Curvature-induced secondary microflow motion in steady electro...

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)


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. Fluids 23, 102004 (2011)


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.



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.



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.



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.



@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.



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.

1T. M. Squires and S. R. Quake, “Microfluidics: Fluid physics at the nanoli-

ter scale,” Rev. Mod. Phys. 77, 977 (2005).2S. Ghosal, “Electrokinetic flow and dispersion in capillary electro-

phoresis,” Annu. Rev. Fluid Mech. 38, 309 (2006).3J. H. Masliyah and S. Bhattacharjee, Electrokinetic and Colloid TransportPhenomena (Wiley, Hoboken, 2006).

4D. J. Harrison, K. Fluri, K. Seiler, Z. Fan, C. S. Effenhauser, and A. Manz,

“Micromachining a miniaturized capillary electrophoresis-based chemical

analysis system on a chip,” Science 261, 895 (1993).5M. Ueda, T. Hayama, Y. Takamura, Y. Horiike, T. Dotera, and Y. Baba,

“Electrophoresis of long deoxyribonucleic acid in curved channels: The

effect of channel width on migration dynamics,” J. Appl. Phys. 96, 2937(2004).

6J. Zhu and X. Xuan, “Particle electrophoresis and dielectrophoresis in

curved microchannels,” J. Colloid Interface Sci. 340, 285 (2009).7W.-H. Tan and S. Takeuchi, “A trap-and-release integrated microfluidic

system for dynamic microarray applications,” Proc. Natl. Acad. Sci.

U.S.A. 104, 1146 (2007).

FIG. 9. The variations of vorticity profile with different values of EDL

thickness for glass shallow channel, where conditions are same with Fig. 8.

Vorticity magnitudes given as numbers on each contour line have an order

of 10�8 s�1.

FIG. 10. The evolution of average magnitude of total vorticity for shallow

and deep geometry at different W/RC with variations of external electric field(D//L), where j�1¼ 965 nm.



http://dx.doi.org/10.1103/RevModPhys.77.977http://dx.doi.org/10.1146/annurev.fluid.38.050304.092053http://dx.doi.org/10.1126/science.261.5123.895http://dx.doi.org/10.1063/1.1776625http://dx.doi.org/10.1016/j.jcis.2009.08.031http://dx.doi.org/10.1073/pnas.0606625104http://dx.doi.org/10.1073/pnas.0606625104

8L. Gervais and E. Delamarche, “Toward on-step point-of-care immuno-

diagnostics using capillary-driven microfluidics and PDMS substrates,”

Lab Chip 9, 3330 (2009).9H. Chen and J. C. Meiners, “Topological mixing on a microfluidic chip,”

Appl. Phys. Lett. 84, 2193 (2004).10M. J. Mitchell, R. Qiao, and N. R. Aluru, “Meshless analysis of

steady-state electro-osmotic transport,” J. Microelectromech. Syst. 9,435 (2000).

11S. K. Griffiths and R. H. Nilson, “Band spreading in two-dimensional

microchannel turns for electrokinetic species transport,” Anal. Chem. 72,5473 (2000).

12B. M. Paegel, L. D. Hutt, P. C. Simpson, and R. A. Mathies, “Turn geome-

try for minimizing band broadening in microfabricated capillary electro-

phoresis channels,” Anal. Chem. 72, 3030 (2000).13J. I. Molho, A. E. Herr, B. P. Mosier, J. G. Santiago, T. W. Kenny,

R. A. Brennen, G. B. Gordon, and B. Mohammadi, “Optimization of

turn geometries for microchip electrophoresis,” Anal. Chem. 73, 1350(2001).

14W.-J. Luo, Y.-J. Pan, and R.-J. Yang, “Transient analysis of electro-

osmotic secondary flow induced by dc or ac electric field in a curved rec-

tangular microchannel,” J. Micromech. Microeng. 15, 463 (2005).15J.-K. Chen, W.-J. Luo, and R.-J. Yang, “Electro-osmotic flow driven by

DC and AC electric fields in curved microchannels,” Jpn. J. Appl. Phys.

45, 7983 (2006).16H. S. Woo, B. J. Yoon, and I. S. Kang, “Optimization of zeta potential dis-

tributions for minimal dispersion in an electroosmotic microchannel,” Int.

J. Heat Mass Transfer 51, 4551 (2008).17M.-S. Chun, T. S. Lee, and N. W. Choi, “Microflow analysis of electroki-

netic streaming potential induced by microflows of monovalent electrolyte

solution,” J. Micromech. Microeng. 15, 710 (2005).18J. H. Yun, M.-S. Chun, and H. W. Jung, “The geometry effect on steady

electrokinetic flows in curved rectangular microchannels,” Phys. Fluids

22, 052004/1-10 (2010).19W. R. Dean, “Note on the motion of fluid in a curved pipe,” Philos. Magn.

4, 208 (1927).20S. V. Patankar, Numerical Heat Transfer and Fluid Flow (McGraw-Hill,

New York, 1980).

21C. L. M. H. Navier, “Mémoire sur les lois du mouvement des fluids,”

Mem. Acad. Sci. Inst. Fr. 6, 389 (1823).22E. Lauga and H. A. Stone, “Effective slip in pressure-driven Stokes flow,”

J. Fluid Mech. 489, 55 (2003).23L. Bocquet and J.-L. Barrat, “Flow boundary conditions from nano- to

micro-scales,” Soft Matter 3, 685 (2007).24D. P. J. Barz and P. Ehrhard, “Model and verification of electrokinetic

flow and transport in a micro-electrophoresis device,” Lab Chip 5, 949(2005).

25D. C. Tretheway and C. D. Meinhart, “Apparent fluid slip at hydrophobic

microchannel walls,” Phys. Fluids 14, L9 (2002).26P. Joseph and P. Tabeling, “Direct measurement of the apparent slip

length,” Phys. Rev. E 71, 035303R (2005).27P. Huang, J. S. Guasto, and K. S. Breuer, “Direct measurement of slip

velocities using three-dimensional total internal reflection velocimetry,”

J. Fluid Mech. 566, 447 (2006).28M. Mala, D. Li, C. Werner, H. Jacobasch, and Y. Ning, “Flow characteris-

tics of water through a microchannel between two parallel plates with

electrokinetic effects,” Int. J. Heat Fluid Flow 18, 489 (1997).29M.-S. Chun and S. Lee, “Flow imaging of dilute colloidal suspension in

PDMS-based microfluidic chip using fluorescence microscopy,” Colloids

Surf. A 267, 86 (2005).30J. Song, J. F. L. Duval, M. A. C. Stuart, H. Hillborg, U. Gunst, H. F.

Arlinghaus, and G. J. Vancso, “Surface ionization state and nanoscale

chemical composition of UV-irradiated poly(dimethylsiloxane) probed by

chemical force microscopy, force titration, and electrokinetic meas-

urements,” Langmuir 23, 5430 (2007).31Y. Yamaguchi, F. Takagi, K. Yamashita, H. Nakamura, H. Maeda, K.

Sotowa, K. Kusakabe, Y. Yamasaki, and S. Morooka, “3-D simulation and

visualization of laminar flow in a microchannel with hair-pin curves,”

AIChE J. 50, 1530 (2004).32L. Joly, C. Ybert, E. Trizac, and L. Bocquet, “Liquid friction on charged

surfaces: From hydrodynamic slippage to electrokinetics,” J. Chem. Phys.

125, 204715/1-14 (2006).33H. Song, Z. Cai, H. Noh, and D. J. Bennett, “Chaotic mixing in microchan-

nels via low frequency switching transverse electro-osmotic flow gener-

ated on integrated microelectrodes,” Lab Chip 10, 734 (2010).



http://dx.doi.org/10.1039/b906523ghttp://dx.doi.org/10.1063/1.1686895http://dx.doi.org/10.1109/84.896764http://dx.doi.org/10.1021/ac000595ghttp://dx.doi.org/10.1021/ac000054rhttp://dx.doi.org/10.1021/ac001127+http://dx.doi.org/10.1088/0960-1317/15/3/006http://dx.doi.org/10.1143/JJAP.45.7983http://dx.doi.org/10.1016/j.ijheatmasstransfer.2008.02.008http://dx.doi.org/10.1016/j.ijheatmasstransfer.2008.02.008http://dx.doi.org/10.1088/0960-1317/15/4/007http://dx.doi.org/10.1063/1.3427572http://dx.doi.org/10.1080/14786440708564324http://dx.doi.org/10.1017/S0022112003004695http://dx.doi.org/10.1039/b616490khttp://dx.doi.org/10.1039/b503696hhttp://dx.doi.org/10.1063/1.1432696http://dx.doi.org/10.1103/PhysRevE.71.035303http://dx.doi.org/10.1017/S0022112006002229http://dx.doi.org/10.1016/S0142-727X(97)00032-5http://dx.doi.org/10.1016/j.colsurfa.2005.06.046http://dx.doi.org/10.1016/j.colsurfa.2005.06.046http://dx.doi.org/10.1021/la063168shttp://dx.doi.org/10.1002/aic.v50:7http://dx.doi.org/10.1063/1.2397677http://dx.doi.org/10.1039/b918213f

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