Numerical Analysis of Dynamic Electro-Osmotic Flows of Non-Newtonian Fluids in Rectangular Microchannels

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: [email protected] 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

1 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

2 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

3 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 electroosmosis. 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

4 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 electroosmosis. 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

5 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

22  e 22 (1) yz 0 r -12 where ε0 is the permittivity of vacuum (ε0=8.854 ×10 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)

zev  evvn n ze 2sinh zen   (2) kTB

where n and n is respectively ionic number concentrations of positive and negative ions in the EDL region.

Introducing the dimensionless groups: yyD / h , zzD / h ,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

22 K 2 sinh (3) yz22 which is subject to the following boundary conditions:

 w (4a) yHD//hh zWD

   0 (4b) yzy0 z 0

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  wvwB ze / kT, the

1/2 1 122 Debye length  is defined as   kTBv/2 ezn  , where n and zv are the bulk

6 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)

V T VVp   V V F (5a) t  

V  0 (5b) where  is the density, p is the pressure, F is the body force vector, v is the velocity

T gradient tensor and v 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   Γ : Γ (6) 2  where Γ is the rate of strain tensor and is given by Γ =V VT  /2. The fluid   viscosity then can be expressed as a function of the magnitude  , namely   . In the

7 present work, a classic non-Newtonian fluid termed as the power-law fluid is assumed, and its dynamic viscosity,  is given by (Deen, 1998)

 m 2  n1 (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 n 1, 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 n 1 . Shear-thickening (also termed as dilatant) behavior is obtained for n 1, 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)

8 (9) FEftxe  0  

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

mt u t  11nn ,u  (10) uDsh us and taking into account the aforementioned considerations, the simplified nondimensional counterpart for equation (4) reads

uu    uKn1 sinh f t (11)   n  ty  y  z  zn w

In which   can be expressed as

n1 2 2 2 n1 uu  2    (12) yz 

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

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

1 1n n n  wE0 uns   (13) m

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

u  0 (14a) t 0

u u  0,  0 (14b) y y 0 z z 0

9 u  0 ,u  0 (14c) yWD / h zHW /

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

20 3 -6 concentration is n∞=6.022×10 /m (c∞=10 M). The dielectric constant εr of the

10 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 103 and 5 104 , 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103 were applied in the study.

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

11 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,

12 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., K 1), 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

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

AC electric field. Then in the simulations, we choose f tt  sin  and the

11nn corresponding dimensionless frequency to be  uDsh/2 m  . Fig. 5 presents the steady-state evolution of the axial velocity contours in the transverse section from phase 20 t  to phase 2 t   . At 20 t  , 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 2/10 t   , 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 2/5 t   , the region of flow in the positive direction expands to occupies the entire microchannel. From phase 23/10 t   to phase

2/2 t   , 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

14 microchannel to continue to increase in the positive direction. At phase 23/5 t   , the velocity of the bulk flow in the positive direction reaches an almost equal value. After phase 27/10 t   , 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 20 t  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 22 t   , the variation of the applied AC electric field intensity is the mirror image of the variation described from 20 t  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 22 t   follows the same process as that described for 20 t  to 2 t   . However, it is noted that the axial flow directions are reversed. Consequently, at 22 t   , the axial velocity contours are identical to those observed at 20 t  .

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 ( yz0 ) 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

15 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

16 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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19 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 z  0 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) t 103 ;(b) t 102 (c) t 101 (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) 20 t  (b)

2/10 t   ; (c) 2/5 t   ; (d) 23/10 t   ; (e) 22/5 t   ; (f) 2/2 t   ; (g)

23/5 t   .; (h) 27/10 t   ; (i) 24/5 t   .; (j) 29/10 t   ; (k) 2 t   .

20 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 ( yz0 ) 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.

E ft 0   y x

2H  w o z 2W

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

1.0 Analytical results (Chang and Wang 2008) Numerical results 0.8

0.6 -3 t =5.010 u

-3 0.4 t=3.010

-3 0.2 t=1.010

0.0 0.0 0.1 0.2 0.3 0.4 y

23 u

z y

(a)

z y

(b)

z y

(c)

z y

(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) t 103 ;(b) t 102 (c) t 101 (d) t .

t  1.0

0.8 t  101

0.6 u

0.4 n=0.6 n=1.0

n=1.4 3 0.2 t  10

0.0

0.0 0.1 0.2 0.3 0.4 y

Figure 4. Comparison of transient evolution characteristics for power-law fluids with different fluid behavior indices along the y axis at z=0.

z y

(a)

z y

(b)

27 u

z y

(c)

z y

(d)

28 u

z y

(e)

z y

(f)

29 u

z y

(g)

z y

(h)

30 u

z y

(i)

z y

(j)

z y

(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) 20 t  (b)

2/10 t   ; (c) 2/5 t   ; (d) 23/10 t   ; (e) 22/5 t   ; (f) 2/2 t   ; (g)

23/5 t   .; (h) 27/10 t   ; (i) 24/5 t   .; (j) 29/10 t   ; (k) 2 t   .

1.0 0.8 0.6 0.4 0.2 u 0.0

-0.2 -0.4 -0.6 -0.8 -1.0 012345 t

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 ( yz0 ) 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.