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Instability of electro-osmotic channel flow with streamwise conductivity gradients
Brian StoreyJose Santos
Franklin W. Olin College of EngineeringNeedham MA
“Electrokinetic instability”2003 Experiments (Mike Oddy of J. Santiago’s group)
1 mm
V
High conductivity fluid
Low conductivity fluid
Model comparison
ExperimentComputation
t = 0.0 s
t = 0.5 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
t = 1.0 s
Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004Storey, Tilley, Lin. Santiago, Phys Fluids 2005Lin, Storey, Santiago, JFM 2008
Hoburg and Melcher (1976)
Unstable EHD in microfluidics
Posner, Santiago, JFM 2006
Chen, Lin, Lele, Santiago JFM 2005
Baygents, Baldessari PoF1998 ElMochtar, Aubry, Batton, LoC 2003
Storey, PhysD 2005 Boy , Storey, PRE 2007
Field Amplified Sample Stacking (FASS)
+t > 0-
-
---
--- -
Stacked Analyte
-
t = 0
High Conductivity bufferLow Conductivity SampleHigh Conductivity buffer
---- --
- - - -+
- -UB US Oi E
ESEB
EEB
Electrokinetic dispersion
•Electroosmotic velocity depends upon the electric field•Electric field is high when conductivity is low•Low conductivity = high EO velocity
High conductivity, E1
ueof, 1 ueof, 2
High conductivity, E Low conductivity, E2
ueof, 1 ueof, 2
1
ueof, 1
High conductivity, E
Red; cond =10 Blue; cond =1
Questions• Can instability and dispersion interact in “stacking”
applications? • Does instability influence stacking efficiency?
Lin, Storey, Santiago, JFM 2008
Generalized governing equations two symmetric species, dilute
Convective diffusion (+) and (-)
0c
c v z Fc E D ct
Convection Electromigration Diffusion
( ) ~ ( )E F z c z c c c Charge Density and Gauss Law
0( )r EE
0v
2( ) E
vv v p v E
t
Navier-Stokes Equations
Note (c+-c-)/(c++c-)~10-5
Electro-neutral bulk assumptionThin double layer approx.
0:Sub
0 :Add
0
0
utral)(Electrone
Ec
cDcvt
c
cDEFccvt
c
cDEFccvt
c
ccc
Final eqns & mechanism for flow
( ) 0E
21,
eRv
t a
21( )
Re E
vv v p v E
t
0v
/EE
0 EE
HS electro-osmotic slip boundary conditions Euslip
Dimensionless parameters
Re evU H
eve
U HRa
D
ev
eov U
UR
L
H
low
high
H
Lsample
Electric Rayleigh number
Reynolds number
Channel aspect ratio
Ratio of electro-osmotic to electroviscous velocity
Electrical conductivity ratio
Ratio of sample length to channel height
HE
U ev
2
Unstable flowE=25,000 V/m, Conductivity ratio=10
Posner, Santiago, JFM 2006
Observations
•“Shock” at the leading edge of the sample.•Vertical velocity at the channel walls pumps fluid toward the centerline.•Unstable flow only inside the sample region.
Stability measureMaximum vertical V
Stability measure as function of applied field
Unstable E field
Role of electric body force
No electro-osmotic slip (zeta=0)E=10,000 V/m (much lower field than with EO)
Phase diagram
Phase diagram
DRaRv
22
D
HERa
22
lo
lohi
1
Conclusions
• Instability can occur in FASS geometry.
• Simple stability map can be used to predict stability within reason.
• Phenomena seems generic when you drive low conductivity into high conductivity.
• Instability doesn’t impact rate of dispersion that much.
• Preliminary – instability doesn’t seem to impact sample concentration as
much as you might think.