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Efficient and Rapid Mixing in a Slip-Driven ThreeDimensional Flow in a Rectangular Channel*
J. Rafael Pacheco1 Kang Ping Chen1 and Mark A. Hayes2
1
Department of Mechanical and Aerospace Engineering
Arizona State University
Tempe, AZ 85287-6106
2
Chemistry and Biochemistry Department &
The Center for Solid State Electronics Research
Arizona State University, PO Box 871604
Tempe, AZ 85287-1604
Abstract
Mixing of a blob of a solute in a slip-driven three-dimensional flow in a
rectangular channel is studied. For large width-to-height aspect ratio channels, this
serves as a good model for mixing in electro-osmotic flows in a three-dimensional
channel. It is demonstrated that under certain conditions, efficient and rapid mixing
can be achieved. A method to implement this operating condition in electro-osmotic
flow is also proposed.
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1. Introduction
Enhanced mixing plays an important role in biological and chemical analysis in
microfluidic systems. The Reynolds number, which measures the importance of the
inertia force relative to the viscous force, is very small for flows in a microfluidic device
due to the small dimensions of the device. Conventional methods used in generating
mixing in macro-scale fluid flows require sufficiently large Reynolds numbers and they
become ineffective when applied to micro-scale flows. Thus, the search for effective
mixing mechanisms suitable for micro-scale flows is becoming an active area of research.
Electroosmosis has proven to be an attractive method for transporting and
manipulating fluids in micro-devices. When an electrolyte solution comes into contact
with a solid surface, conterrions accumulates in a thin layer adjacent to the solid surface.
When an external electric field is applied, the counterions in this thin electric doublelayer (EDL) is set into motion and the viscous force drags the fluid beneath to move
along. In many applications, the EDL is very thin. In this case, electro-osmotic flow in a
two-dimensional channel can be modeled by specifying a “slip velocity” at the solid wall.
The slip velocity is related to the strength of the electric field and the so-called zetapotential which is the static electric potential difference across the EDL. One
characteristic of such a two-dimensional electro-osmotic flow is that it is essentially a
plug flow outside of the EDL, the velocity of which is independent of the dimension of
the channel. This provides the advantage of easy transport of fluids. On the other hand,
mixing in this plug-like electro-osmotic flow is not efficient. This severely limits the
application of such electro-osmotic micro-devices for rapid diagnosis, since rapid
diagnosis requires rapid mixing of samples with reagents, and the reagents used in typical
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applications possess relatively low diffusivity. Several methods have been proposed to
enhance mixing in electro-osmotic flows in micro-channels. These include using periodic
and time-dependent surface charge for a two-dimensional channel (Qian & Bau, 2002),
and using patterned surface charge to induce more complicated multidirectional electroosmotic flows for a three-dimensional channel with infinite transverse span (Stroock et
al. 2000, 2001). In Stroock et al. (2000), the external electric field is applied along the
longitudinal direction of the channel, and periodic, step-like surface charge variation is
imposed in either the longitudinal direction or the transverse direction. These charge
variations generate either multidirectional flows or recirculating cellular flows. The
channel discussed in Stroock et al. (2000) is unbounded in both the longitudinal and
transverse directions, while Qian & Bau (2002) considers a two-dimensional channel
only. An alternative method for generating efficient mixing in electro-osmotic flows in a
long but laterally confined three-dimensional micro-channel is proposed here. Instead of
varying charges on the surface walls, a secondary time-dependent external electric field
transverse to the flow is applied, in addition to the steady electric field along the channel
longitudinal direction which drives the primary flow. The transverse electric filed can be
generated by applying pre-determined voltages across two micro-electrode pairs placed
on the lateral bounding walls of the channel as shown in Figure 3. This additional
transverse electric field generates a vortex flow in the channel cross-section. By carefully
controlling the transverse electric field, efficient and rapid mixing can be achieved even
for low Reynolds number flows.
The idea of applying and controlling transverse electric field to promote mixing
stems from the superficial analogy between the transverse velocity field in the cross-
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section of the channel generated by the transverse electric field and that of a twodimensional cavity flow. A two-dimensional cavity flow is driven by the tangential
movement of the top surface and/or the bottom surface. This is in essence a flow driven
by prescribed “slip-velocities” of the bounding surfaces. Carefully controlled cavity flow
is known to be capable of generating excellent mixing in the cavity (Ottino, 1989;
Boyland & Aref, 2000; Aref, 2002; Vikhansky, 2003). However, when a channel is
laterally bounded, even the primary electro-osmotic flow in the longitudinal direction is
not strictly a flow driven by a slip-velocity due to the variation of the electric potential in
the lateral direction. Likewise the transverse flow is not strictly a cavity flow driven by
the slip-velocities either. In other words, the slip-velocity model is not strictly valid for
three-dimensional electro-osmotic flows. On the other hand, if the width-to-the-height
aspect ratio of the channel cross-section is large, the electric potential is essentially
constant on most of the portion of the cross-section, with variations in the lateral
direction occurring only near the lateral bounding walls. In this case, the slip-velocity
model can still serve as a good approximation for such flows. The advantage of adopting
this simple model is that the insight gained from studies on mixing in the traditional
cavity flow can be directly applied to this three-dimensional flow. Operating parameter
range for achieving good mixing and basic understandings of the mixing features can be
readily obtained. Results from this slip-velocity model are reported here. A subsequent
investigation will remove the limitations posed by the slip-velocity model and study
rigorously the electro-osmotic mixing problem by solving the exact static electric field as
well as the velocity field in a three-dimensional channel.
In the slip-velocity model for three-dimensional channel flows, at vanishing
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Reynolds numbers, the velocity field is the superposition of a plug-like (but not uniform)
flow along the longitudinal direction and a two-dimensional transverse cavity flow.
Convection-diffusion equation can be used to calculate how a blob of species disperses in
this flow field down the channel. Many reagents used in applications have relatively
small diffusion coefficients. In the ideal situation of such diffusion-limited case, the
reagent will be transported along the instantaneous local streamlines and passive tracer
particles can be used to track the movements of the reagents. Efficient mixing is sought
by alternating the transverse field in time, and the effectiveness of this mixing method is
examined by studying the transport of passive particles in this velocity field. A numerical
simulation of the dispersion of a reagent blob employing the full convection-diffusion
equation will also be presented.
2. The model for the velocity field
Consider the motion of a viscous incompressible Newtonian fluid in a threedimensional channel driven by the slip velocities on the top and the bottom walls, as
shown in Figure 1. The steady primary flow is in the longitudinal direction (the xdirection). The density and viscosity of the fluid is  and , respectively. The channel has
a rectangular cross-section with a height of 2h, and a width of 2a. If the characteristic
velocity of the primary flow is U, a Reynolds number for the flow can be defined
as Re  2Uh  , where     is the kinematic viscosity of the fluid. For micro-fluidic
applications, this Reynolds number is very small (<<1), and the flow can be
approximated as a creeping flow. This simplification allows the linearization of the
Navier-Stokes equations and decouples the primary steady flow from the transverse flow.
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We scale velocity with U, length with 2h, pressure with U/2h, and time with 2h/U. In
dimensionless forms, the equations governing the flow are:
divu  0 ,
(1)
p   2 u ,
(2)
where p is the pressure, and the velocity field u  u i  v j  w k . The flow is driven by slip
velocities on the top and bottom walls, and the no-slip conditions on the side-walls are
applied:
y  0:
u  u  , w  w  , v  0;
(3)
y  1:
z  0, b :
u  u  , w  w  , v  0;
u  v  w  0,
(4)
(5)
where b  a h is the aspect ratio. The governing equations are solved using a numerical
scheme based on the finite-volume fractional step method (Zang et al. 1994, Pacheco
2001). Readers may consult the referenced papers for the technical details.
X
Y
2h
Z
2a
Figure 1: Infinite long rectangular channel. The flow is driven by the slip velocities on
the top and the bottom walls of the channel, at y  0, and y  2h.
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Due to the linearity of the boundary-value problem, the velocity field is the
superposition of the steady longitudinal velocity field u ( y, z ) i , driven by the slip
velocities u  and u  , and the transverse velocity field v ( y, z ) j  w ( y, z ) k , driven by
the transverse slip velocities w  and w  . In all the results presented in this paper, the
dimensionless slip velocities along the longitudinal direction on the top and the bottom
surfaces are set to be one, u   u  1 . Although the adoption of the slip-velocity model
requires the aspect ratio b to be large, the qualitative features of the flow field will remain
the same for all finite values of the aspect ratio. Thus, for exploration purposes, we set
the aspect ratio b  2 in all computations presented here.
Even in the absence of the transverse field, the longitudinal velocity in not
uniform in each cross-section due to the no-slip conditions imposed on the lateral walls,
z  0, b . Maximum velocities occur at the top and the bottom surfaces. Figure 2 shows
the three-dimensional velocity field when the top wall has a transverse slip-velocity of
one w   1 , and the bottom wall has zero transverse velocity w  0 . A single vortex cell
is generated in the cross-section, and fluid particles travel down the channel along spiral
streamlines. A similar pattern appears when the bottom wall has a transverse slip-velocity
of one w 1 and top has zero transverse slip-velocity w   0 . When both the top and
the bottom walls are subjected to transverse slip-velocities in the same direction, a double
vortex structures appear in the cross-section. Since passive tracer particles are transported
along the streamlines, in the diffusion-limited cases, no chaotic mixing can be created in
steady flows because these streamlines are fixed in both space and time when the flow is
steady. Thus, a key element to achieve enhanced mixing is to let the transverse slipvelocities to vary in time in a certain fashion. As discussed in Ottino (1989), excellent
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mixing can be achieved by repeatedly stretching and folding the samples. The specific
example examined here has the following transverse slip velocities at the top and bottom
surfaces of the channel: w 1, w  0 for half period T/2, and w  0, w 1 for the other
half period. This operation can be realized in experiments by placing two micro-electrode
pairs on the lateral bounding walls of the channel, as shown in Figure 3, where φ is the
electric potential. By alternatively switching on and off the applied DC voltages across
the top and the bottom electrode pairs for each half period, we can achieve the required
Heaviside function of the slip velocities of the top and the bottom surfaces.
Figure 2: Spiral streamlines for steady flow with u   u  1 , w   1 , w   0 .
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Figure 3: Cross-section of the micro-channel and the micro-electrode arrangements for
generating transverse electric field.
3. Dispersion of a diffusion-limited reagent blob: tracer particle simulation
The dispersion of a reagent in the fluid is governed by the convection-diffusion
equation:
c
1 2
 div cv  
 c,
t
Pe
(6)
where c represents the reagent concentration, and the Peclet number is defined as
Pe  2Uh  , with  being the coefficient of diffusion of the reagent. The Peclet number
is the ratio of the diffusion time scale h2/over the convection time scale h/U. Typical
reagent diffusion coefficient is of the order of 10-11 m2/s, and typical velocity is of the
order of 1 cm/s. For a channel with a height of 20 μm, or h  10 μm, this gives a Peclet
number of Pe  103. Thus, the typical diffusion time scale is much longer than the
convective time scale. As a consequence, convective transport dominates diffusive
transport when the Peclet number is large, and dispersion of the reagent in the flow field
can be greatly enhanced by promoting the convective transport. In such diffusion-limited
cases, the diffusion effects can be neglected (except the initial short period of time), and
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the reagent just convects with the fluid along the local instantaneous streamlines. Study
of the trajectories of passive tracer particles is sufficient to provide the information on the
dispersion of the reagent in the flow field without solving the convection-diffusion
equation (6).
Since passive tracer particles move along the instantaneous streamlines, chaotic
dispersion can only occur in an unsteady flow field. For this purpose, we generate an
unsteady transverse velocity field by varying the transverse slip-velocity with time as
described in Section 2. For a single tracer particle, it moves down the channel in a zig-zag
fashion following the instantaneous streamlines. Thus, the motion of a tracer particle can
be tracked by the kinematic equation
dr  v(r, t ) ,
dt
(7)
where r (t )  x(t ) i  y (t ) j  z (t ) k is the location vector of the tracer particle, and v (r , t ) is
the local velocity field obtained from computations in Section 2. Equation (7) is advanced
from a given initial condition of the tracer particle location. The location of the particle at
the end of each period t T  1, 2, 3, ... is then projected onto the yz-plane to generate the
so-called Poincare maps (Ottino, 1989). If a tracer particle, regardless of its initial
position in the cross-section, samples the entire channel cross-section of the channel
while moving downstream, i.e. if the Poincare map covers the entire cross-section
regardless of the initial cross-sectional position of the particle, chaotic mixing is
achieved. Poincare maps for a single particle for periods T  1, 5, 7 are plotted in Figures
4, 5, 6, respectively. A tracer particle is initially located at the geometric center of the
cross-section at ( y, z )  (0.5, 1.0) , and these Poincare maps are generated till t  500 .
As shown in these figures, for period T  1 , this particle travels in a very confined
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narrow band near the upper half of the cross-section. On the other hand, when the period
is increased to T  5 , the particle scatters to almost all regions of the cross-section
(Figure 5). The holes or exclusion zones in Figure 5 indicate further enhancement may be
achieved, which is shown in Figure 6 when the period is increased to T  7 . Nearly
chaotic motion of the particle is achieved for this large period since the exclusion zones
have been dramatically reduced in size. The mathematical reasoning behind the effect of
the period T has on the nature of mixing is discussed in Qian & Bau (2002). Suffice to
say that for the case of discontinuous/alternating transverse slip velocities, increasing the
period T can induce a chaotic motion for the particle.
Figure 4: Poincare map for a single tracer particle period T  1 . The particle initially is
located at ( y, z )  (0.5, 1.0) .
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Figure 5: Poincare map for a single tracer particle period T  3 . The particle initially is
located at ( y, z )  (0.5, 1.0) .
Figure 6: Poincare map for a single tracer particle period T  7 . The particle initially is
located at ( y, z )  (0.5, 1.0) .
To simulate the dispersion of a blob of reagents, we place 10,000 passive tracer
particles uniformly in a sphere of radius of 0.1 centered at the geometric center of the
rectangular cross-section at t  0 , and follow the motions of all these particles. At
t  0, 5, 10, 15, ..., 100 , we locate the particle positions and then project their positions
onto the yz-plane. This series of snap shots provides a dynamic view of the transverse
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dispersion process of the tracer particles in the channel. As evident from the series for
period T  1 , shown in Figure 7, the mixing is of very low quality, since these tracer
particles only move in a narrow band that covers very limited portion of the crosssection. There are large unfilled “holes” or exclusion zones in the cross-section at
t  100 . For period T  5 , relatively good mixing is achieved, but there still remain
many significant exclusion regions where the particles could not reach. But when the
period is increased to T  7 , the particles reached a major portion of the cross-section,
and excellent mixing is achieved at t  100 (Figure 8). It is apparent from these results
that increasing the period T enhances the mixing, a result well-known for twodimensional cavity flows (Ottino, 1989).
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Figure 7: Transverse dispersion of a spherical blob of reagent initially located at the
center of the channel cross-section. The initial radius of the blob is 0.1. Period T  1 , and
t  0, 5, 10, ..., 100 .
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Figure 8: Transverse dispersion of a spherical blob of reagent initially located at the
center of the channel cross-section. The initial radius of the blob is 0.1. Period T  7 , and
t  0, 5, 10, ..., 100 .
How fast a reagent blob disperses in the fluid also depends on the initial location
of the reagent even for the same flow condition. To this end, we place the blob initially at
a location near the top wall at ( y, z )  (0.7, 1.0) . For period T  7 , our computations
shows that at the same instant, the reagent is more uniformly distributed in the crosssection when the blob is initially placed off the geometric center near the top surface at
( y, z )  (0.7, 1.0) than placed at the geometric center at ( y, z )  (0.5, 1.0) . Figure 9 is a
side-by-side comparison of the dispersions at the instant t  100 for these two different
initial locations of the blob. As can be seen from Figure 9, even though they share similar
features, the exclusion zones on the left panel (centrally placed blob) is narrowed and
intruded when the blob is placed off center (right panel), indicating better dispersion
quality.
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Figure 9: Side-by-side comparison of the dispersion at t  100 for initial blob location of
( y, z )  (0.5, 1.0) (centrally placed, left panel) and ( y, z )  (0.7, 1.0) (right panel).
In a typical application with h  10 μm , U  1 cm/s , the characteristic time
t c  2h U  2  10 3 sec. Period T  7 then corresponds to a dimensional time of 0.014
second, and excellent mixing is achieved in just 0.2 seconds (100 dimensionless time
unit). Rapid on/off switching of the micro-electrode pairs is required to operate with a
period of 0.014 second. Fortunately for this type of on-and-off slip-velocities, increasing
period T improves mixing quality, and one can choose a larger period T in practice to
avoid extremely rapid on-and-off switching of the electrodes.
4. Dispersion of a reagent with small diffusion: full numerical simulation
The miscible dispersion of a solute into a flowing fluid in a non-uniform velocity
field in a circular capillary was first analyzed by Taylor (1953, 1954). A slug of solute, in
the shape of a cylinder of the same diameter as the capillary, is “inserted” into the
capillary where a solvent fluid is moving with a parabolic velocity profile. The major
dispersion mechanism for this problem is due to axial convection as well as radial
diffusion. If the same problem is studied for a two-dimensional electro-osmotic flow in a
channel, dispersion of the solute is due to axial (longitudinal) convection, as well as axial
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diffusion. No lateral diffusion occurs since the velocity is uniform in the cross-section,
and the concentration remains uniform in each cross-section.
The problem studied here is slightly different from Taylor’s. The solute (reagent)
initially is a small blob located at the center of the channel at the location x  1 , instead
of occupying the entire cross-section. The blob is a sphere of radius r  0.1 , and density
of unity. We consider the diffusion-limited case, and set the Peclet number to Pe  1000 ,
so that the dimensionless diffusion coefficient 1 Pe  0.001 . The convection-diffusion
equation (6) is then solved, with non-penetration conditions on the walls of the channel.
The computational domain has a dimensionless length of 20, with periodic conditions
imposed at the entrance and the exit, so that the evolution of this solute blob can be fully
traced.
Since the Peclet number is large, the diffusion effect is only significant for the
initial short period of time due to the very large concentration gradient initially. The
dispersion process is dominated by the axial as well as cross-sectional convective
transport. Figure 10 illustrates the evolution of the blob (density field) from t  11 to
t  20 for the case of period T  7 . The folding and stretching action of the solute blob
is evident from this sequence. At t  12 , the initially spherical solute has been elongated
transversely into a long cylindrical shape by the transverse transport. The appearance and
subsequent disappearance of the long tail in the longitudinal direction is the manifest of
the three-dimensional nature of the dispersion process. As time progresses, the solute is
further stretched in the cross-section, and then folded to form a donut shape. Further
transport force the closure of the hole of the donut ( t  19, 20 ). The sample is also
stretched in the downstream direction, as evident from the increased length of the blob.
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Figure 11 shows the blob (density) at time t  93 . It is obvious that the blob has been
well dispersed throughout the cross section of the channel. For example, nearly half way
between the “head” and the “tail” of the blob, at x  14 , the maximum variation of the
solute density in the cross-section is 24%, indicating very good mixing. The longitudinal
extent of the blob has been stretched from the initial length of 0.2 to nearly 18. Solute
concentration distributions along the mid-planes as well as along the centerline
( y, z )  (0.5, 1.0) are plotted in Figures 12, 13, 14, respectively. It is obvious from these
plots that the solute concentration levels have been dramatically reduced and the solute
has been very well mixed with the surrounding fluids.
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Figure 10: Evolution of the blob of reagent from t  11 to t  20 . Initially the blob is a
sphere of radius 0.1 centered at ( y, z )  (0.5, 1.0) .
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Figure 11: Concentration distribution of the blob of the solute at t  93 . The color bar
gives the concentration levels.
Figure 12: Concentration distribution at the mid-plane y  0.5 .
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Figure 13: Concentration distribution at the mid-plane z  1.0 .
Figure 14: Variation of the solute concentration along the x direction at the centerline
( y, z )  (0.5, 1.0) at t  93 . The maximum concentration along this line has been
reduced to about 2  10-4 from the initial value of 1.0.
5. Discussions
Despite the limitations of the slip-velocity model for three-dimensional electroosmotic flows, the computations reported in this paper reveal that efficient and rapid
mixing can be achieved if the transverse flow field is controlled properly. With the
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simple on-and-off transverse slip-velocity of the top and the bottom surfaces, nearly
chaotic mixing can be achieved when the period T ≥ 7. This operating condition can be
implemented in practice by, for example, placing two micro-electrode pairs on the lateral
walls of the channel, as shown in Figure 3. It is shown that one can use a collection of
passive tracer particles to effectively and economically simulate non-diffusive mixing.
The effectiveness and the quality of the mixing also depend on the initial placement of
the blob of the solute. Direct numerical simulation with small diffusion confirms the
findings of the passive tracer particle computations. Even though the slip-velocity model
is rigorously valid only for large aspect ratio channels, the qualitative features on the
efficiency of the mixing in the model flow provide a guidance for further studies on the
exact electro-osmotic mixing problem in a three-dimensional channel, which is presently
pursued by the authors.
ACKNOWLEDGMENTS
The authors would like to thank Ms. Renate Mittelmann from Department of
Mathematics at ASU for access to computer facilities.
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