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Numerical analysis of dynamic electro-osmotic flows of

non-Newtonian fluids in rectangular microchannels

Cunlu Zhao* and Chun Yang

School of Mechanical and Aerospace Engineering, Nanyang Technological University

50 Nanyang Avenue, 639798, Republic of Singapore

Address correspondence to this author. E-mail: zhao0070@e.ntu.edu.sg

Abstract

Numerical analyses of transient electro-osmosis of a typical non-Newtonian liquid

induced by DC and AC electric fields in a rectangular microchannel are conducted in the

framework of continuum fluid mechanics. The famous power-law constitutive model is

used to express the fluid dynamic viscosity in terms of the velocity gradient. Transient

start-up characteristics of electro-osmotic power-law liquid flow in rectangular

microchannels are simulated by using finite element method. Under a DC electric field, it

is found out and the fluid is more inert to the external electric field and the steady-state

velocity profile becomes more plug-like with decrease of the flow behavior index of the

power-law liquids. The numerical calculations also confirm the validity of the

generalized Smoluchowski slip velocity which can serve as the counterpart for the classic

Smoluchowski slip velocity when dealing with electrokinetic flow of non-Newtonian

power-law fluids. Under AC electric fields, the fluid is more obviously accelerated during

oscillations and the amplitude of the oscillating velocity is closer to the magnitude of the

generalized Smoluchowski velocity as the fluid behavior index increases. These dynamic

predictions are of practical significance for the design of microfluidic devices that

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manipulate non-Newtonian fluids such as biofluids, polymer solutions and colloidal

suspensions.

Keywords: Electro-osmotic pumping, Non-Newtonian power-law fluids, Transient

characteristics, Microfluidics.

1 Introduction

Microfluidic technologies eventually promise a conventional biochemical laboratory to

be constructed on a single, disposable microchip. The generic microfluidic systems

involve buffer fluid and sample manipulations such as pumping, valving, mixing,

injection, dispensing, etc (Bousse et al., 2000; Harrison et al., 1993; Squires and Quake,

2005; Whitesides, 2006). Fundamental understanding of the liquid flow characteristics in

microchannels is thus essential to optimum design and precise control of microfluidic

devices. In general, liquid motion can be generated by either applying a pressure gradient

or imposing an electric field, leading to respective pressure-driven flow or

electrokinetically-driven flow. Traditionally, in large-sized channels flow is often driven

by pressure that is usually generated by mechanical pumps. In microchannels however it

becomes increasingly difficult to utilize pressure-driven flow mode as the channel size

shrinks, especially down to micro-and submicron range. Moreover, some parts like

microvalves and micropumps with moving components are difficult to fabricate, and they

are prone to mechanical failure due to fatigue and fabrication defects. Alternatively,

electro-osmotic flow enjoys numerous advantages (over pressure-driven flow), including

ease of fabrication and control, no need for moving parts, high reliability, no noise etc.

Specifically, a plug-like velocity profile in electro-osmotic flow can result in reduced

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dispersion of sample species, making capillary electrophoresis become one of most

successful technologies for chemical and biomedical analyses (Manz et al., 1994).

When considering fluid motion on micro-scales, viscous relaxation is typically achieved

within milliseconds. The time evolution of microflows is nevertheless highly relevant to a

growing variety of applications of electro-kinetics in microfluidics and colloidal

dynamics in the sub-millisecond range(Yossifon et al., 2009). These include inter alia

biochip operation in high-speed electrophoretic separation processes (Fan and Harrison,

1994; Jacobson et al., 1998; Jacobson et al., 1994), the use of short-duration pulsed

electric fields (Söderman and Jönsson, 1996) in order to distinguish between particle

velocities and background electro-osmotic flow or to suppress thermal-zone broadening

(Dose and Guiochon, 1993) as well as micro-mixing and micro-pumping by means of AC

or modulated DC fields (Ajdari, 2000; González et al., 2000; Ramos et al., 1999). Owing

to the short time scales involved, it is essential to account for the unsteady dynamics of

the electro-kinetic flows in these applications.

Most previous analyses of transient electro-kinetic flow have focused on Newtonian

fluids. They have theoretically studied dynamic characteristics of electro-osmosis

(Campisi et al., 2005; Hanna and Osterle, 1968; Ivory, 1983; Kang et al., 2002; Keh and

Tseng, 2001; Mishchuk and González-Caballero, 2006; Söderman and Jönsson, 1996;

Yang et al., 2002; Yang and Kwok, 2003; Zhang et al., 2006) in response to various

modes of suddenly applied external fields. Obviously, the short sub-millisecond time

scales involved get in the way of an experimental characterization of dynamic behavior

of electro-osmotic flows. Nevertheless, the recent advances in the state of the art µPIV

techniques (Yan et al., 2007b; Yan et al., 2007a) enable experimental characterizations of

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dynamics of electrokinetic flows. However, microfluidic devices are usually used to

analyze biofluids which may not be treated as Newtonian fluids. Thus, the more general

Cauchy momentum equation, instead of the Navier-Stokes equation should be used to

describe the flow characteristics of non-Newtonian fluids provided that proper

constitutive equations are available. The aim of constructing constitutive equations for

non-Newtonian fluids is to find correlations between dynamic viscosity and shear rate.

The Power-law model (Graham and Jones, 1994), Carreau model (Khellaf and Lauriat,

2000) , Moldflow first-order model (Koh et al., 2004), and Bingham model (Das et al.,

2008) have been successfully developed to analyze non-Newtonian fluid flow and

heat/mass transfer. As for electro-osmotic flows of non-Newtonian fluids, they also may

behave differently from their Newtonian counterparts. Understanding the rheology in

such situations is a crucial aspect for the design and operation of microfluidic devices,

thus several authors are currently studying the electrokinetic flow of non-Newtonian

fluids in microchannel (Afonso et al., 2009; Berli, 2009; Das and Chakraborty, 2006;

Park and Lee, 2008; Tang et al., 2009; Zhao and Yang, 2010; Zhao et al., 2008;

Zimmerman et al., 2006). Unfortunately, all the studies listed above for non-Newtonian

electro-osmosis are conducted for the steady state. The only relevant work (Zhao and

Yang, 2009) just gave the analytical solutions for the transient electro-osmosis of

viscoelastic fluids of Oldroyd-B type in a rectangular microchannel, and however did not

report how the rheological properties of the fluids affects the dynamics of the electro-

osmosis. This study reports numerical analyses of the transient electro-osmotic flow of

typical power-law fluids in a rectangular microchannel driven by both DC and AC

electric fields. The numerical simulations are carried out by using finite element method

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which is verified through the comparison with the available exact solutions for

Newtonian fluids. Parametric studies are performed to show the dynamics of

electroosmosis for power-law fluids and examine effects of fluid rheology (fluid behavior

index) on the transient velocity distributions of the electro-osmotic flow of power-law

fluids.

2 Problem Formulations

Figure 1 shows the dimensions of the micro-channel and the coordinate systems

considered in this work. The channel is filled with a liquid solution of dielectric constant,

r . It is assumed that the channel wall is uniformly charged with a zeta potential, w , and

the liquid solution is a typical non-Newtonian fluid whose behavior can be described by the

well-known power-law model. When an external dynamic electric field E0 f (t) is imposed

along the x-axis direction, the fluid in the micro-channel sets in motion due to electro-

osmosis. f (t) is a time-dependent function and characterizes the dynamic behavior of the

applied electric fields In this study, we consider a DC driving electric field (f (t)=1) and a

AC driving electric field (f (t)=sin(ωt), in which ω is the frequency). Because of symmetry,

the analysis is restricted in the first quadrant of the y-z plane.

2. 1 Electric field in double layer

When the liquid in the micro-channel contacts the solid wall, an interfacial charge is

established which causes the free ions in the liquid to rearrange so as to form a thin

region with non zero net charge density. This region is commonly referred to as the

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electrical double layer (EDL). According to electrostatics theory, the electric potential

distribution in the EDL region is governed by the Poisson equation, which is expressed as

2 2

2 20

e

ry z

(1)

where ε0 is the permittivity of vacuum (ε0=8.854 ×10-12 F/m), ρe is the net charge density

in the EDL region and is related to ionic number concentrations and the EDL potential by

using the assumption of Boltzmann distribution (assuming an symmetric electrolyte)

2 sinh ve v v

B

z en n z e z en

k T

(2)

where n and n is respectively ionic number concentrations of positive and negative ions

in the EDL region.

Introducing the dimensionless groups: / hy y D , / hz z D ,K=κDh, and =zveψ/(kBT),

then substituting equation(2) in to equation(1), one can show that the electrical potential

profile in the EDL is governed by the Poisson–Boltzmann equation expressed by

2 22

2 2sinhK

y z

(3)

which is subject to the following boundary conditions:

/ /h hwy H D z W D

(4a)

00

0zy

y z

(4b)

Dh is the hydrodynamic diameter of the rectangular micro-channel and is defined as

Dh=4HW/(H+W), the dimensionless wall zeta potential is given by /w v w Bz e k T , the

Debye length 1 is defined as 1/21 2 2/ 2B vk T e z n , where n and zv are the bulk

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number concentration and the valence of ions, respectively, e is the fundamental charge,

kB is the Boltzmann constant, and T is the absolute temperature.

2. 2 Electro-osmotic flow of power-law fluids

When an external electric field is applied, the liquid flow of an incompressible power-law

fluid induced by electro-osmosis is governed by the general momentum equations and

continuity equation, i.e.(Deen, 1998)

Tp

t

VV V V V F (5a)

0 V (5b)

where is the density, p is the pressure, F is the body force vector, v is the velocity

gradient tensor and vT is the transpose of velocity gradient tensor. is the

dynamic viscosity and its pertinent implications will be presented later.

The difference between non-Newtonian and Newtonian fluids lies in that the viscous

stress is not a linear function of the rate of strain tensor. A number of empirical

expressions have been used to describe variations in the apparent viscosity with the rate

of strain. A scalar measure of the rate of strain suitable for such expression, is the

magnitude of the rate of strain tensor, which is defined as the double dot product of the

rate of strain tensor (Deen, 1998)

1/ 2

1

2

Γ :Γ (6)

where Γ is the rate of strain tensor and is given by / 2T Γ = V V . The fluid

viscosity then can be expressed as a function of the magnitude , namely . In the

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present work, a classic non-Newtonian fluid termed as the power-law fluid is assumed,

and its dynamic viscosity, is given by (Deen, 1998)

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nm (7)

where m is the flow consistency index, and n is the flow behavior index which represents

an apparent or effective viscosity being a function of the shear rate. Shear-thinning (also

termed as pseudoplastic) behavior is obtained for 1n , and it indicates that the fluid

viscosity decreases with increasing the rate of shear. Pseudoplasticity can be

demonstrated by the manner in which shaking a bottle of ketchup causes the contents to

undergo an unpredictable change in viscosity. It is also commonly observed in polymeric

fluids at relatively high values of the rate of shear, as those developed in microchannels.

Newtonian behavior is obtained for 1n . Shear-thickening (also termed as dilatant)

behavior is obtained for 1n , and it shows that the fluid viscosity increases with the rate

of shear. The dilatant effect is unusual and can readily be seen with a mixture of

cornstarch and water, which acts in counter-intuitive ways when struck or thrown against

a surface.

For the unidirectional electro-osmotic flow considered here, the velocity distribution is of

the form

V = u(y,z,t) i (8)

where u is the x-component of velocity and i is the unit vector in the x-direction. Thus

using Eq. (8), the continuity Eq. (1) is satisfied automatically. Furthermore, for electro-

osmotic flow, the only driving force is due to the interaction of the applied electrical field

E0 f(t) and the net charge density e in the EDL region near the channel wall. Such force acts

only along x direction, and is expressed by (Masliyah and Bhattacharjee, 2006)

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0x eF E f t (9)

For an open-end horizontally placed channel, no pressure gradient is induced and hence

the pressure gradient term in the Cauchy momentum equation disappears.

Besides the nondimensional groups used in the previous subsection, introducing

additional nondimensional parameters

1 1n ns h

mtt

u D ,s

uu

u (10)

and taking into account the aforementioned considerations, the simplified

nondimensional counterpart for equation (4) reads

1

sinhn

nw

u u u Kf t

t y y z z n

(11)

In which can be expressed as

1

2 2 21

2

n

n u u

y z

(12)

In Eq.(10), us represents the generalized Smoluchowski velocity for power-law liquids

(María et al., 2009; Zhao et al., 2008), and given by

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0n n

wns

Eu n

m

(13)

The initial and boundary conditions applicable to equation (11) are

00

tu

(14a)

0

0y

u

y

,

0

0z

u

z

(14b)

10

/0

hy W Du

,

/0

z H Wu

(14c)

3 Numerical methods and model validation

In the present analysis, both EDL potential field and electro-osmotic flow field are solved

in the PDE mode embedded in the popular finite element numerical analysis package

COMSOL multiphysics 3.4. In the PDE mode, the general form of PDEs are given in

terms of a lot of coefficients and a source term which are left for the user to specify for

constructions of their models. These coefficients can either be constant or dependent on

spatiotemporal variables and the source term can either be a linear or nonlinear function

of sought quantities, generating great flexibility for handling nonlinear PDEs. In our work,

a PDE governs the EDL potential and a PDE governs electro-osmotic flow field are

constructed from the general form of PDEs. Through the source term sinh in equations

(11), these two PDEs are coupled together.

In order to check the validity of the present model, we compare our numerical results

with the exact results (Chang and Wang, 2008) derived for the velocity distribution for

the starting electro-osmotic flow of Newtonian fluids in a rectangular microchannel.

However, their result only can be used in limited situations due to the linear Debye-

Hückel approximation involved in the analysis, which usually requires the wall zeta

potential to be smaller than -25mV. Thus, in our numerical validation, the channel has a

uniform wall zeta potential of ψw=-25mV and geometric size of the microchannel is

given as 2H=10μm and 2W=15μm. The working fluid flowing in the microchannel is

formed by dissolving a symmetric electrolyte (z:z), say NaCl, in a Newtonian fluid (a

special power-law fluid with flow behavior index n=1), and the bulk ionic number

concentration is n∞=6.022×1020/m3 (c∞=10-6M). The dielectric constant εr of the

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electrolytic solution is taken to be the same as the room temperature water, namely εr

=78.5. In electro-osmotic flows, the velocity experiences drastic changes in the EDL

region near the channel wall. Therefore, in the present analysis, we densify the grids near

the channel wall to ensure that the velocity characteristics in the EDL can be captured.

The calculated solutions are carefully validated to ensure that they are independent of the

computational grid points and time step. Grid-independence was examined for two

different grids whose total cell numbers were 6700 (100 67) and 1500 (100 150)

respectively. We also examined two different time steps, i.e. 1 310 and 5 410 , to

exclude the time-step dependency. It was found that the calculated flow rates differences

under two examinations were both less than 1%. Therefore, the grid and time

independence were corroborated, and the grid with 6700 cells and time step of 1 310

were applied in the study.

Figure 2 shows the velocity profiles for three different times at 0y , respectively

computed by using the analytical formula (Chang and Wang, 2008) and our devised

COMSOL model. It can be perceived from this plot that the numerical results for velocity

distributions obtained from our COMSOL model at three different times agree very well

with the existing analytical model, which indicates that our COMSOL model is of high

accuracy to predict transient electro-osmotic flow of power-law fluids in rectangular

microchannels.

4 Results and discussion

To predict the dynamic behaviors of the transient electro-osmotic flow of power-law

fluids under both DC and AC electric fields, we take all the parameters the same as in the

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Section 3 except that a relatively large wall zeta potential of ψw=-100mV is prescribed for

all the microchannel walls. The resultant dimensionless electrokinetic parameter K=40,

which secures the microchannel has relatively thin EDL and thus the dynamic momentum

transfer between the EDL and bulk flow can be identified.

4.1 Transient electro-osmotic flows of power-law fluids under DC electric fields

Figures 3 shows typical instant electro-osmotic flow velocity profiles of power-law fluids

with n=0.8. When the electric field is initially applied, the fluid within the double layer

responds virtually immediately, but the bulk fluid in the microchannel remains stationary.

The evolution of the velocity distributions throughout the microchannel cross-section

reveals a unique characteristic of the electro-osmotic flow. It is observed that the liquid

flow starts within a thin layer adjacent to the channel wall, i.e., the EDL region. During

elapsed dimensionless time of the order of 10-3, the velocity develops rapidly from zero

and reaches a local maximum velocity inside the EDL region, and then drops gradually to

zero as the distance is away from the channel wall. Because the electro-osmotic flow is

driven by the electrical forces resulting from the action of an external electrical field and

the EDL field, such driving forces exist only inside the EDL region where the electrical

net charge is present. The flow in this region may be viewed as “active.” However, as the

fluid velocity within the double layer continue to increase; the bulk fluid is gradually

dragged into motion through the transfer of momentum from the double layer, finally this

impact extends to the channel center. Hence the flow outside the EDL region may be

considered as “passive” flow caused by viscous forces. At the steady-state situation, the

velocity exhibits a plug flow. Notice also the absolute maximum is near the corner, which

is peculiar to channels with corners. This corner effect was also previously observed (Luo,

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2004) in a numerical study carried out for Newtonian liquid. This phenomenon is

demonstrated during the early stage of the evolution of the electroosmosis. The presence

of such “corner effect” is due to the overlap of the two EDLs formed at the two adjacent

sides of the rectangular channel wall.

Fig.4 characterizes the transient evolution characteristics for electroosmosis for power-

law fluids with different fluid behavior indices along the y axis at z=0. At the same

instant time 10-3, the fluids with larger fluid behavior indices respond more promptly to

the external electric field inside the EDL. As time elapse, momentum transfer from the

EDL to the bulk fluid progress and the fluids with larger fluid behavior more quickly

approach the steady state. At steady state, the normalized velocity in the bulk fluid all

increase to unit one, which reveals that for fluids with different fluid behavior indices, the

electro-osmotic velocities in the bulk liquid all approach their corresponding generalized

Smoluchowski velocities at the steady sate. Furthermore, the present numerical

simulations demonstrate that the generalized Smoluchowski velocity derived in the

parallel plate slit channels (María et al., 2009; Zhao et al., 2008) is also valid for the

electro-osmotic flow of power-law fluids in microchannel with rectangular cross-sections.

For situations where we have large-sized channels or thin EDLs (i.e., 1K ), the steady-

state electro-osmotically driven power-law fluid in the microchannels moves like a plug

with velocity given by generalized Smoluchowski velocity equation (13). Thus, this

generalized Smoluchowski velocity can play the role of the conventional Smoluchowski

velocity (n=1 in equation (1)) in microfluidic applications dealing with non-Newtonian

fluids.

4.2 Electro-osmotic flows of power-law fluids under AC electric fields

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In this particular investigation, the fluid is driven through the microchannel by an applied

AC electric field. Then in the simulations, we choose sinf t t and the

corresponding dimensionless frequency to be 1 1 / 2n ns hu D m . Fig. 5 presents

the steady-state evolution of the axial velocity contours in the transverse section from

phase 2 0t to phase 2 t . At 2 0t , the intensity of the applied electric field is

zero, and the flow in the center of the microchannel moves in the negative direction. It

can be seen that the preceding negative electric field intensity causes this axial velocity

contours. As further time elapses, the fluid within the EDL is rapidly driven in the

positive direction under the influence of the applied AC electric field. Since the peak net

charge density occurs at the corners of the microchannel, the maximum velocity of the

flow also takes place at the corner. At phase 2 /10t , the flow velocity near the two

walls continues to increase, and the resulting momentum transfer causes the region of

flow with a positive velocity to spread gradually from the double layer toward the central

region of the microchannel. Finally, at 2 / 5t , the region of flow in the positive

direction expands to occupies the entire microchannel. From phase 2 3 /10t to phase

2 / 2t , the momentum transfer from the EDL to the bulk flow increase with the

increase of intensity of the external electric field and the velocity in the whole channel

domain continues to grow. As the elapsed time increases further, the intensity of the

electric field gradually decreases, and hence the positive axial velocity of the flow within

the double layer decreases. The peaks in the net charge density cause the fluid in the

corners to respond rapidly to the change in the applied AC electric field, and

consequently there is a slight reduction in the local maximum velocity at the corner.

Meanwhile, inertia forces cause the axial velocity of the flow in the center of the

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microchannel to continue to increase in the positive direction. At phase 2 3 / 5t , the

velocity of the bulk flow in the positive direction reaches an almost equal value. After

phase 2 7 /10t , the positive intensity of the applied AC electric field continues to

decrease, and momentum transfer causes the region of flow with slower positive axial

velocity to expand toward the central region. At phase 2 t , it is noted that the axial-

flow velocity contours strongly resemble those evident at phase 2 0t in terms of

their shape. However, it is important to note that the direction of the axial velocity is

reversed. During the second half of the cycle, from 2 t to 2 2t , the variation

of the applied AC electric field intensity is the mirror image of the variation described

from 2 0t to 2 t , and accordingly, the axial flow velocity in the central region

of the microchannel reverses from a positive direction to a negative direction. Essentially,

the evolution of the axial velocity contours during the cycle from 2 t to 2 2t

follows the same process as that described for 2 0t to 2 t . However, it is noted

that the axial flow directions are reversed. Consequently, at 2 2t , the axial velocity

contours are identical to those observed at 2 0t .

Fig.6 displays the transient evolutions in applied AC electric field and axial velocity of

electro-osmotic flow of power-law fluids with three different flow behavior indices in

transverse center ( 0y z ) of the microchannel. After turning on the AC electric fields,

the fluids with larger fluid behavior index responds more quickly as in the DC electric

field. Generally, the velocity lags behind the applied AC electric field, and the phase lag

increases with the increase of the fluid behavior index, but not astonishingly. We also

note that from one peak to the adjacent trough, the power-law fluids experience more

significant accelerations for the larger fluid behavior indices. In addition, the amplitudes

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of oscillating velocities are closer to the corresponding generalized Smoluchowski

velocity with the increase of fluid behavior index.

5 Conclusions

We conducted numerical analyses of dynamic electrokinetic flow of power-law fluids in

a rectangular microchannel by using finite element method. For the case of transient

electroosmotic flow with an external DC electric field, initially, the impulse provided by

the applied DC electric field drives the fluid within the double layer in the axial direction.

The momentums of the double layer gradually transfer to the central region, and finally

occupying the entire channel. The greater fluid behavior index causes the fluid to respond

more promptly to the influence of the applied electric field. In addition, our numerical

calculation confirms that the Smoluchowski velocity given in the previous works for a

parallel plate microchannel is still valid for microchannels with rectangular cross-sections.

The velocity is of practical importance in non-Newtonian microfluidic manipulations

adopting electroosmosis.

This study also considered the application of a sinusoidal electric field to the flow in the

microchannel. The results indicate that the flow in the center of the microchannel initially

exhibits a transient response when the electric field is applied and then attains a steady-

state oscillation. The fluid with larger fluid behavior index respond tends to have a faster

response to the electric field and a larger phase lag between the applied electric field.

Furthermore, the fluids with larger fluid behavior indices are more susceptible to

acceleration initiated by the oscillating electric field. The greater net charge density in the

corners of the microchannel results in local maximum or minimum axial velocities in the

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corners during increasing or decreasing applied electric field intensity in either the

positive or negative direction. It has been shown that the axial-flow velocity tends to

follow the varying intensity of the applied AC electric field. The evolution process of the

axial velocity contours during a half cycle is actually a mirror image for the preceding

half cycle.

Acknowledgments Z.C.L. would like to thank NTU (Nanyang technological university,

Republic of Singapore) for a PhD scholarship.

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Figure Captions Figure 1.Electro-osmotic flow system in a rectangular microchannel. The cross-section of

the channel has a height of 2H and a width of 2W. All the walls are charged with uniform

zeta potential w , and the dynamic electric field E0 f t is applied along the axial

direction of the microchannel. The zeta potential on the walls induces a near-wall electric

double layer in which the charge density is not zero. Then interaction of the external

electric field and the non-zero charge density drive the electro-osmosis.

Figure 2 Transient velocity profiles along the y direction at 0z under three different

times for Newtonian fluids (n=1.0), predicted by using the analytical formula and the

devised COMSOL model, respectively.

Figure 3. Transient evolutions of axial-velocity contours for power-law fluids with n =0.8

due to an applied DC electric field. (a) 310t ;(b) 210t (c) 110t (d) t .

Figure 4. Comparison of transient evolution characteristics for power-law fluids with

different fluid behavior indices along the y axis at z=0.

Figure 5. Steady-state oscillating axial velocity contours for power-law fluids with n=0.8

under different phases in a period due to an applied AC electric field. (a) 2 0t (b)

2 /10t ; (c) 2 / 5t ; (d) 2 3 /10t ; (e) 2 2 / 5t ; (f) 2 / 2t ; (g)

2 3 / 5t .; (h) 2 7 /10t ; (i) 2 4 / 5t .; (j) 2 9 /10t ; (k) 2 t .

21

Figure 6. Transient evolutions in applied AC electric field and axial velocity of electro-

osmotic flow of power-law fluids with three different flow behavior indices in transverse

center ( 0y z ) of the microchannel. The dash-dot line represents the scaled electric

field (( sin 2 t ), the dash line represents the scaled velocity for Newtonian fluids (n=1),

Thick solid line is the scaled velocity for the power-law fluids with n=0.6, and thin solid

line denotes the scaled velocity for the power law fluids with n=1.4.

22

Figure 1.Electro-osmotic flow system in a rectangular microchannel. The cross-section of

the channel has a height of 2H and a width of 2W. All the walls are charged with uniform

zeta potential w , and the dynamic electric field E0 f t is applied along the axial

direction of the microchannel. The zeta potential on the walls induces a near-wall electric

double layer in which the charge density is not zero. Then interaction of the external

electric field and the non-zero charge density drive the electro-osmosis.

2W

2H z

y x

o

0E f t

w

23

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0

u

t

y

Analytical results (Chang and Wang 2008) Numerical results

=1.010-3

t=3.010-3

t =5.010-3

Figure 2 Transient velocity profiles along the y direction at 0z under three different

times for Newtonian fluids (n=1.0), predicted by using the analytical formula and the

devised COMSOL model, respectively.

24

(a)

(b)

yz

u

yz

u

25

(c)

(d)

Figure 3. Transient evolutions of axial-velocity contours for power-law fluids with n =0.8

due to an applied DC electric field. (a) 310t ;(b) 210t (c) 110t (d) t .

yz

u

yz

u

26

0.0 0.1 0.2 0.3 0.4

0.0

0.2

0.4

0.6

0.8

1.0

110t

310t

t

u

y

n=0.6 n=1.0 n=1.4

Figure 4. Comparison of transient evolution characteristics for power-law fluids with

different fluid behavior indices along the y axis at z=0.

27

(a)

(b)

u

z y

u

z y

28

(c)

(d)

u

z y

u

z y

29

(e)

(f)

u

z y

u

z y

30

(g)

(h)

u

z y

u

z y

31

(i)

(j)

u

z y

u

z y

32

(k)

Figure 5. Steady-state oscillating axial velocity contours for power-law fluids with n=0.8

under different phases in a period due to an applied AC electric field. (a) 2 0t (b)

2 /10t ; (c) 2 / 5t ; (d) 2 3 /10t ; (e) 2 2 / 5t ; (f) 2 / 2t ; (g)

2 3 / 5t .; (h) 2 7 /10t ; (i) 2 4 / 5t .; (j) 2 9 /10t ; (k) 2 t .

u

z y

33

0 1 2 3 4 5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

t

u

Figure 6. Transient evolutions in applied AC electric field and axial velocity of electro-

osmotic flow of power-law fluids with three different flow behavior indices in transverse

center ( 0y z ) of the microchannel. The dash-dot line represents the scaled electric

field (( sin 2 t ), the dash line represents the scaled velocity for Newtonian fluids (n=1),

Thick solid line is the scaled velocity for the power-law fluids with n=0.6, and thin solid

line denotes the scaled velocity for the power law fluids with n=1.4.