-
Curvature-induced secondary microflow motion in steady
electro-osmotictransport with hydrodynamic slippage effect
Jin-Myoung Lim and Myung-Suk Chuna)
Complex Fluids Research Laboratory, National Agenda Research
Division, Korea Institute of Science andTechnology (KIST),
Seongbuk-gu, Seoul 136-791, South Korea
(Received 17 February 2011; accepted 9 September 2011; published
online 19 October 2011)
In order to exactly understand the curvature-induced secondary
flow motion, the steady electro-
osmotic flow (EOF) is investigated by applying the full
Poisson-Boltzmann/Navier-Stokes equa-
tions in a whole domain of the rectangular microchannel. The
momentum equation is solved with
the continuity equation as the pressure-velocity coupling
achieves convergence by employing the
advanced algorithm, and generalized Navier’s slip boundary
conditions are applied at the hydro-
phobic curved surface. Two kinds of channels widely used for
lab-on-chips are explored with the
glass channel and the heterogeneous channel consisting of glass
and hydrophobic polydimethylsi-
loxane, spanning thin to thick electric double layer (EDL)
problem. According to a sufficiently low
Dean number, an inward skewness in the streamwise velocity
profile is observed at the turn. With
increasing EDL thickness, the electrokinetic effect gets higher
contribution in the velocity profile.
Simulation results regarding the variations of streamwise
velocity depending on the electrokinetic
parameters and hydrodynamic fluid slippage are qualitatively
consistent with the predictions docu-
mented in the literature. Secondary flows arise due to a
mismatch of streamline velocity between
fluid in the channel center and near-wall regions. Strengthened
secondary flow results from increas-
ing the EDL thickness and the contribution of fluid inertia
(i.e., electric field and channel curva-
ture), providing a scaling relation with the same slope.
Comparing with and between the cases
enables us to identify the optimum selection in applications of
curved channel for enhanced EOF
and stronger secondary motion relevant to the mixing effect. VC
2011 American Institute of Physics.[doi:10.1063/1.3650911]
I. INTRODUCTION
The electro-osmotic flow (EOF) is a fundamental electroki-
netic phenomenon, which plays an important role for improve-
ments and optimum design of the devices related to
lab-on-chips
(LOCs).1 Exemplary advantages of EOFs over pressure-driven
flows are ease of fabrication, flowing control, and precise
particle
manipulations. A uniform external electric field E applied in
thechannel induces the movement of an electrolyte solution
(with
dielectric constant e and viscosity l) relative to a stationary
sur-face (with zeta potential f). A relationship between EOF
velocityand E is given by the convenient
Helmholtz-Smoluchowski(H-S) formula, vEOF ¼ �e fE=l.1–3 In the case
of very thinelectric double layer (EDL), vEOF for Newtonian
fluidsappears to slip at the wall represented by this formula,
and
then one needs to consider the velocity field outside the
EDL.
It is noted that a plug-like velocity profile in EOF
provides
reduced dispersion of sample species, realizing the
electropho-
resis become one of the most effective technologies for
chem-
ical and biomedical analyses.
A design of microfluidic devices requires the effort to
consider the flow condition and flow pattern depending on
the channel geometry, where a curved rectangular channel
can commonly be dealt with. Curved channels become an in-
dispensable tool in the chip-based electrophoresis system4–6
and the controllable reaction residence to be required for
synthetic applications.7,8 Chaotic advection generating pas-
sive mixing at low Reynolds number (Re) is achieved by
introducing turn geometry, acting to enhance stretching and
breaking of the flow.9
Although lots of theoretical studies have contributed to
elucidating the EOF, those studies were almost confined to
an
analysis with applying straight channel with uniform cross
sections. Simulations of the EOF were performed in various
geometries, demonstrating that the linearized Poisson-
Boltzmann (P-B) referred to as Debye-Hückel (D-H) ansatz
electric field can still produce plug velocity predictions
for
large zeta potentials.10 Note that a few previous studies
have
dealt with compensations for turn-induced broadening and
how channels can be designed to minimize such effects. Grif-
fiths and Nilson11 derived the analytical method of a closed
form expression for the increased band variance induced by a
turn. Paegel et al.12 proposed narrowing the channel
widthupstream of a turn followed by widening the channel once
the
turn is completed. Molho et al.13 tried to come up with opti-mal
shapes for low-dispersion geometries using computer
simulations. Yang and co-workers explored the transient
EOFs in curved channels applying an AC field with a single
frequency14 and suddenly induced DC and AC fields of
various frequencies.15 Later, Woo et al.16 treated the problemof
electro-osmotic flow in a U-turn microchannel to determine
the optimal zeta potential distributions for minimal
dispersion.
In Fig. 1, the external electric field is tangential to the
wall and same potential drop occurs over a shorter distance
a)Author to whom correspondence should be addressed. Electronic
mail:
[email protected]. Fax: þ82-2-958-5205.
1070-6631/2011/23(10)/102004/10/$30.00 VC 2011 American
Institute of Physics23, 102004-1
PHYSICS OF FLUIDS 23, 102004 (2011)
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http://dx.doi.org/10.1063/1.3650911http://dx.doi.org/10.1063/1.3650911http://dx.doi.org/10.1063/1.3650911
-
of arc AB on the inner side than on the outer side of arc
A0B0. Thus, a stronger field is created on the inside edge ofthe
channel wall, driving a faster EOF. Furthermore, the
pressure distribution along the spanwise direction results
from a gap of the flow path between the inner and outer
walls
of a curved channel. By extending previous work on the
explicit model for electrokinetic pressure-driven flow in
straight microchannels,17 Chun and co-workers18 investi-
gated recently the problem of steady flow in a curved
rectan-
gular channel to figure out the skewed pattern in a
streamwise velocity with stressing the geometry effect. They
found the inward skewness in the streamwise axial velocity
due to a dominant effect of the spanwise pressure gradient
over the fluid inertia by centrifugal force. Re and
nondimen-
sional curvature (W/RC) are combined into the Dean
number(De¼Re
ffiffiffiffiffiffiffiffiffiffiffiffiffiW=RC
p) that quantifies the inertial force in the
curvature.19 As De gets higher enough, the axial velocity
profile becomes skewed into the outer wall as can be
observed in the general narrow-bore channels.
One can presume that the spanwise velocity, pressure
gradient, and skewed streamwise velocity are formed by
channel curvature, although those subjects were not
quantita-
tively emphasized in the previous studies. This paper
addresses the numerical investigation of EOFs in curved rec-
tangular microchannels of uniform width that provides quan-
titative information useful in designing either channels or
devices and their fabrications. In pursuit of more effective
approach for the pressure-velocity coupling, an alternating
direction implicit (ADI) method was implemented in the
SIMPLE (semi-implicit method for pressure-linked equa-
tions) algorithm.20 Numerical results of the streamwise
veloc-
ity and vorticity change are reported to examine the unique
behavior of secondary microflow in shallow and deep chan-
nels with either thick or thin EDL thickness, providing
origi-
nal insights. Unlike the thin EDL situation, the H-S slip
velocity at the edge of the EDL cannot be applied in the
thick
EDL case which makes the analysis complicated. Our frame-
work enables to conduct the dissimilar configurations of the
channel wall taking unequal surface charge and fluid
slippage
with adopting hydrophilic glass and hydrophobic polydime-
thylsiloxane (PDMS), which has not been fully attempted.
II. MODEL FORMULATION
A. The geometry and Navier’s fluid slip at curved wall
Let us consider a situation for steady-state EOF through
a curved rectangular microchannel with a uniform curvature.
The geometry of the channel is schematically presented in
Fig. 1. Here, a fluid flows through the channel of width Wand
height (or, referred to as the depth) H that makes a leftturn.
Coordinate transformation is available between the
global toroidal system and the Cartesian system, where r(¼RCþ x)
and y correspond to spanwise and longitudinaldistances. The
curvature radius RC ¼ dz=dh is measuredfrom the axis of curvature,
and z is the streamwise distancealong the axis of a
microchannel.
While no-slip boundary conditions (BCs) are applied to
the wall of hydrophilic surface, the hydrodynamic BC at the
hydrophobic surface undergoes the fluid slippage that is a
function of the wettability indicated by the contact angle.
Fluid slippage at the stationary interface can be
interpreted
as the continuity of shear stress r for the tangential
velocity
component, vt¼m � r/fs, with the interfacial friction
coeffi-cient fs and the unit normal m directed into the fluid.
Com-bining the constitutive equation for the Newtonian fluid
allows a slip length (sometimes, Navier length) b at a uni-form
hydrophobic rigid boundary to linearly relate vt at thewall and the
wall shear strain rate, so-called the Navier’s
fluid slip condition21–23
vt ¼ b m � rvþ rvð ÞTh i
: (1)
Here, b (¼ l/fs) can be regarded as the characteristic
lengthnormal to the wall, representing the degree of absence of
the
shearing force. Since the normal velocity component vn¼ v � mð Þ
is governed by conservation of mass, vn ¼ 0 for a
stationary and impermeable solid referred to as the no-
penetration condition.
The slip length b is a material parameter, and it is thelocal
equivalent distance below the solid surface at which
the no-slip BC (b¼ 0) would be satisfied if the flow fieldswere
extended linearly outside of the physical domain. It is
possible to derive the streamwise velocity by applying Eq.
(1) to the curved region in Fig. 1, expressed as
vh ¼ b1
r
@vr@hþ r @
@r
vhr
� �� �: (2)
Since the normal velocity vr vanishes on the surface, weobtain a
generalized slip condition at a surface with RC as
vh ¼1
bþ 1
RC
� ��1 @vh@r¼ bRC
bþ RCð Þ@vh@r
: (3)
Then, bRC/(bþRC) corresponds to the slip length bC on acurved
surface.
B. Electrokinetic governing equations
We recall here briefly the background of electrokinetic
transport. Once the non-conductive and non-neutral
(charged) surface is in contact with an electrolyte in
solution,
the electrostatic charge would influence the distribution of
nearby ions and the electric field is established.3,24 For a
uni-
formly charged rectangular channel, the nonlinear P-B equa-
tion governing the electric potential w field is given as
FIG. 1. The curved channel with relevant coordinates system and
each
boundary condition.
102004-2 J.-M. Lim and M.-S. Chun Phys. Fluids 23, 102004
(2011)
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r2W ¼ j2 sinh W: (4)
Here, the dimensionless electric potential W denotes Kew/kTand
the EDL thickness (or Debye thickness) j�1 is defined
by j
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nbK
2i e
2=ekTq
with the valence of i ions Ki. Apply-
ing symmetric (K-K) electrolytes provides the same valencesof
cations and anions. The electrolyte ionic concentration in
the bulk solution at the electroneutral state nb (1/m3)
equals
to the product of the Avogadro’s number NA (1/mol)and bulk
electrolyte concentration (mM). The dielectric
constant e is given as a product of the dielectric
permittivityof a vacuum (¼ 8.854� 10�12 C2/J�m) and the
relativepermittivity for aqueous fluid (¼ 78.9), e the
elementarycharge, and kT the Boltzmann thermal energy at
roomtemperature.
The equilibrium Boltzmann distribution of the ionic
concentration of type i (i.e., ni¼ nb exp(�Kiew/kT)) pro-vides a
local charge density Kieni. One can determine the netcharge density
qe (: Ri Kieni¼Ke(nþ � n� )), as follows:
qe ¼ Kenb exp �Wð Þ � exp Wð Þ½ � ¼ � 2Kenb sinh W: (5)
The Dirichlet BCs of constant-potential surface are imposed
on each side of the wall displayed in Fig. 1: W¼Ws,L atx¼�W/2,
W¼Ws,R at x¼W/2, W¼Ws,B at y¼ 0, andW¼Ws,T at y¼H. The subscripts
T, B, R, and L correspondto top, bottom, right, and left-sided
channel wall.
Regarding the velocity field for an incompressible New-
tonian fluid at the steady state, the continuity equation
and
Navier-Stokes (N-S) equation are provided as
r � v ¼ 0; (6)
q v � rvð Þ ¼ �rpþ lr2vþ F; (7)
where q is the density of the fluid. If the flow field exists
byaxially applied pressure-driven, the velocity profile in
straight channel with uncharged (i.e., inert) wall is
described
by the Stokes flow (l52v¼5p). In the case of EOF withoutthe end
effects (i.e., @/@h¼ 0), the fluid velocity andthe pressure due to
curvature effect are expressed as
v ¼ vr r; yð Þ; vy r; yð Þ; vh r; yð Þ
and p¼ p(r,y), respectively.Gravitational forces and the effect
of Joule heating are
neglected. The body force F per unit volume ubiquitouslycaused
by the action of electric field E on the net charge den-sity
qe(r,y) can be written as F¼qeE.
The electric field can be given by the linear superposi-
tion of the external electric potential / imposed by end-channel
electrodes and the inherent electric potential w,providing that
E¼�5(/þw).3 Originally, / satisfies herethe 2-dimensional Laplace
equation (i.e., 52/(r,h)¼ 0) inthe outside region of the EDL. As
the BCs, the Dirichlet con-
ditions can normally be given at the inlet and outlet of the
channel, and the Neumann conditions are assigned at the
inside and outside walls of impermeable solid channel
(@/wall/@r¼ 0). It is acceptable that the spanwise depend-ence
of / is much smaller than its streamwise axial depend-ence, whereas
the streamwise dependence of w is negligible.Then, streamwise
electric field Eh is defined by the externalpotential / as Eh¼�
d/(h)/dh.
With above conditions and pertinent approximations,
the N-S equation readily reduces to
r : q vr@vr@rþ vy
@vr@y� vh
2
r
� �¼�@p
@rþl 1
r
@
@rr@vr@r
� ��
�vrr2þ@
2vr@y2
��qe
@w@r;
(8)
y : q vr@vy@rþ vy
@vy@y
� �¼� @p
@yþ l @
2vy@r2þ 1
r
@vy@r
�
þ@2vy@y2
�� qe
@w@y
;
(9)
h : q vr@vh@rþ vy
@vh@yþvrvh
r
� �¼�qe
r
@/@hþl 1
r
@
@rr@vh@r
� ��
�vhr2þ@
2vh@y2
�: (10)
Utilization of the Cartesian coordinates leads to the
follow-
ing equations:
x : q vx@vx@xþ vy
@vx@y� vz
2
xþ RC
� �
¼ � @p@xþ l 1
xþ RC@
@xxþ RCð Þ
@vx@x
� �� vx
xþ RCð Þ2
"
þ @2vx@y2
�� qe
@w@x
; (11)
y : q vx@vy@xþ vy
@vy@y
� �¼ � @p
@yþ l @
2vy@x2þ 1
xþ RC@vy@x
�
þ @2vy@y2
�� qe
@w@y
; (12)
z : q vx@vz@xþ vy
@vz@yþ vxvz
xþ RC
� �¼ � RC
xþ RCqe@/@z
� �þ
l1
xþ RC@
@xxþ RCð Þ
@vz@x
� �� vz
xþ RCð Þ2þ @
2vz@y2
" #: (13)
For hydrophobic curved surfaces in Fig. 1, each BC is
applied as
vxjy¼0 ¼ bB @vx=@yð Þy¼0; vxjy¼H ¼ �bT @vx=@yð Þy¼H; (14)
vyjx¼�W=2 ¼ bL @vy=@x� �
x¼�W=2;
vyjx¼W=2 ¼ �bR @vy=@x� �
x¼W=2;(15)
vzjy¼0 ¼ bB @vz=@yð Þy¼0; vzjy¼H ¼ �bT @vz=@yð Þy¼H; (16)
vzjx¼�W=2 ¼ bCL @vz=@xð Þx¼�W=2;vzjx¼W=2 ¼ �bCR @vz=@xð
Þx¼W=2:
(17)
Here, bCL and bCR are displayed in Fig. 2 for inner and
outer
walls, respectively, expressed as
bCL ¼ bL RC �W=2ð Þ= RC �W=2ð Þ � bL½ �; (18)
bCR ¼ bR RC þW=2ð Þ= RC þW=2ð Þ � bR½ �: (19)
102004-3 Curvature-induced secondary microflow motion Phys.
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The straight channel (W/RC! 0) takes a change in Eq.
(17),leading to vzjx¼�W=2 ¼ bL @vz=@xð Þx¼�W=2 and vzjx¼W=2¼ �bR
@vz=@xð Þx¼W=2. Since the velocity becomes a maxi-mum at the center
of the channel, the sign of the velocity
gradient changes into negative along the axis.
We substitute Eq. (5) into Eq. (13), and the velocity
fields are normalized by the characteristic velocity l=qdh soas
to elucidate the secondary flow effect along with each ve-
locity component: Vx ¼ vx qdh=lð Þ, Vy ¼ vy qdh=lð Þ, andVz ¼ vz
qdh=lð Þ. Other nondimensionalized parameters areemployed as R0 ¼
RC=dh, X ¼ x=dh, Y ¼ y=dh, Z ¼ z=dh,P ¼ p qdh2=l2
� �, Cw ¼ 2kTnb sinhW qdh2=l2
� �, C/ ¼ 2Kienb
sinhW qdh2wo=l
2� �
, and dH=dZ¼ d/=dzð Þ dh=woð Þ. Here, dhis the hydraulic
diameter, the external potential difference
D/ equals EzL with channel length L, and wo is the
referenceelectric potential. Then, one obtains the
nondimensionalized
equations of motion, such that
Vx@Vx@Xþ Vy
@Vx@Y� V
2z
X þ R0
¼ � @P@Xþ 1
X þ R0@
@XX þ R0ð Þ @Vx
@X
� �� Vx
X þ R0ð Þ2
þ @2Vx@Y2þ Cw
@W@X
;
(20)
Vx@Vy@Xþ Vy
@Vy@Y¼� @P
@Yþ @
2Vy@X2
þ 1X þ R0
@Vy@X
þ @2Vy@Y2þ Cw
@W@Y
;
(21)
Vx@Vz@Xþ Vy
@Vz@Yþ Vx Vz
X þ R0
¼ R0
X þ R0 C/@H@Z
� �þ 1
X þ R0@
@XX þ R0ð Þ @Vz
@X
� �
� VzX þ R0ð Þ2
þ @2Vz@Y2
: (22)
III. COMPUTATIONAL METHODS AND PARAMETERS
A precedent requirement is the electric potential profile
obtained by solving Eq. (4) with the finite difference
method
(FDM), of which basic procedures are analogous to the pre-
vious work.18 The five-point central difference is taken on
the left-hand side of Eq. (4) and we adopt the successive
under-relaxation iterative scheme. The sinh W on the
right-hand side can be linearized with the Taylor formula.
For
the external electric potential, 52H(X,Z)¼ 0 is treated as
aspecial case of Eq. (4) subject to the relevant BCs. Unlike W,the
solution for H shows negligibly small gradient in the span-wise
direction. The fluid velocity profile can be computed by
solving the aforementioned N-S equation of Eqs. (11)–(13).
The well-established SIMPLE algorithm represented by the fi-
nite volume method (FVM) was employed in this study to
solve the unknown pressure terms p(x,y) of the momentumequation
by using the continuity equation as the pressure-
velocity coupling. Detailed discussion of the SIMPLE is
avail-
able in the literature.20 The velocity and pressure
corrections
are accomplished by repeating the sequence of operations
until the convergence is guaranteed. Note that the ADI
method and staggered grid are adopted when the procedure
comes to treat x and y components of discretized N-S equationand
the continuity equation. As mentioned in the previous
work,18 the staggered grid is half-node shifted from the
origi-
nal grid, which prevents the failure of pressure estimations
and assures calculations that are more accurate.
Computations are performed with the 1-1 type electro-
lyte of KCl fluid (q¼ 103 kg/m3, l¼ 10�3 Pa�s), where itsEDL
thickness j�1 in nanometers is expressed as [fluid elec-trolyte
concentration (M)]�1/2/3.278. We consider two kinds
of configurations with setting each side of channel walls as
(I) glass (i.e., glass in all sides) and (II) glass in bottom
and
PDMS in the other sides, which are the most widely prepared
and used in microchip electrophoresis. As the prototype of
channel cross section, the dimension is set 25 lm wide and 5lm
deep for shallow geometry (H/W¼ 0.2, low-aspect-ratio)and 5 lm wide
and 25 lm deep for deep geometry (H/W¼ 5,high-aspect-ratio), with
the hydraulic radius Rh (¼HW/(HþW)) of 4.17 lm.
After the mesh refinement with grid convergence test to
yield satisfactory accuracy level, the computational domain
is discretized into 151� 151 equally spaced grid points
intransverse (x and y) directions. A very fine grid with
largernumber of points is employed for the case of thin EDL
(i.e.,
j�1¼ 9.7 nm), because of steep potential profile near thechannel
wall. Convergence criteria are specified with the rel-
ative variation between two successive iterations to be
smaller than the pre-assigned accuracy order of O(10�8).
The convergence is fast even for small j�1, and 500 itera-tions
were enough for the present channel dimension. The
under-relaxation coefficients were chosen in the range of
0.3–0.7. The velocity prediction matches the analytical
prob-
lem in a straight channel, validating our codes. Axial
veloc-
ity profiles at the horizontal cut (i.e., H/2) along
thestreamwise direction maintain a fully developed state in the
entire region of the curved channel except the entrance. In
order to ensure its axial independency (@vz/@z¼ 0), the
ve-locity and pressure fields are analyzed in the transverse
sec-
tion at the middle of the turn.
To provide an insight for realistic predictions, rigorous
values of material parameters are required. A number of ex-
perimental studies conducted to determine the slip length
b reveal a fact that it depends on the experimental condi-tions,
such as solution environment, shear rate, and surface
roughness.25–27 Here, b of 100 nm is taken for partially
non-
FIG. 2. The fluid slippage at hydrophobic curved surfaces.
102004-4 J.-M. Lim and M.-S. Chun Phys. Fluids 23, 102004
(2011)
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wetting PDMS surfaces (contact angle of ca. 98�). Althoughwe
need pursue its accurate value, a more detailed discussion
of this issue is beyond the scope of this study. According
to
Eqs. (18) and (19), the slip lengths for each BC at
left-side
and right-side walls have rightly been corrected, and
result-
ant changes compared to uncorrected slip lengths are esti-
mated in Fig. 3. It is emphasized that the relative
differences
increase with increasing slip length, presenting scaling
rela-
tions as �ba with the consistent slope a. A higher differencecan
be obtained at either inner wall (i.e., left-side wall) or
deep channel for the specified conditions, where the
increas-
ing ratio of bCL to bCR gets much larger in response to an
increase of W/RC.The magnitude of dimensionless surface
potential Ws
(¼ws e/kT) at medium pH 7 can be taken by the zeta poten-tial
data. According to a great deal of previous reports,28–30
the zeta potential for 1� 10�4mM 1-1 type electrolyte isfound to
vary from� 20 to� 100 mV for PDMS and� 35to� 200 mV for glass
surfaces, depending on the electrolyteconcentration and the
condition of surface treatment. We
take here Ws of PDMS as� 0.78 (ws¼� 20 mV),� 1.56(ws¼� 40 mV),�
2.33 (ws¼� 60 mV), and Ws of glassas� 1.36 (ws¼� 35 mV),� 3.11
(ws¼� 80 mV),� 4.67(ws¼� 120 mV) for 1.0, 10�2, and 10�4mM KCl
electro-lyte, respectively.
IV. SIMULATION RESULTS
A. Spanwise and streamwise velocities in EOF
Figure 4 presents the spanwise velocity and pressure
gradient observed in the glass channel with shallow geometry
for the thick EDL thickness j�1¼ 965 nm. It provides a
strong electrokinetic interaction with low concentration
limit
of 10�4mM KCl electrolyte that is similar to a state of
the deionized and distilled water. Note that jRh> 3 is
validfor applying the P-B equation with acceptable accuracy.
The spanwise velocity becomes minimum in top and bottom
regions, where the negative spanwise velocity means
FIG. 3. Dependence on the slip length for left-side and
right-side walls of
shallow and deep channels, where b¼bL for left-side and b¼bR for
right-side.
FIG. 4. (Color online) The dimension-
less profiles of spanwise velocity and
pressure field for the case of glass with
H¼ 5 lm, W¼ 25 lm, W/RC¼ 0.5,E¼ 100 V/cm, j�1¼ 965 nm (10�4
mMelectrolyte solution), and jRh¼ 4.3.Here, vx,max¼ 1.92 nm/s and
pmax¼ 8.4� 10�10 bar.
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directing to inner wall. The higher pressure in the outer
region results in the spanwise pressure gradient directing
to
inner wall from outer one. As noted before, this spanwise
pressure distribution results from a gap of the flow path
between the inner and the outer walls of a curved channel.
The fluid in the outer region passes relatively slow, due to
a
longer distance of arc A0B0. Since the PDMS/glass channeldoes
not show a qualitative difference, its figures are not pre-
sented here. However, we can observe that the negatively
minimum created in the bottom region of the glass surface
has vanished.
Figure 5 demonstrates how the streamwise velocity pro-
file, corresponding to the EOF velocity, changes for each
case of channels. The magnitude of the streamwise velocity
was found to be greater in the order of about 104 than that
of
the spanwise velocity, supporting the aforementioned
assumption of purely streamwise dependence of Ez (i.e., @//@z in
Eq. (13)). The De number of this study is estimated as
FIG. 5. (Color online) The profiles and contour plots of
streamwise velocity of glass and PDMS/glass channels for (a) thick
and (b) thin EDL thickness, where
other conditions are same with Fig. 4.
102004-6 J.-M. Lim and M.-S. Chun Phys. Fluids 23, 102004
(2011)
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-
very low values in the order of 10�2� 10�3, from which
thebehavior of inward skewness is explained by a dominant
effect of the spanwise pressure gradient over the fluid
inertia
by centrifugal force, similarly as reported in the pressure-
driven flow.18,31 Further computations provide that the mag-
nitudes of spanwise and streamwise velocities as well as
their ratio increase with increasing either dimension of
chan-
nel cross-section or external electric field. Hence, the
ability
to utilize the role of curvature-induced secondary flow by
control both the streamwise and spanwise electric fields can
be linked with the rational design and operation for the
axial
dispersion and the enhanced mixing performance. De also
increases in response to an increase of Rh, implying the
con-tribution of fluid inertia. In the case of thick EDL, the
streamwise velocity becomes decreased in the PDMS/glass
channel due to a lower magnitude of surface potential ws atthe
other sides of PDMS wall, nonetheless the fluid slippage
occurs. Against, PDMS/glass channel has higher streamwise
velocity in the case of thin EDL, except at the bottom of
glass wall. It allows us to figure out that the
electrokinetic
effect is dominant in the lower jb (thick EDL), while thefluid
slippage gets higher contribution in the higher jb(thin EDL).
In order to justify the contribution of surface potential in
the thick EDL case (i.e., lower jb), streamwise velocity
pro-files obtained at the horizontal cut are compared for each
case, as shown in Fig. 6. One needs remark again on the fact
that, regardless of the existence of slip, the higher ws
resultsin the higher EOF velocity. Thickening the EDL results in
an
increase of the electrokinetic strength originated from
long-
range electrostatic repulsion. In Fig. 7, we have changed
the
EDL thicknesses j�1 as 9.7 nm (1.0 mM), 96.5 nm (10�2
mM), and 965 nm (10�4mM), which correspond to jb % 10,1, and
0.1. The fluid slippage in the PDMS/glass channel
leads to an arched shape at the side walls around the
horizon-
tal cut of the profile, and the linear profile is found in
the
glass channel due to the H-S slip alone.
The increasing behavior of streamwise velocity in the
purely glass channel with increasing magnitude of ws is
con-sistent with theoretical predictions according to the H-S
equation for the EOF velocity. The enhancement of stream-
wise velocity by ws is weaker in the PDMS/glass channel,because
the individual effects exerted by PDMS and glass
surfaces, occupying a larger area of top and bottom in shal-
low geometry, are almost equivalent. If the purely PDMS
channel is considered, one can find a qualitative agreement
with the theoretical analysis on non-wetting charged
surfaces
reported in the literature,32 given by
vEOF¼�efEz
l1þjeff b� � ws
f
� �¼�ews Ez
l1þjeff b� �
: (23)
Here, jeffb¼ (f�ws)/ws is the ratio of the slip length to theEDL
characteristic thickness jeff
�1 (¼�ws/5wjs), repre-senting contributions due to fluid slip.
jeff
� 1 is given as
j�1 2Ws=sinh 2Wsð Þð Þ and reduces to j�1 in the case of
D-Happroximation.
B. Vorticity pattern and magnitude of circulation
It is known that curvature introduces a secondary rota-
tional flow-field perpendicular to the flow direction
(namely
Dean flow). From the spanwise and longitudinal velocities,
the stream function S is defined as vr¼ 1=rð Þ @S=@yð Þ andvy¼�
1=rð Þ @S=@rð Þ. Taking the vorticity vector x(¼5� v) yields its h�
and z�components, given by
xh ¼ @vy=@r � @vr=@y; xz ¼ @vy=@x� @vx=@y: (24)
The vorticity provides information on the flow structure
inside the channel. For local rectangular coordinates, it is
obtained after some rearrangement of terms that
FIG. 6. The variations of streamwise velocity profile at the
position of H/2with different values of Ws for glass and PDMS/glass
channels, where otherconditions are same with Fig. 4. Dotted curves
correspond to virtual cases.
FIG. 7. The variations of streamwise velocity profile at the
position of H/2with different values of EDL thickness for glass and
PDMS/glass channels,
where the values of WPDMS and Wglass depend on j�1. Other
conditions are
same with Fig. 4.
102004-7 Curvature-induced secondary microflow motion Phys.
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-
@2S
@x2� 1
xþ RC@S
@xþ @
2S
@y2¼ � xþ RCð Þxz ¼
xþ RCdh
� �r2S:
(25)
Figure 8 provides the vortices results in two channels
with shallow and deep geometry, where the magnitude of dif-
ference between each contour line is all equal as 2� 10�7s�1.
The rotational direction is determined by a sign of the
magnitude of vortex cell, with plus for anticlock-wise and
minus for clock-wise. Evidently, the positions of maximum in
the vorticity profile are located with shifting close to
inner
wall. A pair of counter-rotating vortices placed above and
below the plane of symmetry of the channel is commonly
found in the glass channel. On the contrary, PDMS/glass
channel does not show this trend, but the close pattern
appears
to be developed near the bottom wall of the glass surface.
In
light of the EDL effect displayed in Fig. 7, we obtained the
variations of vorticity pattern in the PDMS/glass channel
with
shallow geometry. In Fig. 9, as j�1 increases, one can see
thehigher magnitude of vortex cell as well as the extended
vortic-
ity layer, representing the strengthened secondary flow. In
the
vorticity profile, the splitting pattern into above and below
the
symmetric plane becomes more obvious.
Estimating the average magnitude of total vorticity
xzj jh i at a specified cross section of the channel allows
oneto predict the circulation strength in the secondary flow.
Such a flow motion provides useful information in the micro-
fluidic mixing and the manipulation of particles or cells.33
In
Fig. 10, xzj jh i increases with increasing either D//L
orchannel curvature W/RC, evidencing the contribution of
fluidinertia. The magnitude of vorticity depends on the slip
length, by noting that actual changes gain even in several
percentages of the relative difference between the corrected
slip length and uncorrected one. The log-log plot provides a
result of scaling relations as xzj jh i!(D//L)a with the slopea.
Each exponent was obtained as almost the same valuefor the same
channel, regardless of the cross sectional geom-
etry and channel curvature W/RC. The effect of W/RC repre-sents
complicated behavior. The shallow geometry keeps
consistency of the higher xzj jh i value in both channels,except
in the glass channel at W/RC¼ 0.1. Moreover, ahigher W/RC leads to
the enhancement of axial velocity
FIG. 8. (Color online) The contour plots of vorticity xzfor (a)
glass and (b) PDMS/glass channels with cross
sections of shallow (lying figures: H¼ 5 lm, W¼ 25lm) and deep
(standing figures: H¼ 25 lm, W¼ 5 lm)geometry, where W/RC¼ 1.0, and
other conditions aresame with Fig. 4.
102004-8 J.-M. Lim and M.-S. Chun Phys. Fluids 23, 102004
(2011)
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-
relative to one in the straight channel. Figure 10 points
out
that, besides the fluid inertia in terms of D//L and W/RC,both
the enhancement of EOF velocity and the secondary
flow are strictly influenced by aspect ratio and surface
prop-
erties of the curved channel.
V. CONCLUSIONS
The curved channel is frequently encountered in the LOC
system because it provides a convenient way for increasing
the effective channel length per unit chip area. The
behavior
of steady EOF with secondary motion in curved microchan-
nels was explored based on a model with P-B/N-S and the nu-
merical framework developed by advanced scheme. Correct
BCs of Navier’s slip derived at curved surfaces were applied
at each hydrophobic wall. Both the glass and the PDMS/glass
channels were considered in the cases of shallow and deep
ge-
ometry with taking the coupled effect of the EDL and the
fluid
slip for jb % 0.1 – 10.Each channel revealed different features
in the velocity
profile, vorticity pattern, and its magnitude, according to
var-
ious conditions. The streamwise velocity profile was domi-
nated by the electrokinetic effect in the thick EDL, while
the
fluid slippage gets higher contribution in the thin EDL
case.
Our simulation results regarding the variations of
streamwise
velocity according to j�1, ws, and b were qualitativelymatched
with the analytical predictions reported in the litera-
ture. Strengthened secondary flow results from increasing
the EDL thickness and the contribution of fluid inertia (D//Land
W/RC), with scaling relations as xzj jh i!(D//L)a. It isfound that,
in a view of the curvature-induced secondary
flow, a channel design of the shallow geometry can take
advantages in either channel if one needs to obtain more
enhanced mixing. The framework established in this study
has an impact on applications to the situation in which the
channel may have arbitrary aspect ratios of cross section as
well as any kinds of wall condition.
ACKNOWLEDGMENTS
This work was supported by the Basic Research Fund
(No. 20100021979) from the National Research Foundation
of Korea.
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s1cor1s2s2AE1E2E3s2BF1E4E5E6E7E8E9E10E11E12s2BE13E14E15E16E17E18E19E20E21E22s3F2s4s4AF3F4F5E23s4BE24F6F7E25F8s5B1B2B3B4B5B6B7F9F10B8B9B10B11B12B13B14B15B16B17B18B19B20B21B22B23B24B25B26B27B28B29B30B31B32B33