Numerical Study on the Mixing Performance of a...

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Copyright copy 2012 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofNanoscience and Nanotechnology

Vol 12 4523ndash4530 2012

Numerical Study on the Mixing Performance of a

Ring-Type Electroosmotic Micromixer with Different

Obstacle Configurations

Hyeon-Seok Seo1 Bongtae Han2 and Youn-Jea Kim1lowast1School of Mechanical Engineering Sungkyunkwan University 300 Cheoncheon-dong Suwon 440-746 Korea

2Department of Mechanical Engineering University of Maryland College Park Maryland 20740 USA

A new type of electrokinetic micromixer with a ring-type channel is introduced for fast mixing Theproposed mixer takes two fluids from different inlets and combines them in a ring-type mixing cham-ber The fluids enter two different inlets (inner radius 25 m and outer radius 50 m) respectivelyThe total channel length is 500 m and four microelectrodes are positioned on the outer wall of themixing chamber The electric potentials on the four microelectrodes are sinusoidal with time havingvarious maximum values of voltage zeta potential and frequency Also in order to compare themixing performance with different obstacle configurations we performed a numerical analysis usinga commercial code COMSOL The concentration of the dissolved substances in the working fluidand the flow and electric fields in the channel were investigated and the results were graphicallydepicted for various flow and electric conditions

Keywords Micromixer Mixing Performance Zeta Potential Electric Potential

1 INTRODUCTION

The operation of microfluidic-based ducts nozzles valves

etc cannot always be predicted from conventional macro-

scopic flow models such as the Navier-Stokes equations

with the no-slip boundary condition at the fluid-solid inter-

face unlike that of larger flow devices which is predicted

by these models routinely and successfully The pressure

gradient in a long microduct is non-constant and the mea-

sured flow rate is higher than that predicted by the con-

ventional continuum flow model Moreover surface effects

dominate small devices such as MEMS (Micro Electro

Mechanical Systems)-based devices which are used in

microfluidics1

Microfluidics systems have common properties they

work with laminar flows under reduced thermal gradi-

ents and with small samplereagent volumes properties

that promote effective process control and reproducibil-

ity Microfluidic devices with these properties have been

built for lab-on-a-chip applications where the functions

such as separation mixing reaction synthesis and analy-

sis are performed2 This micro device concept and tech-

nology which has developed rapidly is being applied

increasingly in biological and chemical applications3ndash5 the

lowastAuthor to whom correspondence should be addressed

fabrication of hydrogel for drug delivery systems chemi-

cal and enzyme reactions synthesis of nucleic acids and

analysis of DNA (deoxyribonucleic acid) and protein In

most microfluidic applications eg fast chemical reac-

tions DNA separation and amplification the performance

of the microfluidic system is governed by its mixing

efficiency

The rapid and efficient mixing is important for many

microfluidic applications But it is a challenging process in

many microfluidic systems that perform complex chemical

synthesis and analysis Most existing microfluidic mixing

systems are limited flows in the low Reynolds number

regimes According to scaling law decreasing the mixing

path can shorten the mixing time and enhance the mix-

ing quality For this reason it is important to understand

the mixing process in micromixers To do this one must

be able to characterize and evaluate the outcomes of the

mixing process6

Micromixers can be classified into two types passive

and active7ndash10 The operation principles of the passive-type

mixers are mainly based on fluid stretch folding breakup

and molecular diffusion Some passive micromixers reduce

the diffusion path between fluid streams by splitting and

recombining them

Because of the methods that can easily integrate

electrodes into microfluidic and nanofluidic devices

J Nanosci Nanotechnol 2012 Vol 12 No 6 1533-48802012124523008 doi101166jnn20126188 4523


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Numerical Study on the Mixing Performance of a Ring-Type Electroosmotic Micromixer Seo et al

electro-kinetically-driven mixing devices are under exten-

sive investigation for use as active micromixers11 Elec-

troosmosis used to generate electrical potential gradients

has been proposed as a propulsion mechanism for bio-

fluids This mechanism would eliminate the need for a

moving apparatus12 The technology of electrokinetically-

driven mixing devices has led to the need for an efficient

fluid control mechanism for micro-fabricated systems that

perform complex chemical synthesis For example local-

ized flow circulations within the bulk flow near regions of

the microchannel wall with oppositely-charged surface het-

erogeneities were investigated and these circulations were

successfully applied to enhance the species mixing in a

T-shaped micro-mixer13 The results indicated that these

heterogeneous regions reduce the required length of the

mixing channel by 70 Sasaki et al14 reported a novel

micromixer based on AC electroosmosis and analyzed its

performance by experiments according to various param-

eters including applied voltage frequency and solution

viscosity Their results were compared with those obtained

from a theoretical model of the AC electroosmosis mixer

The comparison showed that a larger voltage led to more

rapid mixing The results could be used to estimate the

mixing performance of the AC electroosmosis-based mixer

under various experimental conditions They also provided

guidelines on the use of the micromixer in microfluidic

chemical analysis In addition the electroosmotic effect

results in swelling at the cathode side due to water enrich-

ment and shrinkage at the anode side owing to water deple-

tion these phenomena refer to the openclosed control

of IPMC (Ionic Polymer Metal Composite)-based micro-

valve systems15

Bio-fluids which show non-Newtonian fluid flow

behavior are often used in MEMS-fluidics Due to the

Fig 1 Schematic of the modeled electroosmotic micromixer

importance of Bio-MEMS and lab-on-a-chip technolo-

gies many researchers have recently focused on the

non-newtonian fluid behaviors of electrokinetically-driven

bio-fluids16 Das and Chakraborty17 obtained an analyti-

cal solution describing the transport characteristics of a

non-newtonian fluid flow in a rectangular microchannel

under the sole influence of electrokinetic effects As an

illustrative case study they analyzed the flow behavior

of a blood sample Zhao et al18 analyzed the electroo-

somotic flow of power-law fluids in a slit microchannel

by introducing exact and approximate analytical expres-

sions of shear stress effective viscosity and velocity profile

distribution Akgul and Pakdemirli19 presented analytical

and numerical solutions for electroosmotic flow of a third

grade fluid between micro-parallel plates They analyzed

influences of non-newtonian parameters namely the Joule

heating effect viscosity index and electrokinetic effect on

the velocity and temperature profiles of the fluid

In this study the mixing performance of a ring-type

electroosmotic micromixer was numerically investigated

Also the mixing performance of the mixer with different

obstacle configurations in the mixing chamber was exam-

ined by numerical analysis using the commercial code

COMSOL Especially the concentration of the dissolved

substances in the fluid was considered according to volt-

age electrode frequency and zeta potential

2 NUMERICAL DETAILS

21 Governing Equations

The complex flow phenomena in the modeled micromixer

were investigated by numerical methods Two-dimensional

Navier-Stokes equations describing fluid behavior were

employed to simulate the mixing of two aqueous fluids in

4524 J Nanosci Nanotechnol 12 4523ndash4530 2012


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Seo et al Numerical Study on the Mixing Performance of a Ring-Type Electroosmotic Micromixer

Table I The boundary conditions applied to this study

Inlet velocity Electric potential Zeta potential Frequency

Model [ms] (V0 [V] () [V] [Hz]

Rectangular 200 1 2 minus01 minus005 4 8

the designed channel under unsteady-state conditions A

complete description of time-dependent ICEO (Induced-

Charge Electro Osmosis) at large voltages which is

beyond the scope of this study would be required con-

sidering all of the effects together The approximation of

thin double layers which has been applied mostly to

steady-state problems involving non-polarizable objects

is a good starting point Thus an ICEO slip velocity is

very rapidly established in response to an induced zeta-

potential and lsquoouterrsquo tangential field Furthermore despite

double layer and tangential field gradients the classical

Helmholtz-Smoluchowski formula correctly gives the elec-

troosomotic slip velocity20

Because the electroosomotic flow velocity is indepen-

dent of channel size electroosomotic pumping presents a

natural and popular technique for fluid manipulation in

small channels On the other hand when the solidfluid

interface is that of a freely suspended particle the electroo-

somotic slip velocity gives rise to motion of the particle

itself termed electrophoresis20

Fig 2 Grid systems (a) Circular obstacle and (b) Rectangular obstacle

Fig 3 Streamlines in the mixing chamber with circular obstacle and

one cycle of sine curve for V0 = 1 V f = 4 Hz and =minus01 V (a) 0 s

(b) 00625 s (c) 0125 s and (d) 01875 s

The equations used in the above simulations are shown

as follows

[V

t+ V middotV

]=minusp+ 2V +eE (1)

where is the fluid density V is the fluid velocity p is

the static pressure of the flow is the molecular viscosity

of the fluid e is the charge density of the fluid and E is

the intensity of the applied electric field Under a wide

range of conditions the local slip velocity is given by the

Helmholtz-Smoluchowski equation as

Uslip =minusEx

(2)

where is the relative permittivity is the zeta potential

and Ex is the tangential component of the bulk electric

field

Fig 4 Streamlines in the mixing chamber with rectangular obstacle

and one cycle of sine curve for V0 = 1 V f = 4 Hz and = minus01 V

(a) 0 s (b) 00625 s (c) 0125 s and (d) 01875 s

J Nanosci Nanotechnol 12 4523ndash4530 2012 4525


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The ions in the diffuse part of the double layer are

approximately in thermal equilibrium and tangential con-

centration gradients modify the usual electroosmotic slip

by changing the bulk electric field (concentration polariza-

tion) and by producing diffusion osmotic slip20 We can

express the balance equation for the current density with

Ohmrsquos law as

middot minusV = 0 (3)

where denotes the conductivity of solution

In addition the following convection-diffusion equation

describes the concentration of the dissolved substances in

the fluidc

t+ middot minusDc= Rminusu middotc (4)

where c is the concentration D is the diffusion coeffi-

cient and R is the reaction rate pertaining to the working

fluid When R= 0 the concentration is not affected by any

reactions

Fig 5 Concentration distributions in the modeled micromixer with circular obstacle (a) f = 4 Hz =minus01 V t = 05 s (b) f = 4 Hz =minus01 V

t = 25 s (c) f = 4 Hz =minus005 V t = 05 s (d) f = 4 Hz =minus005 V t = 25 s (e) f = 8 Hz =minus01 V t = 05 s (f) f = 8 Hz =minus01 V

t = 25 s (g) f = 8 Hz =minus005 V t = 05 s and (h) f = 8 Hz =minus005 V t = 25 s

The mixing efficiency (m applied to this study is

defined as follows

m equiv(1minus

int cminus cdxint c0minus cdx)times100 (5)

where c is the concentration value of outlet c0 is the initialconcentration value before mixing both of working flu-

ids and c is the completely mixed concentration value

respectively

22 Model and Grid Systems

Figure 1 shows the schematic of an electroosmotic mixer

The mixer takes two fluids from different inlets of 25 min size respectively and combines them into a sin-

gle channel that is 50 m wide The fluids then enter

the central loop where four microelectrodes are posi-

tioned along the outer wall of the loop These micro-

electrodes impose a spatially-varying electric field and

the fluids are manipulated via the electroosmotic slip



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boundary condition before they enter the outlet channel

The electric potentials produced by the microelectrodes

are time-dependent so the fluids are mixed chaotically

in the mixing chamber In this study we performed a

two dimensional numerical analysis using the commercial

code COMSOL

The electric potentials produced by the four microelec-

trodes produced a time-dependent sinusoidal electric field

for the maximum applied voltages of 2 V and 8 Hz Details

of boundary condition applied to this study are shown in

Table I The potentials of electrodes 1 and 3 were denoted

as V0 sin (213ft and the potentials of electrodes 2 and

4 were minusV0 sin (213ft The working fluid was an elec-

trolyte that possessed the same properties as water ie

it had a relative permittivity of = 783 a conductivity

of an ionic solution of = 011845 Sm and a diffu-

sion coefficient of D= 10minus11 m2s The inlet fluid velocity

was described by a parabolic graph with a mean value of

200 ms (Pe= 7times10minus2 Re= 1times10minus2 and the inletupper

Fig 6 Concentration distributions in the modeled micromixer with rectangular obstacle (a) f = 4 Hz =minus01 V t= 05 s (b) f = 4 Hz =minus01 V



and inletlower concentration values were 1 and 0 molm3

respectively The time scale of the transient effect in the

micromixer was set to 012560 s and the total calculation

time was 3 s

In order to investigate the simultaneous effects of

electroosmotic flow and convection-diffusion in the mix-

ing chamber two-dimensional unstructured grids contain-

ing approximately 40000 elements were generated using

COMSOL preprocessor as shown in Figure 2 Especially

a non-uniform grid mesh which is thinner in vicinity of

walls has been used to increase the accuracy of the results

The convergence criteria have been satisfied when the

maximum mass residuals of the grid control volume has

been less than about 10minus7

3 RESULTS AND DISCUSSION

Effective mixing basically involves repeated combinations

of stretching and folding of fluid elements and small



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scale diffusion Results of the streamlines for two dif-

ferent obstacle configurations in the mixing chamber for

V0 = 1 V f = 4 Hz and =minus01 V are shown in Figures 3

and 4 for one cycle of the sine curve In both cases the

streamlines show similar flow patterns They were influ-

enced by the electric fields that were produced by the indi-

vidual electrodes Also eddy flows were formed around

the electrodes in the mixing chamber The fluid particles

that stayed in the central loop stretched and folded for a

long time before they entered the outlet channel These

flow patterns are corresponded with those by Hadigol

et al16 which were performed numerically applied to elec-

troosmotic effect in the microchannel with based on phys-

ical properties of the water This fact indicated a chaotic

advection which in turn expedited the final part of mixing

through molecular diffusion

The concentration distributions in the modeled

micromixer for the two different obstacle configurations

are shown in Figures 5 and 6 It is seen that the perturbed

flows continuously moved along the mixing chamber

and the mixing improved everywhere in the modeled

micromixer because of the stretching and folding behav-

iors of the fluid particles which was induced by the

applied electric field Results also showed that the mixing

Fig 7 Concentration profiles along the horizontal direction of the modeled micromixer from the mixing chamber to the outlet for various conditions

at t = 25 s (a) f = 4 Hz =minus01 V (b) f = 8 Hz =minus01 V (c) f = 4 Hz =minus005 V and (d) f = 8 Hz =minus005 V

performance was influenced by the frequency of elec-

tric field and the zeta potential The decreases of the

frequency and the increases zeta potential improved the

mixing efficiency But from the viewpoint of fluid flow

the streamline produced by the circular obstacle in the

mixing chamber was smoother than that produced by the

rectangular one This is because the fluid flow correspond-

ing to the rectangular obstacle in the mixing chamber was

affected by the resistance between it and the surface of the

obstacle

In order to evaluate the mixing performance for two

different obstacle configurations at various values of volt-

ages frequency and zeta potential we set up the data

line in the modeled micromixer as shown in Figure 1

The dissolved-substances concentration data was obtained

along the width line of the outlet region and also from

the centerline along the horizontal direction of the mixing

chamber

Figure 7 shows the concentration profiles along the hor-

izontal direction of the modeled micromixer from the mix-

ing chamber to the outlet for various conditions at t= 25 sThe concentration efficiency reaches over 90 in all cases

which means that each fluid was sufficiently mixed and

moved toward the exit of the micromixer In particular the



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Fig 8 Concentration profiles along the vertical direction of the outlet region of mixing chamber for various conditions at t = 25 s (a) f = 4 Hz

=minus01 V (b) f = 8 Hz =minus01 V (c) f = 4 Hz =minus005 V and (d) f = 8 Hz =minus005 V

oscillation of flow increases with the increase of frequency

which means that it hinders stable mixing At constant zeta

potential value the concentration distribution is the most

stable for the case of 4 Hz which means that the fluids were

well mixed at the proper frequency rather than at high fre-

quencies At constant frequency the mixing performance

was influenced by the zeta potential value When the elec-

trodes were applied with low voltages the agglomeration of

fluid particle increased with the decrease of the zeta poten-

tial value So we may deduce that the mixing efficiency

with f = 4 Hz and = minus01 V was the most stable com-

pared those of the other cases The concentration value at

the applied voltage of 2 V was higher than that at 1 V

However there was little difference in the mixing efficiency

according to the obstacle configuration

Figure 8 shows the concentration profiles along the ver-

tical direction of the outlet region of the mixing chamber

for various conditions at t = 25 s Numeric number plusmn25

along the horizontal axis denotes the high and low limits of

width (see Fig 1) At the frequency value of 8 Hz the con-

centration distributions indicate a similar pattern through

the channel width in all cases (see Figs 8(b and d)) How-

ever at f = 4 Hz the concentration profiles indicate a

high mixing performance over the channel width and the

mixing efficiency increases about 30 compared with that

of the case of f = 8 Hz (see Figs 8(a and c)) Especially

for the case of f = 4 Hz and = minus01 V the con-

centration distribution is the most stable and the mixing

efficiency remains almost over 85 all over the channel

width Even with the change of the obstacle configuration

the mixing characteristics change very little with respect

to concentration

4 CONCLUSION

An electroosmotic actuated active ring-type micromixer

was designed and investigated numerically for two differ-

ent obstacle configurations in the mixing chamber Of par-

ticular concern was the concentrations of the fluids in each

condition for various values of voltage frequency at the

electrodes and zeta-potential The following conclusions

were obtained

(1) Chaotic behavior was confirmed by the stretching and

folding of the material lines due to the applied electric

fields

(2) In the evaluation of the concentration value accord-

ing to frequency and zeta potential the mixing was the

most active for the case of f = 4 Hz and = minus01 V



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compared with those of the other cases As the applied

voltage at electrodes increased the mixing performance in

the micromixer increased

(3) From the viewpoint of fluid flow the circular obstacle

showed a smoother pattern than the rectangular one but

there was little difference in the mixing efficiency between

the two obstacle configurations

By taking these findings into account one can design

effectively the other types of active micromixer Further

improvements of the mixing efficiency will be possible by

considering a novel design such as locations of electrode

on the obstacle instead of the outer wall of the mixing

chamber

Acknowledgments Financial aid from the Korea

Ministry of Education through the Brain Korea 21

Project [HRD Center for Convergence Mechanical Sys-

tem Design] is gratefully acknowledged The correspond-

ing author (YJK) also acknowledges the financial support

of the Sungkyunkwan University for his sabbatical year

References and Notes

1 M Gad-el-Hak The MEMS Handbook MEMS Introduction and

Fundamentals Taylor amp Francis London (2005)2 H A Stone A D Stroock and A Ajdari Annu Rev Fluid Mech

36 381 (2004)

3 T Khaleque S Abu-Salih J R Saunders and W Moussa

J Nanosci Nanotechnol 11 2470 (2011)4 W Ehrfeld V Hessel and H Lowe Microreactors Wiley-VCH

Boston (2002)5 D R Reyes D Iossifidis P A Auroux and A Manz Anal Chem

74 2623 (2002)6 J Aubin M Ferrando and V Jiricny Chem Eng Sci 65 2065

(2010)7 H H Bau J Zhong and M Yi Sens Actuators B 79 207 (2001)8 L Fu R Yang C Lin and Y Chien Electrophoresis 26 4605

(2005)9 C Hong J Choi and C H Ahn Lab-on-a-Chip 4 109 (2004)

10 S Park J K Kim J Park S Chung C Chung and K J Chang

J Micromech Microeng 14 6 (2004)11 S H Yoon S Kumar G H Kim Y S Choi and T W Kim

J Nanosci Nanotechnol 8 3427 (2008)12 M Longaretti G Marino B Chini J W Jerome and R Sacco

J Nanosci Nanotechnol 8 3686 (2008)13 D Gobby P Angeli and A Gavriilidis J Micromech Microeng

11 126 (2001)14 N Sasaki T Kitamori and H B Kim Anal Sci 26 815 (2010)15 J S Yun K S Yang N J Choi H K Lee S E Moon and D H

Kim J Nanosci Nanotechnol 11 5975 (2011)16 M Hadigol R Nosrati A Nourbakhsh and M Raisee J Non-

Newtonian Fluid Mech 166 965 (2011)17 S Das and S Chakraborty J Anal Chim Acta 559 15 (2006)18 C Zhao E Zholkovskij J H Masliyah and C Yang J Colloid

Interf Sci 326 503 (2008)19 M B Akgul and M Pakdemirli J Non-Linear Mech 43 985

(2008)20 T M Squires and M Z Bazant J Fluid Mech 509 217 (2004)

Received 28 October 2011 Accepted 14 February 2012



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electro-kinetically-driven mixing devices are under exten-

sive investigation for use as active micromixers11 Elec-

troosmosis used to generate electrical potential gradients

has been proposed as a propulsion mechanism for bio-

fluids This mechanism would eliminate the need for a

moving apparatus12 The technology of electrokinetically-

driven mixing devices has led to the need for an efficient

fluid control mechanism for micro-fabricated systems that

perform complex chemical synthesis For example local-

ized flow circulations within the bulk flow near regions of

the microchannel wall with oppositely-charged surface het-

erogeneities were investigated and these circulations were

successfully applied to enhance the species mixing in a

T-shaped micro-mixer13 The results indicated that these

heterogeneous regions reduce the required length of the

mixing channel by 70 Sasaki et al14 reported a novel

micromixer based on AC electroosmosis and analyzed its

performance by experiments according to various param-

eters including applied voltage frequency and solution

viscosity Their results were compared with those obtained

from a theoretical model of the AC electroosmosis mixer

The comparison showed that a larger voltage led to more

rapid mixing The results could be used to estimate the

mixing performance of the AC electroosmosis-based mixer

under various experimental conditions They also provided

guidelines on the use of the micromixer in microfluidic

chemical analysis In addition the electroosmotic effect

results in swelling at the cathode side due to water enrich-

ment and shrinkage at the anode side owing to water deple-

tion these phenomena refer to the openclosed control

of IPMC (Ionic Polymer Metal Composite)-based micro-

valve systems15

Bio-fluids which show non-Newtonian fluid flow

behavior are often used in MEMS-fluidics Due to the

Fig 1 Schematic of the modeled electroosmotic micromixer

importance of Bio-MEMS and lab-on-a-chip technolo-

gies many researchers have recently focused on the

non-newtonian fluid behaviors of electrokinetically-driven

bio-fluids16 Das and Chakraborty17 obtained an analyti-

cal solution describing the transport characteristics of a

non-newtonian fluid flow in a rectangular microchannel

under the sole influence of electrokinetic effects As an

illustrative case study they analyzed the flow behavior

of a blood sample Zhao et al18 analyzed the electroo-

somotic flow of power-law fluids in a slit microchannel

by introducing exact and approximate analytical expres-

sions of shear stress effective viscosity and velocity profile

distribution Akgul and Pakdemirli19 presented analytical

and numerical solutions for electroosmotic flow of a third

grade fluid between micro-parallel plates They analyzed

influences of non-newtonian parameters namely the Joule

heating effect viscosity index and electrokinetic effect on

the velocity and temperature profiles of the fluid

In this study the mixing performance of a ring-type

electroosmotic micromixer was numerically investigated

Also the mixing performance of the mixer with different

obstacle configurations in the mixing chamber was exam-

ined by numerical analysis using the commercial code

COMSOL Especially the concentration of the dissolved

substances in the fluid was considered according to volt-

age electrode frequency and zeta potential

2 NUMERICAL DETAILS

21 Governing Equations

The complex flow phenomena in the modeled micromixer

were investigated by numerical methods Two-dimensional

Navier-Stokes equations describing fluid behavior were

employed to simulate the mixing of two aqueous fluids in



IP 115145156117Mon 09 Jul 2012 070343

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(b) 00625 s (c) 0125 s and (d) 01875 s


as follows

[V

t+ V middotV








Uslip =minusEx

(2)



field



(a) 0 s (b) 00625 s (c) 0125 s and (d) 01875 s



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Ohmrsquos law as





the fluidc





reactions





defined as follows

m equiv(1minus




respectively











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code COMSOL




















time was 3 s
















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obstacle








chamber








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to concentration

4 CONCLUSION







were obtained



fields






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chamber










36 381 (2004)



















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(b) 00625 s (c) 0125 s and (d) 01875 s


as follows

[V

t+ V middotV








Uslip =minusEx

(2)



field



(a) 0 s (b) 00625 s (c) 0125 s and (d) 01875 s



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Ohmrsquos law as





the fluidc





reactions





defined as follows

m equiv(1minus




respectively











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code COMSOL




















time was 3 s
















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obstacle








chamber








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to concentration

4 CONCLUSION







were obtained



fields






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chamber










36 381 (2004)



















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Ohmrsquos law as





the fluidc





reactions





defined as follows

m equiv(1minus




respectively











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code COMSOL




















time was 3 s
















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obstacle








chamber








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to concentration

4 CONCLUSION







were obtained



fields






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chamber










36 381 (2004)



















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code COMSOL




















time was 3 s
















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obstacle








chamber








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to concentration

4 CONCLUSION







were obtained



fields






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chamber










36 381 (2004)



















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obstacle








chamber








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to concentration

4 CONCLUSION







were obtained



fields






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chamber










36 381 (2004)



















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to concentration

4 CONCLUSION







were obtained



fields






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chamber










36 381 (2004)



















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chamber










36 381 (2004)


















Numerical Study on the Mixing Performance of a...

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