31

0.0

1.0

2.0

3.0

4.0

5.0

6.0

-2.0 -1.0 0.0 1.0 2.0 3.0

Axial Position

Axi

al V

eloc

ity

De = 0De = 0.02De = 0.20De = 0.30De = 0.38


• Vary the WR Vary the Hencky strain:

• For the same FR, the geometry with lower WR yields:– a higher strain rate,– a more pronounced overshoot.

Re = 0FR = 1 WR = 1

WR = 0.5

De = 0 De = 0.31

De = 0 De = 0.38

( )1'3

3ln ln 1 22H

D VR WRD

ε⎛ ⎞ ⎡ ⎤= = + ⋅ ⋅⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠

WR = 1

WR = 0.5

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Effect of De on the Flow Patterns

De = 0 De = 0.2

De = 0.40 De = 0.44

De = 0.34Re = 0WR = 1 (D2 = D1)VR = 20 (U2 = 20 U1)0 ≤ λ ≤ 0.022 s

FLOW BECOMES

TIME-DEPENDENT

De ≥ 0.47

De = 0.2

De = 0.44

De = 0.2

Onset of asymmetry


Axial Velocity Profiles (y = -0.5)

-15

-10

-5

0

5

10

15

-0.5 -0.3 -0.1 0.1 0.3 0.5

Lateral Position

Axi

al V

eloc

ity

De = 0.2De = 0.32De = 0.34De = 0.44

Re = 0WR = 1 (D2 = D1)VR = 20 (U2 = 20 U1)0 ≤ λ ≤ 0.022 s

Flow is asymmetric and bi-stablebeyond De = 0.34

• Axial Velocity Profiles at y = -0.5

• Increasing De Flow becomes increasingly asymmetric until it becomes time-dependent

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Axial Velocity Profiles along the centerline

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

-2.0 -1.0 0.0 1.0 2.0 3.0

Dimensionless Axial Position

Dim

ensi

onle

ss A

xial

Vel

ocity

De = 0De = 0.20De = 0.32De = 0.40De = 0.44

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-2.0 -1.5 -1.0 -0.5 0.0


Dim

ensi

onal

ess

Axi

al V

eloc

ity De = 0De = 0.20De = 0.32De = 0.40De = 0.44

Re = 0WR = 1 (D2 = D1)VR = 20 (U2 = 20 U1)0 ≤ λ ≤ 0.022 s

• Fully developed flow upstream and downstream of the converging region.

• Fluid accelerates as it approaches the converging region.

• Fluid starts accelerating further upstream for high De.• For higher De the fluid starts decelerating earlier and

the flow takes longer to fully develop.


-100

-80

-60

-40

-20

0

20

40

-2.0 -1.0 0.0 1.0 2.0 3.0


Dim

enso

nles

s P

ress

ure

De = 0De = 0.20De = 0.32De = 0.40De = 0.44

Pressure and stress profiles

0

10

20

30

40

50

60

70

80

90

-2.0 -1.0 0.0 1.0 2.0 3.0


Dim

ensi

onle

ss N

orm

al S

tress De = 0

De = 0.20De = 0.32De = 0.40De = 0.44

Re = 0WR = 1 (D2 = D1)VR = 20 (U2 = 20 U1)0 ≤ λ ≤ 0.022 s

• Increasing De τ22 increases• When the flow is symmetric, the extra pressure

drop increases significantly with De.• After the asymmetry, there is no longer a strong

increase in the extra pressure drop.

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Comparison to forced symmetry

De = 0.44De = 0.2• For De = 0.2 Profiles are identical• For De = 0.44 Profiles are different• Excess pressure drop (relative to fully

developed flow) is larger for the symmetric case: ΔP = ΔPtotal - ΔPfd

• Flow becomes asymmetric to reduce pressure losses and dissipate less energy.

-80.0

-60.0

-40.0

-20.0

0.0

20.0

40.0

-2.0 -1.0 0.0 1.0 2.0 3.0

Dimensionless Axial Position D

imen

sion

less

Pre

ssur

e

Imposed Symmetry

Asymmetric

De = 0.2

De = 0.44

Full Domain


Flow Map

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250

VR = FR

Crit

ical

De

Symmetric Flow

Asymmetric Flow

Oscillatory Flow0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250

VR = FR

De

SymmetricAsymmetricTime-dependent

• For low FR asymmetric flow not observed• At certain values of De, increasing VR causes the

flow to change from:oscillatory symmetric asymmetric

• For high VR the critical Deborah number becomes approximately constant: Dec ~ 0.335

Re = 0WR = 1 (D2 = D1)1 ≤ VR ≤ 200 0 ≤ λ ≤ 0.4 s

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Effect of WR on the onset of asymmetry

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 50 100 150 200 250

FRC

ritic

al D

e

WR = 0.4WR = 0.5WR = 1WR = 2

WR = 0.5

WR = 2

De = 0.36 De = 0.37

De = 0.37De = 0.36

• Asymmetries also develop for other WR• At high FR Dec becomes constant.• Lowest Dec occurs for WR = 1.• Increasing or decreasing WR from 1 causes an

increase in Dec.• At very low or very high WR, no asymmetry is seen.

Re = 0VR = 100


Effect of WR on the onset of asymmetry

WR = 0.5

WR = 2

De = 0.36 De = 0.37

De = 0.37De = 0.36

( ) 2c lnDe a WR b= +⎡ ⎤⎣ ⎦

Critical De in the limit of high FRRe = 0VR = 100

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Vortex formation near centerline

De = 0 De = 0.2

De = 0.37 De = 0.52De = 0.32

Vortexappearance

Re = 0WR = 0.5 (D2 = 0.5 D1)VR = 200 (U2 = 200 U1)0 ≤ λ ≤ 0.0026 s


VR = 1

Effect of VR on Newtonian Fluid Flow Patterns

VR = 100 VR = 159

VR = 1000

VR = 20

VR = 200 VR = 106 VR = ∞

Re = 0De = 0WR = 1 (D2 = D1)FR = VR1 ≤ VR ≤ 106

Vortexformation

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Comparison with cross slot flow

VR = ∞

Re = 0De = 0WR = 1 (D2 = D1)

VR = 200 VR = 1000

Present Geometry – 3 inlets, 1 outlet

Typical Cross Slot – 2 inlets, 2 outlets

VR ↑↑

VR ↑↑


Onset of Vortex appearance - Effect of WR

WR = 0.3

VR = 100VR = 220VR = 400

VR = 400VR = 450VR = 500

WR = 3

• Transition envelop exhibits a minimum.

• Two competing effects:

- Increase of flow rate with WR.

- Decrease in streamline curvature with WR.

Crit

ical

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Streamline curvature

WR = 0.1FR = 10

WR = 0.5FR = 50

WR = 1FR = 100

Re = 0De = 0VR = 1000.1 ≤ WR ≤ 510 ≤ Q2 ≤ 500

WR = 2FR = 200

WR = 5FR = 500

WR = 3FR = 300


Effect of WR on Axial flux

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

-0.5 -0.3 -0.1 0.1 0.3 0.5

Lateral Position

Axi

al F

lux

. WR

WR = 0.3WR = 1WR = 2WR = 5

Measure Axial Flux

(upward and downward)

along y = -D2 / 2

Scaled downward flux is changing

non-monotonically with WR

Re = 0De = 0VR = 1000.3 ≤ WR ≤ 530 ≤ Q2 ≤ 500

Sca

led

Axi

al F

lux

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Relationship between flux and vortex appearance

VR = 100

Onset of vortex appearance

Variation of scaled downwardflux at constant VR

• Both curves show similar qualitativebehaviour.

• Higher downward flux, favours vortexformation.

Vortices

NO Vortices

Crit

ical

Scal

ed D

ownw

ard

Flux


Summary

• Numerical study of viscoelastic flow (UCM) through a 2D flow-focusing device.

• Hencky Strain and strain rate may be controlled by changing operational parameters.

• Flow-focusing geometry can generate nearly constant extensional rate along the centerline.

• At high De, a purely elastic asymmetry is observed:– For high VR, the critical De for the onset of asymmetry is approximately constant;– Depends non-trivially on WR.

• For high VR, vortices are generated near the centerline, even for Newtonianfluids.

• Vortex onset depends on streamline curvature and is related to the downward flux through the y = -D2/2 plane.

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Ackowledgements

• Fundação para a Ciência e a Tecnologia (Portugal):– POCI/EME/58657/2004– PPCDT/EME/59338/2004– PTDC/EQU-FTT/71800/2006

• CRUP/BC

• Miguel Jorge and António Martins

Elastic flow asymmetries in microfluidic flow-focusing devicesfpinho/pdfs/MSNO_Liverpool2008.pdf ·...

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