imece, november 15 th , 2004, anaheim, ca
DESCRIPTION
IMECE, November 15 th , 2004, Anaheim, CA. A D EPTH -A VERAGED M ODEL F OR E LECTROKINETIC F LOWS I N A T HIN M ICROCHANNEL G EOMETRY Hao Lin, 1 Brian D. Storey 2 and Juan G. Santiago 1 Mechanical Engineering Department, Stanford University - PowerPoint PPT PresentationTRANSCRIPT
STANFORD MICROFLUIDICS LABORATORY
A DA DEPTHEPTH-A-AVERAGED VERAGED M MODELODEL F FOROR EELECTROKINETIC LECTROKINETIC F FLOWS LOWS I IN N A T A THINHIN
MMICROCHANNEL ICROCHANNEL G GEOMETRYEOMETRY
Hao Lin,1 Brian D. Storey2 and Juan G. Santiago1
1. Mechanical Engineering Department, Stanford University2. Franklin W. Olin College of Engineering
IMECE, November 15th, 2004, Anaheim, CA
STANFORD MICROFLUIDICS LABORATORY
Motivation: Generalized EK flow with conductivity gradients
Field amplified sample stacking (FASS)
Electrokinetic instability (EKI)
Rajiv Bharadwaj
Michael H. Oddy
STANFORD MICROFLUIDICS LABORATORY
Previous WorkLin, Storey, Oddy, Chen & Santiago 2004, Phys. Fluids. 16(6): 1922-1935
– Instability mechanism: induced by bulk charge accumulation; stabilized by diffusion (Taylor-Melcher-Baygents)
– 2D and 3D linear analyses– 2D nonlinear computations
Storey, Tilley, Lin & Santiago 2004 Phys. Fluids, in press.
– Depth-averaged Hele-Shaw analysis (zeroth-order)Chen, Lin, Lele & Santiago 2004 J. Fluid Mech., in press
– Instability mechanism: induced by bulk charge accumulation; stabilized by diffusion (Taylor-Melcher-Baygents)
– Depth-averaged linear analyses– Convective and absolute instability
Experiment
2D Computation
STANFORD MICROFLUIDICS LABORATORY
Thin-Channel ModelPracticality Consideration
– 2D depth-averaged model significantly reduces the cost of 3D computation
– Model well captures the full 3D physics
Develop flow model for generalized electrokinetic channel flows
– Eletrokinetic instability and mixing
– Sample stacking– Other EK flows which involves
conductivity gradients
x
yz
dH
E
1
2
STANFORD MICROFLUIDICS LABORATORY
Full 3D Formulation (Lin et al.)
H. Lin, Storey, B., M. Oddy, Chen, C.-H., and J.G. Santiago, “Instability of Electrokinetic Microchannel Flows with Conductivity Gradients,” Phys. Fluids 16(6), 1922-1935, 2004. C.-H. Chen, H. Lin, S.K. Lele, and J.G. Santiago, “Convective and Absolute Electrokinetic Instabilities with Conductivity Gradients,” J. Fluid Mech., in press, 2004.
eveU HRaD
( ) 0E
21 ,eR
vt a
21 ( )Re E
v v v p v Et
0v
evU HRe
STANFORD MICROFLUIDICS LABORATORY
Depth Averaged ModelAsymptotic Expansion based on the aspect ratio = d/H which is similar to lubrication/Hele-Shaw theory
20 1 2 ...f f f f
Equations are depth-averaged to obtain in-plane (x,y) governing equations
1
1
1( , ) ( , , )2
f x y f x y z dz
Flows in the z-direction are integrated and modeled
2 427( , , ) ( , )
4 30 2eRa zx y z x y U z
x
u z
x
212 2eo
zU
u u
STANFORD MICROFLUIDICS LABORATORY
Depth Averaged Equations
2 221 2[ ( )]
105 e
e
H H H HRat Ra
u U U
( ) 0H H eo U u u
Convective dispersion: Taylor-Aris type
0H u
2 22 23H H H H H HRe pt
u
u u U u
Momentum: Darcy-Brinkman-Forchheimer
H. Lin, Storey, B., and J.G. Santiago, “A depth-averaged model for electrokinetic flows in a thin microchannel geometry,” to be submitted, 2004.
STANFORD MICROFLUIDICS LABORATORY
Field Amplified Sample Stacking (FASS)
+t > 0--
---
--- -
Stacked Analyte
-
t = 0
High Conductivity bufferLow Conductivity SampleHigh Conductivity buffer
---- --
- - - -+
- -UB US Oi E
ESEB
EEB
Rajiv Bharadwaj
STANFORD MICROFLUIDICS LABORATORY
1D Simplification (y-invariant)
( ) ( )I E x x constant
( ) [ ( ), ( )]eo
eo
u U u constant
U x u u E x x
22 2 2
2
1 2( , )
105 e
e
u Ra U x tt x Ra x x x
y
E
x
Dispersion effects include:
•EOF variation in x
•Vertical circulation in z
u ueo, 1 eo, 2
w
z
xHigh Conductivity Low Conductivity
STANFORD MICROFLUIDICS LABORATORY
FASS: Model vs DNS
2 427( , ) ( )
4 30 2eRa zx z x U z
x
DNS
Model
DNS
Model
Model w/o Dispersion
DNS
Model
Model w/o Dispersion
STANFORD MICROFLUIDICS LABORATORY
FASS: Model vs DNS
2 2
2 2 2
2 [ ( )]105
2 | |105
e H H
e
Ra
Ra
U U
Un n
DNS
Model
Model w/o Dispersion
Model RM
S
Time (s)
DNS
Model w/o Dispersion
STANFORD MICROFLUIDICS LABORATORY
Motivation: Electrokinetic Instability (EKI)
No gradient = 10
Stable, conductivity matched condition
50 m
Unstable, fluctuating concentrations in high-conductivity-gradient case
50 m
50 m
1 mm
(Michael H. Oddy)
(C.-H. Chen)(Rajiv Bharadwaj)
STANFORD MICROFLUIDICS LABORATORY
Linear Analysis: 2D vs 3D 3D Linear Analysis
Stable
Ecr,experiment ~ 0.3 kv/cm, Ecr,2D ~ 0.04 kv/cm, Ecr,3D ~ 0.18 kv/cm
H. Lin, Storey, B., M. Oddy, Chen, C.-H., and J.G. Santiago, “Instability of Electrokinetic Microchannel Flows with Conductivity Gradients,” Phys. Fluids 16(6), 1922-1935, 2004.
2D Linear Analysis
Stable
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EKI: Linear Analysis
Model
3D Linear
2 2 23( )H H H E H eo HRe pt
u
u u u u u
zeroth-order momentum1 ( )3eo Ep u u
STANFORD MICROFLUIDICS LABORATORY
EKI: Nonlinear Simulation
ExperimentModel
t = 0.0 s
t = 0.5 s
t = 1.5 s
t = 2.0 s
t = 2.5 st = 3.0 s
t = 4.0 s
t = 5.0 s
t = 1.0 s
STANFORD MICROFLUIDICS LABORATORY
Conclusions and Future Work
Developed depth-averaged model for general EK flows in microchannelsModel validated with DNS and experimentsFuture work:– Modeling and optimization of realistic FASS
applications
– Modeling and optimization of EKI mixing