Transcript Document

IMECE, November 15th, 2004, Anaheim, CA
A DEPTH-AVERAGED MODEL FOR
ELECTROKINETIC FLOWS IN A THIN
MICROCHANNEL GEOMETRY
Hao Lin,1 Brian D. Storey2 and Juan G. Santiago1
1. Mechanical Engineering Department, Stanford University
2. Franklin W. Olin College of Engineering
STANFORD MICROFLUIDICS LABORATORY
Motivation: Generalized EK flow with
conductivity gradients
Field amplified sample stacking (FASS)
Rajiv Bharadwaj
Electrokinetic instability (EKI)
Michael H. Oddy
STANFORD MICROFLUIDICS LABORATORY
Previous Work
Lin, Storey, Oddy, Chen & Santiago 2004, Phys.
Fluids. 16(6): 1922-1935
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–
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Instability mechanism: induced by bulk charge
accumulation; stabilized by diffusion (Taylor-MelcherBaygents)
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-MelcherBaygents)
Depth-averaged linear analyses
Convective and absolute instability
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Experiment
2D Computation
Thin-Channel Model
Practicality 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
z
y
H
x
d
s2
E
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s1
Full 3D Formulation (Lin et al.)
s
1 2
 v s 
 s,
t
Rae
Rae 
U ev H
D
 (s E )  0
v
1
 v  v 
( p   2v   E E )
t
Re
Re 
U ev H

 v  0
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.
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Depth Averaged Model
Asymptotic Expansion based
on the aspect ratio d = d/H
which is similar to
lubrication/Hele-Shaw theory
f  f0  d f1  d 2 f 2  ...
Raed 2 s
s ( x , y , z )  s ( x, y ) 
U
4
x
Flows in the z-direction are
integrated and modeled
 1 z2 
u  u eo  U   
2 2 
z
u
 7
z4 
2
  z  
2
 30
x
Equations are depthaveraged to obtain in-plane
(x,y) governing equations
1
1
f ( x, y )   f ( x, y, z )dz
2 1
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Depth Averaged Equations
Convective dispersion:
Taylor-Aris type
s
1
2
2
2 2
 u  Hs 
Hs 
Rae d  H  [ U( U   H s )] 
t
Rae
105

 H  (s H )  0

U  u  ueo
Momentum: Darcy-BrinkmanForchheimer
u


2
2
2
ReH d 
 u   H u    H p   H  H   3U  d  H u 
 t

2
 H  u  0
H. Lin, Storey, B., and J.G. Santiago, “A depth-averaged model for electrokinetic flows in a thin
microchannel geometry,” to be submitted, 2004.
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Field Amplified Sample Stacking (FASS)
t=0
- -- - - - - -
+
High Conductivity buffer
E
t>0
+
Low Conductivity Sample
High Conductivity buffer
ES
EB
UB
-
US
-
EB
--- -- - -Stacked Analyte
Rajiv Bharadwaj
STANFORD MICROFLUIDICS LABORATORY
iO  s E
-
1D Simplification (y-invariant)
E
y
x
 
2 s  
u


Rae d
 U ( x, t )
 
2

t
x
Rae  x
105
x 
x 
s
s
1  s
2
2
2
2
Dispersion effects include:
•EOF variation in x
I  E ( x )s ( x )  constant
•Vertical circulation in z
ueo, 1
u  U  ueo  constant
 U ( x )  u  ueo [ E ( x ), s ( x )]
ueo, 2
z
w
x
High Conductivity
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Low Conductivity
FASS: Model vs DNS
Model
DNS
DNS
Model w/o Dispersion
Model
Model
DNS
Model w/o Dispersion
Raed 2 s
s ( x, z )  s ( x ) 
U
4
x
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 7
z4 
2
  z  
2
 30
FASS: Model vs DNS
DNS
Model
Model w/o
Dispersion
sRMS
2
Rae2d 2 H  [U( U   H s )]
105
2
2 2  
2 s 

Rae d
| U |

105
n 
n 
Model w/o
Dispersion
Model
DNS
Time (s)
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Motivation: Electrokinetic Instability (EKI)
50 m
1 mm
Stable, conductivity
matched condition
50 m
50 m
Unstable, fluctuating
concentrations in highconductivity-gradient
case
(Rajiv Bharadwaj)
(Michael H. Oddy)
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No gradient
g = 10
(C.-H. Chen)
Linear Analysis: 2D vs 3D
2D Linear Analysis
3D Linear Analysis
Stable
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.
STANFORD MICROFLUIDICS LABORATORY
EKI: Linear Analysis
zeroth-order momentum
1
u  ueo  (p   E )
3
3D Linear
Model
ReH d
2
  u  u   u    p      3( u  u )  d 2 2 u

H 
H
E H
eo
H
 t

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EKI: Nonlinear Simulation
t = 0.0 s
t = 0.5 s
t = 1.0 s
t = 1.5 s
t = 2.0 s
t = 2.5 s
t = 3.0 s
t = 4.0 s
t = 5.0 s
Model
Experiment
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Conclusions and Future Work
Developed depth-averaged model for general
EK flows in microchannels
Model validated with DNS and experiments
Future work:
–
Modeling and optimization of realistic FASS
applications
–
Modeling and optimization of EKI mixing
STANFORD MICROFLUIDICS LABORATORY