Transcript Document

Instability of electro-osmotic channel flow
with streamwise conductivity gradients
Brian Storey
Jose Santos
Franklin W. Olin College of Engineering
Needham MA
“Electrokinetic instability”
2003 Experiments (Mike Oddy of J. Santiago’s group)
High conductivity fluid
1 mm
Low conductivity fluid
V
Model comparison
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
Computation
Experiment
Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004
Storey, Tilley, Lin. Santiago, Phys Fluids 2005
Lin, Storey, Santiago, JFM 2008
Hoburg and Melcher (1976)
Unstable EHD in microfluidics
Baygents, Baldessari PoF1998
Chen, Lin, Lele, Santiago JFM 2005
Posner, Santiago, JFM 2006
Storey, PhysD 2005
ElMochtar, Aubry, Batton, LoC 2003
Boy , Storey, PRE 2007
Field Amplified Sample Stacking (FASS)
t=0
- -- - - - - -
+
High Conductivity buffer
Low Conductivity Sample
High Conductivity buffer
ES
E
t>0
-
EB
+
UB
-
US
--- -- - -Stacked Analyte
-
EB
iO   E
-
Electrokinetic dispersion
ueof, 1
ueof, 1
ueof, 2
High conductivity, E1
Low conductivity, E
2
High conductivity, E1
•Electroosmotic velocity depends upon the electric field
•Electric field is high when conductivity is low
•Low conductivity = high EO velocity
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 (-)
c
  c v   z Fc E  D c   0
t
Convection
Charge Density and
Gauss Law
Navier-Stokes
Equations
Note (c+-c-)/(c++c-)~10-5
Electromigration
Diffusion
 E  F ( z c  z c ) ~ (c  c )
 ( r 0 E )   E
 v  0
 v

 (v )v   p  2v   E E
 t


Electro-neutral bulk assumption
Thin double layer approx.
c   c   c (Electroneutral)



c

   v c  FcE  Dc  0
t



c
    v c  Fc E  Dc   0
t


c


    v c  Dc   0
t



Sub :   cE  0
Add :
 
Final eqns & mechanism for flow
v
1
 v  v 
( p   2v   E E )
t
Re
 v  0
 ( E )  0


  E    E  0

E    E / 

1 2
 v  
 ,
t
Rae
HS electro-osmotic slip boundary conditions
uslip  E
Dimensionless parameters
U ev H
Rae 
D
Re 
U ev H

 high

 low
U eo
Rv 
U ev

H
L
Lsample
H
Electric Rayleigh number

E 2 H 
U ev 

 

Reynolds number
Electrical conductivity ratio
Ratio of electro-osmotic to electroviscous velocity
Channel aspect ratio
Ratio of sample length to channel height
Unstable flow
E=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 measure
Maximum 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
 1 
 hi   lo
 lo
E 2 H 2
Ra 
D
 2
Rv Ra 
D
2
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
.