the role of fluid inertia on streamwise velocity and vorticity...

Korea-Australia Rheology Journal September 2010 Vol. 22, No. 3 211

Korea-Australia Rheology JournalVol. 22, No. 3, September 2010 pp. 211-218

The role of fluid inertia on streamwise velocity and vorticity pattern in

curved microfluidic channels

Myung-Suk Chun1,*, Jin-Myoung Lim

1 and Dae Young Lee

2

1Complex Fluids Laboratory, Energy Div., Korea Inst. of Sci. and Tech. (KIST), Seongbuk-gu, Seoul 136-791, Republic of Korea2Energy Mechanics Center, Korea Inst. of Sci. and Tech. (KIST), Seongbuk-gu, Seoul, 136-791, Republic of Korea

(Received May 31, 2010; final revision received August 11, 2010; accepted August 30, 2010)

Abstract

Recently, we introduced a secondary Dean flow in curved rectangular microchannels by applying the finitevolume scheme with a SIMPLE (semi-implicit method for pressure-linked equations) algorithm for thepressure-driven electrokinetic transport (Yun et al., 2010). This framework is based on the theoretical modelcoupled with the full Poisson-Boltzmann, Navier-Stokes, and the Nernst-Planck principle of net charge con-servation. To explore intensively the effect of fluid inertia on the secondary flow, both the applied pressuredrop ∆p/L and the channel curvature W/RC are changed for three kinds of rectangular channel cross sectionwith considering the electric double layer and fluid slip condition. Simulation results exhibit that the squarechannel (i.e., channel aspect ratio 1) gets the higher axial velocity, compared to the others. The changeof its skewed velocity profile from inward to outward was found with increasing fluid inertia caused byincreasing ∆p/L, due to the reduced spanwise pressure gradient. The curvature introduces the presence ofpairs of counter-rotating vortices perpendicular to the flow direction. Although the square channel showsa different feature of very close pattern in the vorticity profile, the total magnitude of average vorticityincreases commonly in all cases with increasing either ∆p/L or W/RC, providing scaling relations with thealmost same value of exponent 2. It is obvious that the role of fluid inertia should explicitly be understoodfor a precise design of microfluidic chips taking arbitrary channel aspect ratios.

Keywords : microfluidics, curved channel, secondary flow, fluid inertia, vorticity, streamwise velocity, elec-

trokinetics

1. Introduction

Because the turn geometry in microchannels provides a

convenient scheme for increasing the effective channel

length per unit chip area, it becomes an essential tool in lab-

on-chips (LOC) systems especially for single cell or par-

ticle manipulations via the trap-and-release mechanism

(Tan and Takeuchi, 2007; Russom et al., 2009). It also

allows chaotic advection generating passive mixing at low

Reynolds number (Re) and the controllable reaction res-

idence for synthetic applications (Paegel et al., 2000; Liu et

al., 2000; Gervais and Delamarche, 2009). Likewise, the

improved efficiency in microfluidic devices implying pres-

sure-driven flows can be attained by folding the straight

channel into the shape of a serpentine curved channel. The

ability to figure out the role of channel curvature in the

axial dispersion and the enhanced performance arising from

the curved channel is pertinent to either the rational design

or operation of integral components of LOC devices.

Microfluidics and nanofluidics involving electrokinetic

phenomena have attracted attention, because the electric

double layer (EDL) makes the transport behavior deviate

from that described by the laminar flow equation in general

(Squires and Quake, 2005; Schoch et al., 2008). Since the

earlier works traced back to 1960s and 1970s (Rice and

Whitehead, 1965; Levine et al., 1975), many studies have

contributed to elucidating the electrokinetic flow phenom-

ena. However, note that those studies were almost confined

to an analysis with applying straight channels (Yang and

Li, 1997; Hu et al., 1999; Chun et al., 2005a; Chun et al.,

2005b; Gong et al., 2008). Chun et al. (2005a; 2005b)

developed the explicit model accompanying the numerical

scheme for electrokinetic flows in cylindrical as well as

rectangular microchannels. The model was based on the

analysis of the Navier-Stokes (N-S) momentum equation

coupled with the nonlinear Poisson-Boltzmann (P-B) field,

taking further consideration of the Nernst-Planck (N-P)

equation for charge conservation. As a continued study,

they have treated recently the problem of electrokinetic

flow in a curved microchannel (Yun et al., 2010).

In practical applications, a microchannel may frequently*Corresponding author: [email protected]© 2010 by The Korean Society of Rheology

Myung-Suk Chun, Jin-Myoung Lim and Dae Young Lee

212 Korea-Australia Rheology Journal

be fabricated with the substrate and covering plate and pos-

sesses different rectangular cross sections of the channel.

The flow pattern and species transport depend on the chan-

nel aspect ratio as well as the curvature, therefore, explicit

analyses should be accomplished along the various curved

microchannel geometry shown in Fig. 1. The Re and non-

dimensional curvature are combined into the Dean number

(De = ) that has physical significance quantify-

ing the inertial force in the curved channel (Dean, 1927).

As mentioned in our previous study (Yun et al., 2010), the

streamwise axial velocity tends to shift toward the inner

wall caused by a stronger effect of the spanwise pressure

gradient, if De is sufficiently low. This feature is contrary

to the case of narrow-bore channels taking a hydraulic

diameter above millimeter scale, where the axial velocity

becomes skewed into the outer wall as De gets higher

enough (Thangam and Hur, 1990).

In this paper, we examined the combined effect of span-

wise pressure gradient and fluid inertia on the secondary

Dean flow behavior of electrokinetic transport in shallow,

square (to be exact, nearly square), and deep channels with

ranging complementary aspect ratios (i.e., H/W=0.2−5.0).

The variations of fluid inertia are achieved by changing the

pressure drop and the channel curvature, where dimension

of microchannels applied in this study is commonly used in

actual micro total analysis system (µTAS). In order to

solve the present problem, the SIMPLE (semi-implicit

method for pressure-linked equations) algorithm was

employed, that is considered as the well-established

numerical technique (Patankar, 1980). Our numerical

framework enables to consider exactly the arbitrary con-

figurations of the channel wall taking unequal surface

charge and fluid slip at solid surfaces, which lead to irreg-

ular flow patterns and nonuniform species transport. To

focus on the fluid inertia effect, however, we choose the

uniformly charged surface of microchannels created with

hydrophobic non-wetting material only. Numerical results

of the skewed streamwise velocity and vorticity changes

were reported for different cases of channel cross section,

and relevant discussions were addressed.

2. Model Considerations

2.1. The microchannel geometry We consider a situation for pressure-driven and steady

state electrokinetic flow through a curved rectangular

microchannel of the width W and the height (or, referred to

as depth) H with a uniform curvature, as presented in Fig. 2.

It is available to transform a toroidal coordinate system to

Cartesian one. The curvature radius is mea-

sured from the axis of curvature, and x (= r−RC) and z are

the spanwise and streamwise axial distances along the

channel axis, respectively. Providing the spanwise and lon-

gitudinal velocities, the stream function S is defined as vr

= and vy = . Recognizing the

vorticity vector ω (= ) perpendicular to the axial direc-

tion yields its θ - and z-components, given by

. (1)

For rectangular coordinates, it is obtained after some rear-

rangement of terms that

. (2)

2.2. The electrokinetic microflow The charge property of channel surface is characterized

by the electric surface potential, which is generally deter-

mined by electrokinetic zeta potential measurement. When

the non-conductive and charged surface is in contact with

an electrolyte in solution, the electrostatic charge would

influence the distribution of nearby ions and the electric

Re W RC⁄

RC dz dθ⁄=

1 r⁄( ) ∂S ∂y⁄( ) 1 r⁄( ) ∂S ∂r⁄( )–

∇ v×

ωθ ∂vy ∂r ∂vr ∂y⁄ ;–⁄ ωz ∂vy ∂x ∂vx ∂y⁄ –⁄= =

∂2S

∂x2

-------- 1

x RC+-------------

∂S

∂x------–

∂2S

∂y2

--------+ x RC+( )ωz–x RC+

dh

-------------⎝ ⎠⎛ ⎞∇2

S= =

Fig. 1. The basic concept of contributions of fluid inertia and

pressure gradient to a secondary flow in the curved channel.

Fig. 2. The curved channel with surface properties and relevant

coordinates system.

The role of fluid inertia on streamwise velocity and vorticity pattern in curved microfluidic channels


field is consequently established (Chun and Bowen, 2004).

The nonlinear P-B equation governing the electric potential

ψ field is given as

(3)

where the dimensionless electric potential Ψ denotes Λeψ/

kT and the inverse EDL thickness . Here,

the electrolyte concentration in bulk solution at the elec-

troneutral state nb [1/m3] equals to a product of the

Avogadro’s number NA and bulk electrolyte concentration

[mM]. Λi is the valence of i ions, ε the elementary charge,

e the dielectric constant given as a product of the dielectric

permittivity of a vacuum and the relative permittivity for

aqueous fluid (=78.9), and kT the Boltzmann thermal

energy at room temperature (RT).

The Boltzmann distribution of the ionic concentration of type

i (i.e., ni=nb exp(− ieψ/kT)) provides a local charge density

ieni. Since the net charge density ρe is defined by

, one can determine ρe = .

The constant-potential surface boundary conditions (BCs)

are imposed on each side of the uniformly charged surface

displayed in Fig. 1: Ψ = Ψs,L at x = 0, Ψ = Ψs,R at x = W,

Ψ = Ψs,B at y = 0, and Ψ = Ψs,T at y = H. The subscripts

T, B, R, and L correspond to top, bottom, right, and left-

sided channel wall. Due to the uniform wall condition here,

Ψs,L = Ψs,R = Ψs,B = Ψs,T.

The velocity field for an incompressible Newtonian

aqueous fluid at the steady state obeys the N-S equation,

(4)

where ρ and µ are the density and viscosity of the fluid, respec-

tively. The velocity profile for straight channel with uncharged

wall is described by the Stokes flow . Since the

end effects are negligible , the laminar flow velocity

and the pressure are expressed as

and , respectively (Yun et al., 2010). The body

force F per unit volume ubiquitously caused by the θ -

directional action of flow-induced electric field Eθ on

ρe(r,y) can be written as Fθ = ρeEθ. From the fact that the

spanwise dependence of Eθ is much smaller than its stream-

wise axial dependence, it is defined by the flow-induced

streaming potential φ as Eθ(θ) = −dφ(θ)/dθ. Utilizing the

transformation of toroidal into Cartesian coordinates, each

component of the N-S equation is determined as

:

, (5)

:

, (6)

:

(7)

where the fluid velocity in Eq. (7) is coupled with φ.

Another question concerning the BC at the channel wall is

fluid slip that is a function of the wettability indicated by the

contact angle. According to the Navier’s fluid slip condition

(Navier, 1823), a definition of a slip length β at a hydro-

phobic (solvophobic) rigid boundary, with unit normal vec-

tor directed into the fluid, linearly relates the velocity at the

wall to the wall shear rate (Lauga and Stone, 2003; Bocquet

and Barrat, 2007). The slip length β is a material parameter,

and it is the local equivalent distance below the solid surface

at which the no-slip BC (β = 0) would be satisfied if the flow

field were extended linearly outside of the physical domain.

Since each side of the wall in Fig. 2 is specified by uniform

hydrophobic surfaces with b, each BC is applied as:

. Since the velocity

becomes a maximum at the center of the channel, the sign

of the velocity gradient changes into negative along the

axis.

Next, the net current conservation ( ) is applied in

the microchannel taking into account the N-P equation. The

ionic flux in terms of a number concentration [1/m2·s] is

possible to describe by the contribution owing to convec-

tion, diffusion, and migration resulting from the pressure

difference, concentration gradient, and electric potential

gradient, respectively (Werner et al., 1998; Masliyah and

Bhattacharjee, 2006). Ions in the mobile region of the EDL

are transported along with the flow through the channel

length L, commonly inducing the electric convection cur-

rent (i.e., streaming current) IS for arbitrary cross section

( ). The accumulation of ions sets up an electric

field Ez with the streaming potential difference ∆φ (= EzL).

Subsequently, this field causes the conduction current IC

( ) to flow back in the opposite direction. The total

resistance consists of the surface resis-

tance Rs and the fluid resistance

Rf in parallel, expressed as

, (8)

. (9)

Here, the specific surface conductivity λs depends on the

property of channel wall and λf

indicates the local fluid conductivity. Here, the

mobility Ki (= Di/kT) is defined as the velocity of ion spe-

∇2Ψ κ2 Ψsinh=

κ 2nbΛi

2e2εkT⁄=

Λ

Λ Λi ieni∑i Λe n + n ––( )=( ) 2Λenb Ψsinh–

ρ v v∇⋅( ) p µ∇2v F+ +∇–=

µ∇2v ∇p=( )

∂ ∂θ⁄ 0=( )v vr r y,( ) vy r y,( ) vθ r y,( ),,[ ]=

p p r y θ, ,( )=

r x→

ρ vx

∂vx

∂x------- vy

∂vx

∂y-------

vz

2

r----–+⎝ ⎠

⎛ ⎞ ∂p

∂x------– µ

1

r---∂∂x----- r

∂vx

∂x-------⎝ ⎠

⎛ ⎞ vx

r2

----–∂2

vx

∂y2

---------++=

y y→

ρ vx

∂vy

∂x------- vy

∂vy

∂y-------+⎝ ⎠

⎛ ⎞ ∂p

∂y------– µ

∂2vy

∂x2

--------- 1

r---∂vy

∂x-------

∂2vy

∂y2

---------+ ++=

θ z→

ρ vx

∂vz

∂x------- vy

∂vz

∂y-------

vxvz

r--------+ +⎝ ⎠

⎛ ⎞ RC

r------

∂p

∂z------ ρe

∂φ∂z------+⎝ ⎠

⎛ ⎞–=

µ+1

r---∂∂x----- r

∂vz

∂x-------⎝ ⎠

⎛ ⎞ vz

r2

----–∂2

vy

∂y2

---------+

vx y 0 &H=β= ∂vx ∂y⁄( )y 0 &H=

vy x W±=, β ∂vy ∂x⁄( )x W±=

vz x W ±=,= =

β ∂vz ∂x⁄( )x W±= vz y 0 &H=, β ∂vz ∂y⁄( )y 0 &H=

=

∇ I⋅ 0=

ρevzArea Ad∫=

φ RTot⁄∆=

RTot Rs

1–Rf

1–+( )

1–

=

Rs T,

1–Rs B,

1–Rs R,

1–Rs L,

1–+ + +[ ]

1–

=( )

Rs

L

W λs T, λs B,+( ) H λs R, λs L,+( )+-------------------------------------------------------------------=

Rf

L

λf x y,( ) xd yd0

W

∫0

H

∫---------------------------------------=

x y,( )])



cies i in the direction of an electric field of unit strength

with diffusivity Di for cations (+) and anions (−). At the

steady state, the net current I consisting of IS and IC should

be zero, viz. I IS + IC =0. Following this relationship in

generic form, one can derive the streaming potential

as .

3. Computational Methods

The electric potential is first obtained by solving Eq. (3)

based on the finite difference method (FDM), of which

basic procedures are analogous to the previous work (Chun

et al., 2005a). Five-point central difference method enables

one to solve the finite difference equation by successive

under-relaxation iterative calculation. The fluid velocity

profile can be computed by solving the aforementioned N-

S equation of Eqs. (5)~(7) together with charge conserva-

tion derived from N-P principle. The well-established SIM-

PLE algorithm represented by the finite volume method

(FVM) was employed to solve the unknown pressure terms

p(x,y) of the N-S equation by using the continuity equation

( ) as the pressure-velocity coupling. Detailed dis-

cussion of the SIMPLE is available in the literature (Patan-

kar, 1980). Accomplishing the velocity and pressure

corrections, we repeat the sequence of operations until the

convergence is guaranteed. The staggered grid and an alter-

nating direction implicit (ADI) method are adopted when

the procedure comes to treat x and y components of the dis-

cretized N-S equation and the continuity equation. As men-

tioned in the previous study (Yun et al., 2010), the

staggered grid is half-node shifted from the original grid for

momentum equation, which prevents the failure of pressure

estimations and assures calculations that are more accurate.

After the mesh refinement with grid convergence test to

yield satisfactory accuracy, the computational domain is

discretized into 151×151 equally spaced grid points in

both x and y directions. Convergence criteria are specified

with the relative variation between two successive itera-

tions to be smaller than the pre-assigned accuracy level of

10−8. The convergence is so fast, and 250~300 iterations

were enough to achieve convergence for the present chan-

nel dimension. The under-relaxation coefficients were cho-

sen in the range 0.5~0.7.

4. Results and Discussion

4.1. Choice of conditionsIllustrative computations are performed considering the 1-

1 type electrolyte of KCl fluid (ρ=103 kg/m3, µ=10-3 Pa·s)

with ∆p/L of 1.0 bar/m. From the molar conductance of infi-

nite dilution related to the ion migration (Chun et al., 2005a),

the individual mobilities of ions Ki in aqueous solutions at

RT are evaluated as 7.91×10-1 and 8.19×10-1 pM·s/kg for K+

and Cl−, respectively. We set the dimension of 200 µm

wide and 40 µm deep, 80 µm wide and 100 µm deep, and

40 µm wide and 200 µm deep as the prototype of shallow,

square, and deep channels, respectively. Their cross-sec-

tional areas are same, but the hydraulic radius Rh (= HW/

(H+W)) is 33.3 µm for shallow and deep channels and

44.4 µm for square channel.

The electrokinetic strength originated from long-range

electrostatic repulsion increases by either thickening the

EDL or increasing the electric surface potential. We con-

sider a strong electrokinetic interaction with 10-4 mM KCl

electrolyte that is presumably similar to a state of the

deionized and distilled water. It provides the greatest EDL

thickness κ−1 =965 nm. The magnitude of dimensionless

surface potential has presumably been

taken as 2.0 ( = 51 mV). For the low potential (nor-

mally, ≤30 mV), the errors of using the linearized P-B

equation (i.e., Debye-Hückel ansatz) are quite small. How-

ever, employing the full P-B version is still necessary,

unless this study is confined to the case of low potential. To

take the condition of κβ 1, we assume the slip length β

as 1 µm. Experimentally, β is determined by particle streak

velocimetry with epi-fluorescence microscope (Chun et al.,

2005b). The most recent measurement performed in our

group results in µm for partially non-wetting poly-

dimethylsiloxane (PDMS) surfaces (cf. contact angle of ca.

98o) with less than the critical shear rate.

The numerical code has been validated from the fact that

the velocity prediction matches the analytical problem of

inert flows in a straight channel (W/RC → 0). Axial velocity

profiles at the horizontal cut (H/2) along the streamwise

direction maintain a fully developed state in the entire

region of the curved channel except the entrance. To ensure

the axial independency of the velocity , the

velocity and pressure fields are analyzed in the transverse

section (i.e., r and y directions) at the middle of the turn.

4.2. Streamwise velocity with variations of fluid

inertia Fig. 3 demonstrates how the streamwise velocity profile

changes for each geometry of channel cross section with

H/W =0.2, 1.25, and 5. Comparing each case shows a dif-

ferent pattern of the axial velocity, however, its behavior of

inward skewness is identical. This trend is caused by a

dominant effect of the spanwise pressure gradient over the

fluid inertia by centrifugal force, as reported in the liter-

ature (Snyder and Lovely, 1990; Yamaguchi et al., 2004;

Yun et al., 2010). Note that the degree of inward skewness

is different, where its maximum and minimum values are

acquired from shallow and deep channels, respectively. As

the H/W gets lower, the axial velocity profile becomes hor-

izontal by spreading to both sides of the wall, providing the

fact that the portion of the maximum axial velocity in the

shallow channel is more readily forced to the inner wall

compared to the case of deep channel. In addition, one

≡

φ ISRTot–=( )∆ RTot ρe x y,( )vz x y,( ) xd yd0

W

∫0

H

∫–

∇ v⋅ 0=

Ψs ψs e kT⁄=( )ψs

ψs

≅

β 1≈

∂vz ∂z 0=⁄( )



needs remark on the effect of fluid inertia representing by

De. As the channel aspect ratio H/W becomes 1.0 (i.e., per-

fect square channel) for the same cross-sectional area, both

Rh and average axial velocity <vz> get the highest values,

indicating the highest De. In Fig. 3, the highest <vz> is

2.9 cm/s for square channel.

By noting that the axial velocity profile becomes skewed

in the curved channel, we need determine the shifting of the

spanwise position (x/W) of maximum axial velocity accord-

ing to the variations of fluid inertia. The variations of fluid

inertia are obtained by changing the pressure drop and cur-

vature, as provided in Fig. 4. The shallow channel reveals

greater magnitude of velocity shift toward the inner wall

than the others, and the degree of inward shifting with

increasing curvature ratio W/RC (rapid turn) is much more in

the shallow channel. The shifting tendency toward the inner

wall mostly does not depend on ∆p/L in deep and shallow

channels, whereas the shifting direction with increasing ∆p/

L changes to outer wall from inner wall in the square chan-

nel. Besides the attributed reason due to inertial force already

discussed in Introduction section, the skewed velocity profile

is also influenced by the channel aspect ratio.

Figs. 5 and 6 present a difference in the spanwise pres-

sure gradient and velocity observed between inward and

outward skewed cases. This pressure distribution along the

spanwise direction results from a gap of the flow path

between the inner (arc AB) and the outer (arc A B ) walls

of a curved channel in Fig. 2. The fluid in the outer region

passes relatively slow, due to its longer distance (arc

A B ) than the fluid in the inner region (arc AB). As a

result, the pressure in the out region becomes higher than

′ ′

′ ′

Fig. 3. (Color online only) The contour plots of streamwise

velocity for (a) shallow (H=40 µm, W =200 µm), (b)

square (H=100 µm, W=80 µm), and (c) deep (H=

200 µm, W=40 µm) channel, where W/RC=0.5 and ∆p/

L=1.0 bar/m. All units of the legends are cm/sec.

Fig. 4. The evolution of nondimensional spanwise position of

maximum axial velocity for shallow, square, and deep

channels at different W/RC with variations of ∆p/L.



that in the inner region. The maximum spanwise velocity

is located at the central region of the channel, and min-

imum velocity is divided into top and bottom regions. The

magnitude of the spanwise velocity is smaller in the order

of 10-3~10-5 than that of the axial velocity, verifying the

aforementioned assumption of purely streamwise depen-

dence of Ez (i.e., φ/ z in Eq. (7)). De ( , where

Re = ) is estimated as 0.27 and 24.4 for the con-

ditions of Figs. 5 and 6. The major contribution of outward

skewness of axial velocity can be identified as lower pres-

sure gradient directing to inner wall from outer shown in

Fig. 6, compared to that of Fig. 5.

The electric potential profile moves toward a center

region of the channel by repulsive screening with increas-

ing κ−1, though its figure is not presented here. As

explained the streaming potential, the counterions in the

diffuse layer drag the surrounding liquid molecules, and

consequently the overall flow rate will be reduced. The

apparent viscosity enhancement pertaining to a rheological

property can be estimated from the ratio of the charged

fluid viscosity to that of the inert case. It is referred to as the

electroviscous effect corresponding to the concept of fric-

tion factor. The average velocity decreases in accordance

with the increasing electrokinetic interaction attained by

either higher Ψs or weaker screening with larger κ−1. How-

ever, this electroviscous effect can hardly be observed in

this study, because of the very small relative EDL thickness

(i.e., κRh=35 and 46) as well as the presence of fluid slip.

∂ ∂ Re W RC⁄=

2Rhρ vz⟨ ⟩ µ⁄

Fig. 5. (Color online only) The dimensionless profiles of (a) pressure field and (b) spanwise velocity for the case of inward skewed axial

velocity in square channel at W/RC=1.0 and ∆p/L=0.1 bar/m.

Fig. 6. (Color online only) The dimensionless profiles of (a) pressure field and (b) spanwise velocity for the case of outward skewed

axial velocity in square channel at W/RC =1.0 and ∆p/L=10 bar/m.



4.3. Vorticity pattern with variations of fluid inertia In the previous study (Yun et al., 2010), streamline results

reveal that curvature introduces a secondary rotational flow-

field perpendicular to the flow direction (namely Dean

flow) with upper anticlock-wise and lower clock-wise. With

the streamline, the vorticity provides information on the

flow structure inside the channel. Fig. 7 provides the vor-

tices results of three cases for two pressure drops, where the

magnitude of difference between each contour line is all

equal as 0.01 s-1. It is evident that the vorticity profile

becomes much more close patterned with increasing pres-

sure drop, representing the greater effect of fluid inertia. A

pair of counter-rotating vortices placing above and below

the plane of symmetry of the channel is common in all

cases. The vorticity ωz profile of shallow channel demon-

strates indeed the loose-distribution pattern as a whole.

Although this loose pattern is found in the deep channel, the

close pattern appears nearby top and bottom regions and the

positions of a pair of maximum vortices approach more

close to top and bottom surfaces. Against these one, the

contour of square channel presents a quite different feature,

showing the very close pattern of vorticity profile.

Estimating the total magnitude of average vorticity

at the specified cross section of channel allows to predict

the circulation strength in the secondary flow. Such a flow

motion provides a useful tool in the microfluidic mixing

system and the manipulation of particles or cells. In Fig. 8,

square, deep, and shallow channels represent the order of

the higher for the same conditions. This trend is

directly related to the fact that the square channel takes the

highest average value of axial velocity. The total magni-

tude of average vorticity increases with increasing either

∆p/L or W/RC, evidencing again the contribution of fluid

inertia. The log-log plot of Fig. 8 provides an interesting

result of scaling relations as , where least

square fit gives the slope α. This exponent was obtained as

almost the same value of 2. Figs. 4 and 8 point out that,

besides the fluid inertia represented by ∆p/L and channel

curvature W/RC, the magnitudes of axial velocity and vor-

ticity are strictly influenced by the channel aspect ratio.

5. Conclusions

On the basis of our numerical framework developed

recently, the analysis of the secondary Dean flow behavior

with variations of fluid inertia effect has been performed

for a precise design of microfluidic chip. The pressure-

ωz⟨ ⟩

ωz⟨ ⟩

ωz⟨ ⟩ p L⁄∆( )α∝

Fig. 7. The contour plots of vorticity ωz for (a) shallow

(H=40 µm, W=200 µm), (b) square (H=100 µm, W=

80 µm), and (c) deep (H=200 µm, W=40 µm) channel,

where W/RC=0.5, and ∆p/L=0.1 bar/m (left ones) and

1.0 bar/m (right ones).

Fig. 8. The evolution of total magnitude of average vorticity for

shallow, square, and deep channels at different W/RC with

variations of ∆p/L.



driven steady electrokinetic flow in curved rectangular

microchannels was fully explored by establishing theoret-

ical model with relevant coupled equations. The electro-

kinetic body force incorporated with both the nonlinear P-

B electrostatic field and the flow-induced electric field was

implemented in the N-S equation, accompanying the net

charge conservation. Unknown pressure terms of the N-S

equation were solved by using the continuity equation as

the pressure-velocity coupling achieves convergence.

Shallow, square, and deep channels were considered with

taking the electric double layer of κ−1 =965 nm as well as

fluid slip condition of β=1 µm. As the channel aspect ratio

becomes 1.0 for the same cross-sectional area, the higher

axial velocity and the directional change of its skewed pro-

file were found with a secondary motion. The inward

skewed profile changes to outward one with increasing

fluid inertia caused by increasing ∆p/L, due to the reduced

spanwise pressure gradient. Although a pair of counter-

rotating vortices is commonly placed above and below the

symmetry plane of the channel, the square channel shows

particularly a different feature of very close pattern in vor-

ticity profile. The total magnitude of average vorticity

increases with increasing either ∆p/L or W/RC, pro-

viding scaling relations as . The numerical

framework established in this study has a remarkable

impact on applications to the situation in which the channel

may have arbitrary shapes of cross section.

Acknowledgments

This research was supported by the Converging Research

Center Program through the National Research Foundation

of Korea funded by the Ministry of Education, Science and

Technology (Grant No. 2009-0082136).

References

Bocquet, L. and J.-L. Barrat, 2007, Flow boundary conditions

from nano- to micro-scales, Soft Matter 3, 685.

Chun, M.-S. and W.R. Bowen, 2004, Rigorous calculations of

linearized Poisson-Boltzmann interaction between dissimilar

spherical colloids and osmotic pressure in concentrated dis-

persions, J. Colloid Interface Sci. 272, 330.

Chun, M.-S., T.S. Lee and N.W. Choi, 2005a, Microfluidic anal-

ysis of electrokinetic streaming potential induced by micro-

flows of monovalent electrolyte solution, J. Micromech.

Microeng. 15, 710.

Chun, M.-S., T.S. Lee and K. Lee, 2005b, Microflow of dilute

colloidal suspension in narrow channel of microfluidic-chip

under Newtonian fluid slip condition, Korea-Australia Rhe-

ology J. 17, 207.

Dean, W.R., 1927, Note on the motion of fluid in a curved pipe,

Philos. Mag. 4, 208.

Gervais, L. and E. Delamarche, 2009, Toward on-step point-of-

care immunodiagnostics using capillary-driven microfluidics

and PDMS substrates, Lab Chip 9, 3330.

Gong, L., J. Wu, L. Wang and K. Cao, 2008, Streaming potential

and electroviscous effects in periodical pressure-driven micro-

channel flow, Phys. Fluids 20, 063603.

Hu, L., J.D. Harrison and J.H. Masliyah, 1999, Numerical model

of electrokinetic flow for capillary electrophoresis, J. Colloid

Interface Sci. 215, 300.

Lauga, E. and H.A. Stone, 2003, Effective slip in pressure-driven

Stokes flow, J. Fluid Mech. 489, 55.

Levine, S., J.R. Marriott, G. Neale and N. Epstein, 1975, Theory

of electrokinetic flow in fine cylindrical capillaries at high

zeta-potentials, J. Colloid Interface Sci. 52, 136.

Liu, R.H., M.A. Stremler, K.V. Sharp, M.G. Olsen, J.G. Santiago,

R.J. Adrian, H. Aref and D.J. Beebe, 2000, Passive mixing in

a three-dimensional serpentine microchannel, J. MEMS 9, 190.

Masliyah, J.H. and S. Bhattacharjee, 2006, Electrokinetic and

Colloid Transport Phenomena, John Wiley & Sons, Hoboken.

Navier, C.L.M.H., 1823, Mémoire sur les lois du mouvement des

fluides, Mémoires de l’Académie Royale des Sciences de

l’Institut de France 6, 389.

Paegel, B., L. Hutt, P. Simpson and R. Mathies, 2000, Turn

geometry for minimizing band broadening in microfabricated

capillary electrophoresis channels, Anal. Chem. 72, 3030.

Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow,

McGraw-Hill, New York.

Rice, C.L. and R. Whitehead, 1965, Electrokinetic flow in a nar-

row cylindrical capillary, J. Phys. Chem. 69, 4017.

Russom, A., A.K. Gupta, S. Nagrath, D.D. Carlo, J.F. Edd and

M. Toner, 2009, Differential inertial focusing of particles in

curved low-aspect-raio microchannels, New J. Phys. 11, 075025.

Schoch, R.B., J. Han and P. Renaud, 2008, Transport phenomena

in nanofluidics, Rev. Mod. Phys. 80, 839.

Snyder, B. and C. Lovely, 1990, A computational study of devel-

oping 2-D laminar flow in curved channels, Phys. Fluids A 2,

1808.

Squires, T.M. and S.R. Quake, 2005, Microfluidics: Fluid physics

at the nanoliter scale, Rev. Mod. Phys. 77, 977.

Tan, W.-H. and S. Takeuchi, 2007, A trap-and-release integrated

microfluidic system for dynamic microarray applications,

Proc. Natl. Acad. Sci. U. S. A. 104, 1146.

Thangam, S. and N. Hur, 1990, Laminar secondary flows in

curved rectangular ducts, J. Fluid Mech. 217, 421.

Werner, C., H. Körber, R. Zimmermann, S. Dukhin and H.-J.

Jacobasch, 1998, Extended electrokinetic characterization of

flat solid surfaces, J. Colloid Interface Sci. 208, 329.

Yamaguchi, Y., F. Takagi, K. Yamashita, H. Nakamura, H.

Maeda, K. Sotowa, K. Kusakabe, Y. Yamasaki and S.

Morooka, 2004, 3-D simulation and visualization of laminar

flow in a microchannel with hair-pin curves, AIChE J. 50,

1530.

Yang, C. and D. Li, 1997, Electrokinetic effects on pressure-

driven liquid flows in rectangular microchannels, J. Colloid

Interface Sci. 194, 95.

Yun, J.H., M.-S. Chun and H.W. Jung, 2010, The geometry effect

on steady electrokinetic flows in curved rectangular micro-

channels, Phys. Fluids 22, 052004.

ωz⟨ ⟩ωz⟨ ⟩ p L⁄∆( )2∝

the role of fluid inertia on streamwise velocity and vorticity...

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