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M. S. N. Oliveira1, F. T. Pinho2, M. A. Alves1
Elastic-driven instabilities in
microfluidic flow-focusing devices
1: Departamento de Engenharia Química, CEFTFaculdade de Engenharia da Universidade do PortoRua Dr. Roberto Frias, 4200-465 Porto, Portugale-mail: {mmalves, monica.oliveira}@fe.up.ptweb: http://www.fe.up.pt/~ceft
2: Departamento de Engenharia Mecânica, CEFTFaculdade de Engenharia da Universidade do PortoRua Dr. Roberto Frias, 4200-465 Porto, Portugale-mail: [email protected]: http://www.fe.up.pt/~ceft
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Instabilities at the microscale: Cross-Slot
� Numerical� Experimental
PAA Boger fluid:
Re < 10-2 (De=4.5)Newtonian:
Re < 10-2
UCM model
2D, Creeping Flow
650 µm
500 µm
650 µm
500 µm
2
Newtonian De=0.3
De=0.32 De=0.4
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Instabilities at the microscale: T-Channel
Without Cavity
Pinned stagnation point
0.075 wt. % PEO in glycerol/water solution (60/40 wt. %)
3
With Cavity
Free stagnation point
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� Operational Variables
� Dimensionless Variables
Flow focusing geometry
Q1, Q2
Q3 = 2 x Q2+Q1
FR = Q2/Q1
Q
Outflow
Q
Q3
D
L
4
� Channel dimensions:
� kept constant for all experiments and numerical calculations
� Equal dimensions for all inlet/outlet arms
Cross-slot geometrywith 3 inlets and 1 outlet
VR = U2/U1 (= FR)
Q1
Inflow
Q2
Inflow
Inflow
Q2
h
λ= 2UDe
D
ρη
= 2
0
U DRe
= DeEl
Re
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Fabrication process: soft lithography
100 µµµµm
Outflow
Depth = 100µµµµm
1. Silicon Wafer
2. Spin coat photoresist SU-8and prebake
3. Spin coat barrier coat(CEM-BC7.5) and contrast enhancer (CEM 388SS) (vertical walls).
4. Chrome Mask over coated wafer
UV Light
� Microchannels:
� Planar geometry� Square cross-section� Nearly vertical walls� Accuracy of channels to within 5%
5
Inflow
100 µµµµm
Inflow
Inflow
Outflow5. UV Exposure – cross-link
SU-8
6. Wash barrier coat and contrast enhancer
7. Post-bake and develop SU-8
8. Pour PDMS over substrate and cure (80oC, 25 mins)
9. Peel off substrate
10. Seal with glass slide covered with PDMS
Y. Xia and G. M. Whitesides, Annu Rev. Mater. Sci. 1998, 28:153-84
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Experimental set-up
Visualizations
Microscope:Leica, DMI LED
Camera:Leica, DFC 350X
Objective lens:10× objectiveNA=0.3 (δz=30µm)
Illumination:Mercury lamp (λ=532nm)
Filter Cube:Emission filter: BP 530-545 nm Dichroic mirror: 565 nmBarrier filter: 610-675 nm
Hamilton gastight syringes
6
Barrier filter: 610-675 nm
Tracers:1.1µm Nile red fluorescentparticles (Ex/Em: 520/580 nm)
Additives:SDS (0.1 wt.%)
Flow
Syringe pump:CETONI, neMESYSwith 3 independent modules
Flow rate:0.001 ≤ Q ≤ 3 ml/h
Syringes:Hamilton Gastight syringes
Three module Syringe pump
Inflow
Outflow
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Shear Rheology
� Viscoelastic Fluid:
Aqueous solution of PAA (Mw=18x106 g/mol)
� PAA 125 + NaCl125 ppm PAA+ 1% NaCl (wt) + 0.1% SDS (wt)
� Newtonian Fluid:
Steady Shear
Anton Paar, model Physica MCR 301
Cone-and-plate geometry (d=75mm, 1º)
T = 20ºC
7
Newtonian Fluid:
� WaterWater+ 0.1% SDS (wt)
Onset of inertialinstabilities
Based on 20× the minimum measurable torque (1×10-7 Nm)
.
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Extensional Rheology
PAA 125 + NaCl
Zero-shear rate viscosity
η0 [Pa s] 0.00131
CaBER relaxation time
λCaBER [ms] 12.4±0.2
Density ρ [kg/m3] 1005
Fluid properties
T = 20ºC
Extensional RheologyHaake CaBER 1, Thermo Electron Corporation
(Dp = 6 mm)
Aspect ratios: Λi = Di/Dp=0.33
Λf = Df/Dp=1.56
Imposed strain: ε = ln(Λf/Λi) = 1.53
8
0
exp3
D t
D λ ∝ −
Minimum resolvable diameter
1
3λ
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� Isothermal Incompressible Flow
� Constitutive Equations
Governing equations
0∇ ⋅ =u Conservation of mass
Conservation of momentum
2sp η−∇ + ∇ ⋅ + ∇ =τ u 0
9
� Constitutive Equations
1 tr( ) 2 pp
λε λ ηη
∇ + + =
τ τ τ D
2µ=τ D
Simplified PTT model
Newtonian fluid
ε= 0
2 pλ η∇
+ =τ τ D
λ = 0
Oldroyd-B model
2λ η∇
+ =τ τ Dηηηηs= 0
UCM model
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� Finite Volume Method, using a time-marching algorithm*
� Governing equations discretized in time over a small time step, δt� Differential equations integrated over control volumes, CV
*Oliveira et al., JNNFM,79 (1998) 1-43
� Log-conformation approach** to solve the equivalent form of the constitutive equation containing an evolution equation of the conformation tensor.
**Fattal and Kupferman, JNNFM, 123 (2004) 281-285
Afonso et al., JNNFM, submitted (2008)
� Implicit first-order Euler scheme for time-derivative discretization.
Numerical method
10
� 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
� 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.
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Numerical mesh and boundary conditions
� Boundary Conditions:
� INLET� Fully-developed velocity profile� Null stress components
� OUTLET� Vanishing streamwise gradients
of velocity and stress components� Constant pressure gradient
� WALLS� No-slip conditions
� Mesh Characteristics:
� Two-dimensional� Structured, orthogonal and non-uniform� Dimensions equal to the experiments
U3
11
Geometry NCells ∆xmin, ∆ymin
∆zmin
2D 23001 0.02D
Standard Mesh used
(more refined meshes also used)U2
U2
U1
y
x(0,0)
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Converging Flow
D’3
D1
VR=1 VR=2 VR=10
Water
12
� Two opposing lateral streams shape a third inlet stream
� Converging flow region
� Separation streamlines naturally evolve to a nearly hyperbolic shape
� Hencky strain controlled by operating parameters:
� Approximately constant strain rate at the centerline
( )ε = = + ⋅
1'3
3ln ln 1 2
2H
DVR
D
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Converging Flow
D’3
D1
VR=1 VR=2 VR=10
Water
13
� Two opposing lateral streams shape a third inlet stream
� Converging flow region
� Separation streamlines naturally evolve to a nearly hyperbolic shape
� Hencky strain controlled by operating parameters:
� Approximately constant strain rate at the centerline
( )ε = = + ⋅
1'3
3ln ln 1 2
2H
DVR
D
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De = 0.1UCM model
2D, Creeping Flow
Instabilities at the microscale
Ext.
14
Onset ofasymmetry
De = 0.34
� For low VR: asymmetric flow not observed
� For high VR: symmetric � asymmetric � oscillatory
� The critical Dec becomes approximately constant: Dec ~ 0.33
� Bistable flow
Shear
Rot
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Effect of VR on the degree of asymmetry
3
* W EF FF
F
−=
2W 2E
1
3
Effect of VRUCM Model2D, Creeping Flow
15
� Bistable Flow
� For high VR:
� Dec independent of VR
� Evolution of asymmetry independent of VR
� Supercritical pitchfork bifurcation:
F*=0 (symmetric flow)F* progressively deviates from F* = 0as the flow becomes increasingly asymmetric.
= −
= −
*
0.59 0.33
cF a De De
De
Oliveira et al., JNNFM, 160 (2009) 31-39
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Effect of β on the degree of asymmetry
3
* W EF FF
F
−=
Effect of ββββOldroyd-B Model2D, Creeping Flow
η ηβη η η
≡ =+0
s s
s P
16
2W 2E
1
3
F*=0 (symmetric flow)F* progressively deviates from F* = 0as the flow becomes increasingly asymmetric.
� β has a stabilizing effect on the flow:
� Increase in the critical Dec
� For β ≥ 6/9, no steady asymmetry is observed.
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Effect of ε on the degree of asymmetry
3
* W EF FF
F
−=
2W 2E
3
Effect of εεεεSPTT Model2D, Creeping Flow
17
F*=0 (symmetric flow)F* progressively deviates from F* = 0as the flow becomes increasingly asymmetric.
� Increase in the critical Dec
� Decrease in the degree of asymmetry (ε < 0.04)
� For ε ≥ 0.04, the steady asymmetry no longer observed. The flow transitions directly from steady symmetric to unsteady.
1
Oliveira et al., JNNFM, 160 (2009) 31-39
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Axial Normal Stress Profiles
2W 2E
Axial normal stress profiles along the centerline (x = 0)
18
Oliveira et al., JNNFM, 160 (2009) 31-39
• Similar levels of normal stresses achieved near critical conditions.
• Extensional properties decisive for the onset of flow asymmetry.
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Viscoelastic Fluid: PAA125+NaCl
Vis
coela
stic
Q1 = 0.01 ml/h
increasing Q2
19
Q2 = 0.05 ml/h, VR = 5 Q2 = 0.2 ml/h, VR = 20 Q2 = 0.5 ml/h, VR = 50
Re = 0.23, De = 0.38 Re = 0.87, De = 1.41 Re = 2.15, De = 3.479
Vis
coela
stic
Symmetric Steady Asymmetric Unsteady 3D
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Viscoelastic Fluid: PAA125+NaCl
Vis
coela
stic
Q2 = 0.05 ml/h, VR = 5 Q2 = 0.1 ml/h, VR = 10 Q2 = 0.2 ml/h, VR = 20
Re = 0.23, De = 0.38 Re = 0.45, De = 0.723 Re = 0.87, De = 1.41
Q1 = 0.01 ml/h
20
Vis
coela
stic
New
tonia
n
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Viscoelastic Fluid: PAA125+NaCl
Vis
coela
stic
Experim
enta
lQ1 = 0.01 ml/h
21
UCM
2D
Calc
ula
tions
Q2 = 0.05 ml/h, VR = 5 Q2 = 0.1 ml/h, VR = 10 Q2 = 0.2 ml/h, VR = 20
Re = 0.23, De = 0.38 Re = 0.45, De = 0.723 Re = 0.87, De = 1.41
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Viscoelastic Model
UCM
2D
Calc
ula
tions
Unsteady 3D
Q1 = 0.01 ml/h
22
2D
Calc
ula
tions
Old
royd-B
2D
Calc
ula
tions
Q2 = 0.05 ml/h, VR = 5 Q2 = 0.2 ml/h, VR = 20 Q2 = 0.35 ml/h, VR = 35
Re = 0.23, De = 0.38 Re = 0.87, De = 1.41 Re = 0.87, De = 1.41
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Experimental Flow Map
ExperimentalPAA 125 + NaCl
23
AsymmetricFlow
NumericUCM Model2D, Creeping Flow
UnstableFlow
SymmetricFlow
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Summary
� Elastically-driven asymmetries at the microscale.
� Symmetric-breaking bifurcation at high De and VR.
� Unsteady 3D flow above a critical De: useful for mixing purposes.
24
� Numerical 2D calculations reproduce qualitatively the experimental results.
Ongoing work:� Experimental: µPIV
� Numerical: 3D simulations
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Acknowledgements
� Collaborations:� G. H. McKinley, MIT
� P. J. Oliveira, UBI
� R. J. Poole, U Liverpool
� P. C. Sousa, FEUP
� Funding:
25
� Fundação para a Ciência e a Tecnologia and FEDER (Portugal) under projects:
� PTDC/EQU-FTT/71800/2006
� PTDC/EQU-FTT/70727/2006