simple, coupled algorithms for solving creeping flows and their application to electro-osmotic flows

26
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2011; 66:1248–1273 Published online 15 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2310 Simple, coupled algorithms for solving creeping flows and their application to electro-osmotic flows Yong Kweon Suh , and Sangmo Kang Department of Mechanical Engineering, Dong-A University, 840 Hadan-dong, Saha-gu, Busan 604-714, Republic of Korea SUMMARY In this study we developed simple, coupled algorithms for solving low-Reynolds-number flows applicable to micro-scale flows such as electro-osmotic flows. The most popular scheme, i.e. the projection method, is not suitable for such flows because of its undesirable slip effect on boundaries at low-Reynolds-numbers. In our method, the velocity and pressure are strongly coupled, and the momentum and pressure equations are solved iteratively by using the successive over relaxation (SOR) method while exchanging the unknown variables as soon as they have been updated. The developed methods are applied to a model flow for evaluating their performance. It was found that the coupled schemes are indeed superior to a projection method, i.e. the fractional-step method, in both numerical accuracy and CPU time. The code is then applied to a dc electro-osmotic flow within a cavity driven by electrical force acting on the ions spread in the fluid. In this application, the system of equations for the fluid flow and that for the ion transport are solved in a decoupled way, but each system is solved by using fully implicit schemes. From the simulations and by introducing the concept of vorticity source, we can identify two roles of the body force, one contributing to build-up of the osmotic pressure and the other to the fluid flow. The interesting reverse flow occurring after the external potentials applied on the electrodes have been shut off is also investigated in terms of the vorticity source. Copyright 2010 John Wiley & Sons, Ltd. Received 28 September 2009; Revised 29 January 2010; Accepted 29 January 2010 KEY WORDS: velocity–pressure coupling; low-Reynolds-number flow; SOR method; staggered grids; electro-osmotic flow 1. INTRODUCTION The role of computational fluid dynamics (CFD) in the design of micro- or nano-fluidic devices has now become more important than ever before. Microflows are governed by Navier–Stokes (N–S) equations together with the continuity equation. Small spatial scales involved in such problems imply that the Reynolds number is very low so that the non-linear convection terms, adding more difficulty in the simulation of N–S equations, may be neglected. On the other hand, in microfluidic applications, various non-mechanical forces must be utilized to control fluid flow, such as electrical and magnetic forces; see, e.g. [1–3]. These forces are to be determined by solving additional equations, so that the number of variables is accordingly increased. Moreover, these forces appear Correspondence to: Yong Kweon Suh, Department of Mechanical Engineering, Dong-A University, 840 Hadan-dong, Saha-gu, Busan 604-714, Republic of Korea. E-mail: [email protected] Contract/grant sponsor: Ministry of Education, Science and Technology; contract/grant number: 2005-01091 Contract/grant sponsor: NRF; contract/grant number: 2009-0083510 Copyright 2010 John Wiley & Sons, Ltd.

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2011; 66:1248–1273Published online 15 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2310

Simple, coupled algorithms for solving creeping flows and theirapplication to electro-osmotic flows

Yong Kweon Suh∗,† and Sangmo Kang

Department of Mechanical Engineering, Dong-A University, 840 Hadan-dong, Saha-gu,Busan 604-714, Republic of Korea

SUMMARY

In this study we developed simple, coupled algorithms for solving low-Reynolds-number flows applicableto micro-scale flows such as electro-osmotic flows. The most popular scheme, i.e. the projection method, isnot suitable for such flows because of its undesirable slip effect on boundaries at low-Reynolds-numbers.In our method, the velocity and pressure are strongly coupled, and the momentum and pressure equationsare solved iteratively by using the successive over relaxation (SOR) method while exchanging the unknownvariables as soon as they have been updated. The developed methods are applied to a model flow forevaluating their performance. It was found that the coupled schemes are indeed superior to a projectionmethod, i.e. the fractional-step method, in both numerical accuracy and CPU time. The code is thenapplied to a dc electro-osmotic flow within a cavity driven by electrical force acting on the ions spreadin the fluid. In this application, the system of equations for the fluid flow and that for the ion transportare solved in a decoupled way, but each system is solved by using fully implicit schemes. From thesimulations and by introducing the concept of vorticity source, we can identify two roles of the bodyforce, one contributing to build-up of the osmotic pressure and the other to the fluid flow. The interestingreverse flow occurring after the external potentials applied on the electrodes have been shut off is alsoinvestigated in terms of the vorticity source. Copyright q 2010 John Wiley & Sons, Ltd.

Received 28 September 2009; Revised 29 January 2010; Accepted 29 January 2010

KEY WORDS: velocity–pressure coupling; low-Reynolds-number flow; SOR method; staggered grids;electro-osmotic flow

1. INTRODUCTION

The role of computational fluid dynamics (CFD) in the design of micro- or nano-fluidic devices hasnow become more important than ever before. Microflows are governed by Navier–Stokes (N–S)equations together with the continuity equation. Small spatial scales involved in such problemsimply that the Reynolds number is very low so that the non-linear convection terms, adding moredifficulty in the simulation of N–S equations, may be neglected. On the other hand, in microfluidicapplications, various non-mechanical forces must be utilized to control fluid flow, such as electricaland magnetic forces; see, e.g. [1–3]. These forces are to be determined by solving additionalequations, so that the number of variables is accordingly increased. Moreover, these forces appear

∗Correspondence to: Yong Kweon Suh, Department of Mechanical Engineering, Dong-A University, 840 Hadan-dong,Saha-gu, Busan 604-714, Republic of Korea.

†E-mail: [email protected]

Contract/grant sponsor: Ministry of Education, Science and Technology; contract/grant number: 2005-01091Contract/grant sponsor: NRF; contract/grant number: 2009-0083510

Copyright q 2010 John Wiley & Sons, Ltd.

ALGORITHMS FOR SOLVING CREEPING FLOWS 1249

in the N–S equations in non-linear form so that N–S equations are not linear in a strict sense forthe usual microfluidic phenomena, such as electro-osmosis.

Micro- or nano-fluidic flows typified by low-Reynolds-number flows are in many situationssolved by using commercial software. The main reason for this is that the users involved in devicedevelopment or scientific research are not familiar enough with CFD and they need quick resultsfor use in their specific applications. However, sometimes the CFD results are not completelyreliable, and that is the primary reason why the CFD data should only be used as references inthe initial stages of device designs. Since the device scale becomes smaller and smaller and theexperiment requires more sophisticated measurement tools and longer period of time, we mustdevelop efficient and reliable codes for the simulation of the small-scale flows. We also need to havesuch reliable in-house codes in our fundamental research associated with the multi-disciplinaryphenomena.

Nowadays, the most popular numerical method of solving the N–S equations regarding thevelocity–pressure coupling scheme is the so-called projection (or decoupling or splitting) methodintroduced by Chorin [4]. In this scheme, the momentum equations for the intermediate velocitycomponents are first solved iteratively with the pressure terms treated as known. Since the interme-diate velocity field does not satisfy the continuity equation, the pressure equations are then solvediteratively in the second step and the velocity components are corrected by using the pressure field.In this method, both the velocity and the pressure fields are not treated as unknown simultaneouslyin solving the equations, and it is called the decoupled method. We assert that this decoupledmethod is not suitable for low-Reynolds-number flows, such as microfluidic applications, becauseof the erroneous numerical boundary layers adjacent to walls (see, e.g. [5, 6]) and the undesirableslip motion at the boundaries. In addition, the numerical error is proportional to the inverse ofthe Reynolds number (see, e.g. [7]). For turbulent flows in macro scales, this does not exhibita significant problem, because the Reynolds number is high [8, 9]. For micro scales, however,this error can be significant and even the no-slip condition on the walls cannot be satisfied withthe projection methods. In microfluidic applications, the range of the time scales is broad, andso requiring a small time step for numerical accuracy or stability can create a bottle-neck andthe whole simulation method is rendered ineffective or even useless. Therefore, we must find asuitable scheme that is not only reliable but also accurate and fast for microfluidic applications.

In this paper, we propose a simple but fully coupled scheme for solving the N–S equations and thecontinuity equation. In our method the momentum and pressure equations are solved simultaneouslyusing the successive over relaxation (SOR) method. Aside from the standard algorithm of thecoupled method, we also developed some variants and compared the performance of each schemefor an unsteady flow model whose exact solution is prescribed. The famous third-order Runge–Kutta fractional-step method representative of the decoupled scheme is also applied to the sameproblem for comparative study. We then applied our coupled scheme to the dc electro-osmoticflow within a cavity and addressed the fundamental mechanism of the fluid motion.

2. EQUATION OF MOTION

We consider a viscous incompressible fluid with the density � and the viscosity � in a region �confined by boundaries ��. The fluid flow may be driven by various kinds of forces, such as theshear force exerted from a tangentially moving wall, the normal force caused by momentum fluxinput or output through the openings, or body force spread over the region such as the electricforce. Equations of fluid motion in a dimensionless form then read as

∇ ·u=0, (1)

�u�t

=−∇ p+ 1

Re∇2u+f, (2)

where u=u(x, t)=(u(x, t),v(x, t)) is the fluid velocity, p(x, t) is the pressure, f(x, t) is a givenbody force, t is the time and x=(x, y) are spatial coordinates, all the quantities being dimensionless.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1250 Y. K. SUH AND S. KANG

In this paper, we confine ourselves to two-dimensional problem. The Reynolds number may bedefined from a suitable choice of the reference velocity U and the reference length L as Re=UL/�. In this study the non-linear convective terms are not considered, assuming that Re remainslow enough.

We need initial conditions and boundary conditions (BCs) for the unknown variables in Equations(1) and (2). As the initial conditions, we may provide known data if available or arbitrary valuessuch as zero. At each boundary, we must provide the BCs for the velocity vector and/or the pressurein either the Dirichlet or the Neumann form. In this study we consider two types of BCs; the firsttype is the Dirichlet form, i.e. u=uw on a part of ��, and the second type is the homogeneousNeumann form, i.e. �us/�n=0 on the other part, where n and us denote local coordinates normalto the boundary and the tangential component of u, respectively.

The continuity equation (1) looks simple but in practice it involves the pressure whose role isto couple the continuity equation with the momentum equations. Depending on how the velocitycomponents and the pressure are coupled, the numerical methods can be classified into the coupledand the decoupled methods. One of the primary goals of this study is to develop simple, coupledmethods, but a typical decoupled method is also used to evaluate the performance of the developedmethods for the low-Reynolds-number flow tested.

3. DISCRETIZATION AND NUMERICAL METHODS

In this study, we confine ourselves to a rectangular-type domain for � and Cartesian variable grids,so that the boundary walls coincide with grid lines. We use the standard staggered-grid system;that is, the pressure is defined at the geometric center of the p-cell, surrounded by grid lines, andeach of the velocity components is defined at the center of each edge of the cell, as shown inFigure 1. All the spatial derivatives are discretized by using the central difference scheme. Thetime derivatives are discretized by using the Crank–Nicolson scheme.

The discretized continuity and N–S equations take the following form:

Dun≡ uni, j −uni−1, j

�x pi

+ vni, j −vni, j−1

�y pj=0, (3)

uni, j −un−1i, j

�t=−(Gp)i, j + 1

2Re(Lun+Lun−1)i, j +fi, j , (4)

where D, G and L denote the divergence operator, the gradient operator and the Laplace operator,respectively, in the discretized version. The superscript n attached to u and its components u andv indicate the time level, and the time level for p and f can be understood to be n−1/2.

We can categorize two numerical methods for the coupling of the velocity and pressure. Themain difference between the two can be described by using the flow charts shown in Figure 2. Inthe decoupled method, Figure 2(a), each of the momentum equations is first solved iteratively untilthe variables converge. In this calculation the pressure is taken to be known. Next, the pressure

pixΔ

pjyΔ

uixΔ

vjyΔ

jip , jiu ,

jiv ,

Figure 1. Schematic illustrating the definition points of flow variables and the grid spacings.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1251

1+= nn

converge?

?endtt >

uforsweep1−

vforsweep1−

pforsweep1−

stop

start

yes

yes

no

no

1+= nn

converge?

?endtt >

converge?

converge?

uforsweep1−

vforsweep1−

pforsweep1−

stop

start

yes

yes

yes

yes

no

no

no

no

(a)

(b)

Figure 2. Flow charts for solving the velocity and pressure equations with(a) decoupled and (b) coupled methods.

equation is solved iteratively until it converges. In this calculation, the velocity components aretaken to be known. The above procedure constitutes one time-step computation. In the coupledmethod, on the other hand, the momentum and the pressure equations are solved simultaneously.Hence, while the coupled method solves a big linear system of equations, the decoupled methodsolves three smaller linear systems of equations. Usually the diffusion terms are treated implicitly,whereas the convection terms are treated explicitly (semi-implicit method) or implicitly (fullyimplicit method; e.g. Kim et al. [10]) after a suitable linearization. Since we are interested increeping flows, we neglect the convection terms and our method corresponds to a fully implicitmethod.

3.1. Decoupled method

In the decoupled method, the numerical calculation of the unknown variables during one time step�t is basically divided into two steps. In the first step, the momentum equations are solved implicitlyfor the velocity vector with the pressure being specified from a suitable temporal extrapolation;e.g. simple second-order extrapolation p=2pn−1/2− pn−3/2. Since the velocity field obtained inthis way does not satisfy the continuity equation, it must be corrected in the second step. Thismeans that the momentum equations (4) should be written in terms of an intermediate velocityvector u∗ and the extrapolated pressure p∗ as follows:

u∗−un−1

�t=−Gp∗+ 1

2Re(Lu∗+Lun−1)+f, (5)

where the subscripts i , j have been dropped for simplicity. Note that hereinafter the Crank–Nicolson method is employed for discretization of the diffusion terms. In solving this equation,the simplest choice for the BC of u∗ is setting u∗ =uw on ��. In the second step, the velocity

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1252 Y. K. SUH AND S. KANG

field u∗ is corrected from the solution of the following equation:

�u�t

≡ un−u∗

�t=−G�, (6)

where �u corresponds to the velocity correction and � is the so-called pseudo pressure, its rolebeing to make the field un satisfy the incompressibility constraint (3). Applying the operator D toEquation (6) then gives the Poisson equation for � as follows:

DG�= 1

�tDu∗. (7)

For this equation we can apply the homogeneous Neumann BC; e.g. ��/�n=0 on �� since thenormal velocity component is assumed to be zero there. The numerical procedure is as follows.Solve Equation (5) for u∗; next, solve Equation (7) for �; get �u from (6); then correct thevelocity from un=u∗+�u. The pressure must also be updated for use in the next time step. Thecorresponding formula can be obtained by summing Equations (5) and (6) and comparing the resultwith the original one (4). Thus we have

p= p∗+(1− �t

2ReL

)�. (8)

Taking summation of the formulas (5) and (6) and applying (7) and (8), we reproduce the originalform of the momentum equations (4), and the continuity equation is also satisfied by un. However,if we look into Equation (6) we can readily understand that the prescribed tangential velocity BCis not satisfied. The error amounts to �t ·G� as seen from (6). Hence, we expect a certain amountof slip velocity at the region of the non-zero tangential gradient of the pseudo pressure on theboundaries. Kim and Moin [8] suggested using a modified BC for the tangential component of u∗,but that must be dependent on the type of BC [11].

The above algorithms represent the very basic idea of the decoupled methods. In practice,however, there are many variants of decoupled methods developed under different names such asthe ‘fractional-step method’ or the ‘projection method’. In this study, we applied the fractional-stepmethod with third-order Runge–Kutta algorithm for the temporal integration to the model flowsto comparatively estimate the performance of the coupled method, see, e.g. Kim and Choi [12].In this method, calculation for one global time-step is composed of three sub-step calculations. Ineach sub-step, we obtain un,k , �n,k , un,k and pn,k by using the following equations:

un,k−un,k−1

�t=−2�kGpn,k−1+ �k

Re(Lun,k+Lun,k−1)+2�kf n,k−1/2, (9a)

DG�n,k = 1

2�k�tDun,k, (9b)

�un,k

�t≡ un,k− un,k

�t=−2�kG�n,k, (9c)

pn,k = pn,k−1+(1− �k�t

ReL

)�n,k, (9d)

where the superscript k denotes the sub-step’s index ranging from 1 to 3; the variable with k=0, i.e.�n,0, should be understood as �n,0=�n−1,3. The coefficients �k are: �1= 4

15 , �2= 115 and �3= 1

6 .

3.2. Coupled method—original form

We can write the momentum equations (4) for the computational efficiency as follows:

u∗−un−1

�t= 1

2ReLun−1+f, (10a)

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1253

un−u∗

�t=−Gp+ 1

2ReLun, (10b)

where u∗ is a kind of intermediate velocity, which can be obtained explicitly from Equation (10a).Note that no approximation has been applied to this set-up; adding Equation (10b) to Equation (10a)exactly reproduces Equation (4). In the coupled method, the continuity equation (3) is replacedby the pressure Poisson equation given by substituting (10b) for the velocity components into (3).Then we obtain

DGp= 1

2ReDLun+ 1

�tDu∗, (11)

where we have set Dun=0 as required. The operators L and D can commute with each other,even with the variable grids of course, so that we can also set DLun=0 without restriction, exceptfor the boundary cells where compatible BC for the pressure must be imposed such that we canstill apply DLun=0 there; detailed derivation of the pressure BC will be given later. Then thepressure Equation (15) is simplified as follows:

DGp= 1

�tDu∗. (12)

In summary, the algebraic equations to be solved are (10b) and (12) with suitable BCs. Theboundary conditions for (10b) must be given for both the normal and tangential velocity componentsat each boundary. It is easy to apply them to the solver of Equation (10b). For instance, for thecalculation of uni, j at j =2, we can relate the ghost velocity uni,1 with the real one uni,2 to the secondorder as follows:

uni,1=2uw−uni,2,

where uw is the prescribed wall tangential velocity. In solving Equation (12) care must be given tothe p-cells adjacent to walls, where special consideration should be given to setting the pressureBC so that the restriction DLun=0 is still satisfied and accordingly the regular form (12) can stillbe applicable there. To derive the compatible BC for p, we start with the full discretization ofEquation (11) applicable to the interior p-cells as follows:

1

�x pi

(�x pi+1/2, j −�x pi−1/2, j )+ 1

�y pj(�y pi, j+1/2−�y pi, j−1/2)

1

2Re�x pi

[1

�xui(�xu

ni+1/2, j −�xu

ni−1/2, j )+

1

�y pj(�yu

ni, j+1/2−�yu

ni, j−1/2)

− 1

�xui−1(�xu

ni−1/2, j −�xu

ni−3/2, j )−

1

�y pj(�yu

ni−1, j+1/2−�yu

ni−1, j−1/2)

]

+ 1

2Re�y pj

[1

�x pi

(�xvni+1/2, j −�xv

ni−1/2, j )+

1

�yvj(�yv

ni, j+1/2−�yv

ni, j−1/2)

− 1

�x pi

(�xvni+1/2, j−1−�xv

ni−1/2, j−1)−

1

�yvj−1

(�yvni, j−1/2−�yv

ni, j−2/2)

]

+ 1

�t

(u∗i, j −u∗

i−1, j

�x pi

+ v∗i, j −v∗

i, j−1

�y pj

), (13)

where �x�i, j and �y�i, j denote the central difference of ��/�x and ��/�y around the point where�i, j is to be evaluated; e.g. �x pi+1/2, j =(pi+1, j − pi, j )/�xui and �yui, j+1/2=(ui, j+1−ui, j )/�yv

j .

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1254 Y. K. SUH AND S. KANG

All the terms within [ ] after combination vanish in the interior p-cells reproducing Equation (7)or the following in an explicit form:

1

�x pi

(�x pi+1/2, j −�x pi−1/2, j )+ 1

�y pj(�y pi, j+1/2−�y pi, j−1/2)

= 1

�t

(u∗i, j −u∗

i−1, j

�x pi

+ v∗i, j −v∗

i, j−1

�y pj

). (14)

For the p-cells adjacent to the boundary, however, Equation (13) no longer leads to Equation (14)when the BC is set inappropriately. In order to use the same form as (14) for the pressureequation in the boundary p-cell, we must apply the so called compatible BC for the pressure. Wedemonstrate the derivation of the compatible BC for the pressure at a boundary p-cell located atj =2. Derivation of the pressure equation for this cell should be slightly different from that for theinterior cell. As for the velocity on the boundary edge, vi, j−1, we substitute the known boundaryvalue vw, instead of Equation (4), into the continuity Equation (3); recall that in this study vw wasset at zero for simplicity. Then we get

1

�x pi

(�x pi+1/2, j −�x pi−1/2, j )+ 1

�y pj(�y pi, j+1/2−0)

= 1

2Re�x pi

[1

�xui(�xu

ni+1/2, j −�xu

ni−1/2, j )+

1

�y pj(�yu

ni, j+1/2−�yu

ni, j−1/2)

− 1

�xui−1(�xu

ni−1/2, j −�xu

ni−3/2, j )−

1

�y pj(�yu

ni−1, j+1/2−�yu

ni−1, j−1/2)

]

+ 1

2Re�y pj

[1

�x pi

(�xvni+1/2, j −�xv

ni−1/2, j )+

1

�yvj(�yv

ni, j+1/2−�yv

ni, j−1/2)

− 0 − 0

]

+ 1

�t

(u∗i, j −u∗

i−1, j

�x pi

+ v∗i, j −0

�y pj

), (15)

where the zeros are intentionally added to explicitly demonstrate which terms in Equation (13)have been set at zero. It can be shown by applying Equation (3) to Equation (15) that this isequivalent to the following:

1

�x pi

(�x pi+1/2, j −�x pi−1/2, j )+ 1

�y pj(�y pi, j+1/2−0)

= 1

2Re�x pi �y pj

[−�yuni, j−1/2+�yu

ni−1, j−1/2]+

1

�t

(u∗i, j −u∗

i−1, j

�x pi

+ v∗i, j −0

�y pj

), (16)

where again the boundary velocity vi, j−1 is set at zero because we have assumed vw=0 forsimplicity. Comparing this with the regular form (14), we obtain the following formula for thecompatible pressure BC on the solid wall:

�y pi, j−1/2= 1

2Re�x pi

[−�yuni, j−1/2+�yu

ni−1, j−1/2]+

v∗i, j−1

�t. (17)

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1255

This can be used to evaluate the ghost pressure pi, j−1. When we do not need to get this ghostpressure aside from the interior pressure, we can simply set zero for v∗

i, j−1 not only in Equations (17)but also in Equation (14), regardless of its actual value. The normal pressure gradient �y pi, j−1/2on the boundary may be different from the true value, but it does not affect the numerical resultsin the interior cells at all.

It can be seen from (17) that, when the Reynolds number is very high, setting zero for thenormal gradient �y p should result in a small error. But at low Reynolds numbers, the first term onthe RHS of (16) contributes to the pressure significantly and so it must be included in the pressureBC; in fact this is the key factor for the success of the coupled method with the original form ofthe equations. This method will be referred to as CMO. Further, it is the first term on the RHS ofEquation (17) that requires Equation (12) to be coupled with Equation (10b).

3.3. Decoupled method—some variants

Referring to the basic idea of the decoupled method, we can design some variants of the originalcoupled method in order to simplify the pressure BC. As the first variant (to be referred to asCMV1), we consider the following algorithm:

u∗−un−1

�t= 1

2Re(Lu∗+Lun−1+L�u)+f, (18a)

�u�t

≡ un−u∗

�t=−Gp, (18b)

where the pressure on the RHS of Equation (18b) is to be given from the solution of the followingPoisson equation:

DGp= 1

�tDu∗. (19)

Numerical procedure for the one time step is the same as in Figure 2(b); we first operate 1-sweepcalculation of Equation (18a) for the auxilary velocity u∗, with �u being treated as known; next,1-sweep calculation of Equation (19) for p is performed; then �u, to be used in the next sweep ofEquation (18a), is calculated from Equation (18b); this comprises one global sweep, and we iteratethe sweep until the variables converge. We need BCs for u∗, and the most naive way is to simplyapply the same BCs as those given to the original variables un. In this case, the BCs for �u mustbe homogeneous but should follow the prescribed BC type; that is, for the Dirichlet type BC wejust set �u=0, and for the Neumann type BC we also set zero for the normal gradient of �u there.Even if the no-slip BC is applied to u∗ on the boundary, however, its tangential component showsa velocity profile of slip-flow style near the boundary, and this can be understood from Equation(18b) in the same way as before. Even in this situation, the profile of un reveals the desired no-slipkind, since �u also shows the slip-velocity profile but in the other way, as shown in Figure 3. Thisimplies that we can apply different kinds of BC for u∗ and so for �u. We tried various kinds ofBCs for u∗ and it turned out that the results did not change regardless of the kinds of BC used.BC for the pressure in Equation (19) is very simple. For instance, for the p-cell at j =2 we canjust apply the homogeneous Neumann-type condition (�y p)i, j−1/2=0; of course we must also setzero for the term v∗

i, j−1 appearing on the RHS of Equation (19) for the same reason as before.As the second variant (to be referred to as CMV2), we consider the following algorithm:

u∗−un−1

�t=−G(pn−1+�p)+ 1

2Re(Lu∗+Lun−1+�u)+f, (20a)

�u�t

≡ un−u∗

�t=−G�p, (20b)

DG�p= 1

�tDu∗. (20c)

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1256 Y. K. SUH AND S. KANG

n *un

un nu

(a) (b) (c)

Figure 3. Superposition of the intermediate velocity u∗ and the velocity correction �u, each showing slipmotion, to make the nominal velocity un without slip.

Here �p corresponds to the pressure correction. The pressure at the new time level n is obtainedfrom pn= pn−1+2�p. The numerical procedure and implementation of BCs is basically the sameas those for the first variant.

4. TEST PROBLEM—OSCILLATING TOP LID

As a test problem we consider the fluid flow driven by a top lid’s tangentially oscillating motionwithin a square cavity, 0�x�1,0�y�1, as shown in Figure 4. For the top lid’s tangential velocity

uw(x, t)=sin�t sinx, (21)

we can derive the exact solution for Stokes’ flow as follows:

u(x, y, t)=C

(�F�y

−F(1, t)coshy

sinh

)sinx, (22a)

v(x, y, t)=−C

(F(y, t)−F(1, t)

sinhy

sinh

)cosx, (22b)

p(x, y, t)=[exp(a)sin(�t+b−)−exp(−a)sin(�t−b−)]�C coshy

sinhcosx . (22c)

Here the function F(y, t) and the constant C are defined as follows:

F(y, t)=exp(ay)cos(�t+by−)−exp(−ay)cos(�t−by−),

C=(K 21 +K 2

2 )−1/2,

�=�Re, (23)

a=√0.5(2+

√4+�2), b=

√0.5(−2+

√4+�2),

K1=2[(−a sinha+cothcosha)sinb−bcosha cosb],K2=2[(a cosha−cothsinha)cosb−b sinha sinb],

= tan−1(K2/K1).

We can see from Equations (22a) and (23) that the solution is characterized by a thin layer near thetop wall at high � (angular frequency) and is composed of two traveling waves: one propagatingdownward [first term on the RHS of Equation (23)] and the other upward [second term on theRHS of Equation (23)].

In this test calculation, we employed four kinds of methods. The first one is CMO, whichsolves Equations (10b) and (12) directly with the compatible pressure BC such as Equation (17).The second one is CMV1 solving Equations (18a) and (19). The third one is CMV2 solving

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1257

x

y

),( txuw

Figure 4. Unsteady driven-cavity flow with top lids’ oscillatory motion.

Equations (20a) and (19). The last one is the fractional-step method with the third-order Runge-Kutta scheme (RK3) developed by Kim and Moin [8] and Choi and Moin [9], representing the mosttypical decoupled method. We will evaluate each method’s performance by comparing accuracyand CPU time.

Accuracy is determined by calculating the rms error for the velocity components defined by

εrms= 12 〈(unumi, j −uexai, j )rms+(vnumi, j −vexai, j )rms〉,

where the superscripts ‘num’ and ‘exa’ mean the numerical and the exact solutions, respectively,and the subscript ‘rms’ on the RHS means root-means-square operation of the correspondingvariable over the domain. The symbol 〈〉 indicates the temporal average.

As the initial conditions, we impose the exact solutions given in Equations (22a), (22b) and(22c). As the linear solver, we use the SOR method for CMO, CMV1 and CMV2, and theincomplete Cholesky conjugate gradient (ICCG) method for RK3; the reason of using differentelliptic solvers for the coupled and decoupled methods will be explained later. For the coupledmethod, the iteration during one time-step is judged to have converged when each of the residualsεum, εvm and εdiv defined as

εum=maxi, j

|un,ki, j −un,k−1

i, j |, εvm=maxi, j

|vn,ki, j −v

n,k−1i, j |, εdiv=max

i, j|Dun,k |i, j

is less than a limit value where the superscript k denotes the iteration counter. We set, as the limitresiduals, εuvlim=10−6 for εum and εvm, and εdivlim=10−4 for εdiv, elsewhere mentioned. It wasfound that these limit values are small enough that for all the parameter values treated in thisstudy, the numerical results yield the errors not caused by these limit values. For RK3, we useεum and εvm for the convergence check of the u- and v-equations, respectively. For the pressureequation, however, we cannot use εdiv because it turned out that εdiv never approaches zero buta finite value upon iteration; we assume that the decoupled method inevitably suffers from thisdrawback, although there could be some remedy. Instead of εdiv, we use εpm defined as

εpm=maxi, j

|pn,ki, j − pn,k−1

i, j |

as the convergence checker. In this study we set εplim=10−6 as the limit residual for the ICCGroutine for the pressure equation in the decoupled method.

As is well known, the SOR factor of the SOR method used as the linear solver in our coupledschemes has a significant effect on CPU time, and there exists an optimum value for a given methodwith a given parameter set such as the grid system and the time step. Hence, before evaluatingthe performance of each method, we must find the optimum SOR factor in each method. We havefound that the SOR factors for the momentum equations, i.e. wu and wv , exert far less influencethan the SOR factor for the pressure equation, i.e. wp. Hence, we fix wu =wv =1 for all the runsand study the dependence of wp on CPU time.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1258 Y. K. SUH AND S. KANG

wp

wp

wp

1.5 1.6 1.7 1.8 1.9 20

20

40

60

80

CP

U ti

me

[s]

CP

U ti

me

[s]

CP

U ti

me

[s]

1.5 1.6 1.7 1.8 1.9 20

10

20

30

40

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

20

40

60

80

100

(a)

(b)

(c)

Figure 5. Effect of SOR factor wp on the CPU time spent for two-period calculation with CMO(circles), CMV1 (squares) and CMV2 (deltas) at the parameter sets: (a) �=50, �t=0.001; (b) �=50,

�t=0.0001; and (c) �=5, �t=0.001; Re=1, I = J =51, εum=εvm=10−6, wu =wv =1.

Figure 5 shows the effect of the SOR factor wp on CPU time for two-period calculation fordifferent time steps and angular frequencies. In this study we used a desktop computer with Intel�

CoreTM 2 Duo CPU at 2.66GHz. Hereinafter, we fix the Reynolds number at Re=1. We see that,for all the three methods, at large � and large �t (Figure 5(a)) the variation of CPU time versus wpis not so different from the usually experienced pattern; that is, CPU time slowly increases whenwp is decreased from the optimum value whereas it increases sharply when it is increased beyondthe optimum value (Figure 5(a)). It also shows that CPU time at wp =wopt is not so much differentfor different methods. At high frequency and low time-step (Figure 5(b)), however, CMV2 takesmore time than the others, and CPU time with CMO remains at the lowest level among threemethods over the wide range of wp. It should be also mentioned that the SOR solver for CMV2overflows at the SOR factor larger than 1.5. At low � and large �t (Figure 5(c)), the pattern of

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1259

x

wop

t

rms

0.02 0.04 0.06 0.08 0.11.4

1.6

1.8

2

0.005

0.01

0.015

Figure 6. Optimum SOR factor wopt (solid symbols) and the corresponding rms error εrms (open symbols)versus grid size for CMO (circles), CMV1 (squares) and CMV2 (deltas) methods; �=50, �t=0.001.The solid line represents the optimum SOR factor obtained from the theoretical formula. The dashed lineindicates εrms obtained from CMO at lower �t’s so that εrms is independent of �t . A line segment drawn

below the εrms curve indicates the second-order trend as for the accuracy.

variation is not so much different from that of Figure 5(a), except that the optimum values inFigure 5(c) are slightly smaller than those of Figure 5(a).

Figure 6 shows wopt, the optimum SOR factors for the pressure solver, and the correspondingrms error εrms for three methods calculated at �=50 and �t=0.001. The methods of CMO andCMV1 produce wopt not far from the theoretical prediction given by the following formula for theuniform grids [13]:

wopt= 2(1−√1−�)

�,

�=√

�y2 cos[/(I −1)]+�x2 cos[/(J−1)]�x2+�y2

.

The method of CMV2 shows a much lower value of wopt in particular at coarse grids. On the otherhand, there is no difference in the numerical accuracy among the three methods at every value ofwopt. The dashed line in Figure 6 denotes the highest limit of the accuracy at each grid resolutionobtained by applying �t small enough that the accuracy should be independent of �t . It clearlyindicates that the numerical scheme is also of second order in space.

We also investigated the feasibility of the ICCG method as the solver of the momentum as wellas the pressure equations in the coupled method. We recall that in each sweep of Equation (10b)for calculating un, the pressure p on the RHS of (10b) is treated as known and vice versa ineach sweep of Equation (11) for p. This means that during iterative calculation, the residual ofeach linear system is dependent on the solution of the other system. Therefore, the classical ICCGmethod cannot be applicable in this case. In this study, we modified the original ICCG algorithmand tested it for a simpler but non-linear system. The conclusion is that ICCG is not preferable;refer to Appendix for the details of numerical methods as well as the results. Therefore, in thisstudy we use only the SOR solver for coupled methods. On the other hand, we have found that,in the decoupled method, ICCG is superior to SOR in particular with smaller grid spacings, asdemonstrated in the Appendix. Hence, in the following all the results obtained from RK3 are givenwith ICCG.

Effect of the grid size on CPU time and rms error is then studied and the results are shownin Figure 7 for the CMO and the RK3 methods. Variation of the rms error indicates that bothschemes are of second-order accuracy. Overall, CMO is superior to RK3 because CPU time is lessand accuracy is higher (about three times) for CMO than for RK3. The fundamental reason for

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1260 Y. K. SUH AND S. KANG

Δx

ε rm

s

cpu

time

[s]

0.02 0.04 0.06 0.08 0.110-4

10-3

10-2

10-1

101

100

103

102

Figure 7. Effect of the grid size �x(=�y) on CPU time (solid lines with symbols) and rms error (dashedlines with symbols) for CMO (circles) and RK3 (diamonds); �=50, �t=0.0005. A line segment drawn

between two dashed lines indicates the second-order trend as for the accuracy.

T

ε rm

s

cpu

time

[s]

0.2 0.4 0.6 0.8 1

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

100

200

300400

Figure 8. Effect of the period of the top lid’s motion on CPU time (solid lines with symbols) and rmserror (dashed lines with symbols) for CMO (circles) and RK3 (diamonds); I = J =51, �t=0.001.

such lower accuracy for RK3 is due to the slip effect of the fluid on the boundaries caused by theinsufficient interaction between variables at each time step as explained previously.

Such difference in performance between CMO and RK3 is more pronounced at a smaller �, asshown in Figure 8. At �=2 (or at T =1), CPU time for RK3 is almost 10 times that for CMO,and the numerical accuracy for CMO is more than double the one for RK3. Difference in the codeperformance is also more pronounced at a smaller time step as shown in Figure 9, in which again10 times difference in CPU time is shown at �t=0.0001.

5. ELECTRO-OSMOTIC FLOWS WITHIN A CAVITY

5.1. Formulation and numerical methods

As the second but main flow problem, we consider an electro-osmotic flow around a pair ofelectrodes receiving DC attached to the top and the bottom walls of a rectangular cavity of height Hand length L . An anode maintained at the potential �0 is located at the top wall spatially covering

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1261

Δt

ε rm

s

cpu

time

[s]

10-4 10-3 10-2

0.001

0.002

0.003

0.004

0.0050.0060.0070.008

101

102

Figure 9. Effect of the time step on CPU time (solid lines with symbols) and rms error (dashed lines withsymbols) for CMO (circles) and RK3 (diamonds); I = J =51, �=50.

the whole range of the channel length and a cathode maintained at zero potential is located at thebottom wall with a smaller length, Le. The fluid has the density �, the kinematic viscosity � andthe electric permittivity ε. A pair of monovalent ions of the diffusivity D0 are spread in the fluidwith the concentration C0 in (mole/m3) for each ion.

The dimensionless governing equations for this flow problem can be written as follows; see[14, 15]:

∇ ·u=0, (24a)

�uSc�t

=−∇ p+∇2u− �e∇�, (24b)

�C±

�t+Pe∇ ·(uC±)=∇ ·(∇C±±�∗

0C±∇�), (24c)

∇2�=− �e, (24d)

�e=C+−C−. (24e)

Here the spatial coordinates x and lengths are scaled by H , the time t by H2/D0, the velocityu by Ue0=ε�2

0/(��H), the pressure p by ��Ue0/H , the concentration C± by C0, the potential� by the external potential �0 and the charge density �e by FC0, where F=96485C/mole isthe Faradaic constant. Figure 10 shows the geometry of the flow model and the BCs in terms ofdimensionless variables. Dimensionless parameters shown in Equations (24b), (24c), (24d) andgeometric relation are defined as follows:

Sc= �

D0, Pe= HUe0

D0, = 1

2�2�∗0, �∗

0= �0

�T, xm = L/H, L∗

e = Le/H,

where Sc and Pe are the Schmidt number and the Peclet number, respectively. Further, thedimensionless Debye length � is defined as

�= DH

= 1

H

√εRT

2F2C0.

Here, D is the Debye length, R=8.3145J/Kmole the gas constant and T the temperature.Equations (24a) and (24b) are the continuity and the momentum equations, respectively, and

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1262 Y. K. SUH AND S. KANG

Figure 10. Boundary conditions for the fluid velocity, the ion concentration and the electric potential of adc electro-osmotic flow problem within a two-dimensional space filled with an electrolyte having a pair

of electrodes one at the top wall and the other at the bottom wall.

Equation (24c) corresponds to the Nernst–Planck equation describing the transport of the cationconcentration C+ and the anion concentration C−. Equation (24d) is the Poisson equationrelating the potential and the charge density. For more detailed explanation to the formulation,refer to [14–16].

In discretization of the additional Equations (24c) and (24d), we follow a procedure similar towhat was done to the momentum equation (24b); that is, a central difference scheme for the spatialderivatives and the Crank–Nicolson scheme for the time integration are employed. Depending onthe procedure to couple those equations during one global time step, we used the CMO schemeas the coupled method and the RK3 scheme as the decoupled method.

In CMO, one global time step is composed of two sub-steps. In the first sub-step, the fluid flowto be determined from Equations (24a) and (24b) is solved by using the coupled method describedin Section 3.2; in this calculation, the body-force term �e∇� is set as known. In the secondsub-step, the ion transport problem to be determined from Equations (24c) and (24d) is solvedusing a method similar to the CMO developed for the flow problem. The numerical procedure forthe second sub-step can be described as follows. During the second sub-step, the velocity field isset as known as determined from the first sub-step. First, one-sweep calculation of � is conductedby applying the SOR method to Equation (24d), where the concentrations C± are treated as known.Second, one-sweep calculations of C± are executed by applying the SOR method to Equation(24c), where � is now treated as known. We check the convergence and repeat the above sequentialthree-sweep calculations until the residuals become small enough.

In RK3, one global time step is composed of three sub-steps as described previously. In eachof the sub-steps, Equations (24b)–(24d) are solved separately; that is, each equation is solvediteratively without interaction with the other equations. As the linear solver we used the ICCGmethod for the momentum, pressure and potential equations and the conjugate gradient squared—stabilized (CGSTAB) method for the ion transport equations; see, e.g. Ferziger and Peric [17]for the details of both the methods. More detailed description of the numerical methods is givenin [14].

Along the y-direction, non-uniform grids clustered near the top and bottom walls are designedby using the following function:

y= 1

2

{1+ tanh[�y(�−1/2)]

tanh(�y/2)

},

where grids are uniform in the �-space and �y controls the degree of clustering; a higher value of�y means more clustering. In this study we selected �y =4. Along the x-direction, uniform gridsare used for both the CMO and the RK3 methods and, additionally, non-uniform grids for CMO.For the non-uniform grids along the x-direction, we employed locally linear functions for thederivative dx/d�, as shown in Figure 11, where grid spacing is uniform in the � space. Parametersinvolved in the grid design include xd , half of the range of x near the electrode edges where gridsare to be clustered, and �x controlling the degree of clustering, i.e. �x ∼=(dx/d�)max/(dx/d�)min. Inthis study, we selected xd =0.2L∗

e and �x =10. The number of grids used was I × J =201×101.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1263

dx

d

1m 2m

1 /m

/ 2mx=

a=

dx dx

Figure 11. Sketch of the derivative of the function x= x(�) used in the designof variable grids along the x-direction.

In this model flow, we set the following parameters as fixed:

H =0.5�m, L=1.5�m, Le=0.5�m, C0=0.1mM, D0=10−10m2/s,

ε=6.93×10−10 C/Vm, �=10−6m2/s, �=1000kg/m3, T =293K.

For this set-up, the Schmidt number is calculated to be Sc=104, and the Debye length D =30.1nmand its dimensionless quantity �=0.0602. For the case with the external potential �0=0.2V, wecalculate the important dimensional and dimensionless parameters as follows:

Ue0=0.055m/s, Pe=277, =17.4, �∗0=7.92.

For �0=1V, we obtain

Ue0=1.39m/s, Pe=6930, =3.48, �∗0=39.6.

For all the simulation results presented in this paper, we keep the dimensionless time step constantat �t=5×10−5.

5.2. Numerical results at a low external potential

Figure 12 shows typical streamlines of the electro-osmotic flows given with �0=0.2V. Near thebottom electrode, fluid flows from the leading edges toward the central region, and then it goesup to reach the top wall and recirculates forming a pair of large vortices; see Figure 12(a). Thisflow pattern is typically observed in the ac electro-osmotic flows too; see, e.g. Green et al. [18].Later, there occurs a pair of smaller vortices near the edges of the bottom electrodes as shown inFigure 12(b). These vortices are, however, thought to be spurious, because when more refined gridsare used near the edges of the bottom electrode with the non-uniform grids they never appear, asshown in Figure 12(c). Later, a pair of small, flat vortices appear near the top electrode as shownin Figure 12(d).

Figure 13 presents the spatially averaged fluid velocity V (t) defined as

V (t)=[1

A

∫A(u2+v2)dA

]1/2obtained numerically by using the methods of CMO and RK3. Here, A denotes the domain area.We note that initially the velocity increases rapidly because of charging in the electric doublelayer (EDL) of the electrodes; we call it the ‘acceleration stage’. It reaches a maximum value att∼=0.011 and then decreases slowly; we call this the ‘deceleration stage’. Decrease in the fluidvelocity indicates that charging is being completed. The charging time may be understood to berelated to the diffusion time of the ion concentrations. We have two diffusion times; the first one,�D , characterizing the diffusion within the Debye layer, is obtained from �D =2D/D0=9�s or�∗D =0.0036 in the dimensionless quantity, and the second one, �H , characterizing the diffusion

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1264 Y. K. SUH AND S. KANG

x

y

0 0.5 1 1.5 2 2.50

0.5

1

x

y

0 0.5 1 1.5 2 2.50

0.5

1

(a)

(b)

x

y

0 0.5 1 1.5 2 2.5 30

0.5

1

x

y

0 0.5 1 1.5 2 2.50

0.5

1

(c)

(d)

Figure 12. Streamlines of the electro-osmotic flows obtained for �0=0.2V from the method of CMO;given (a) at t=0.01 with uniform grids; (b) at t=0.07 with uniform grids; (c) at t=0.07 with non-uniform

grids; and (d) at t=0.25 with non-uniform grids.

within the bulk, is given from �H =H2/D0=0.0025s or �∗H =1 in the dimensionless quantity.

The approximate charging time is measured from Figure 13 to be �∗C =0.1, and thus the geometric

mean of �∗D and �∗

H , i.e.√

�∗D�∗

H =0.06 is close to the dimensionless charging time. This isconsistent with the assertion of Bazant et al. [19].

In Figure 13, we can also see that the magnitude of V given by RK3 is slightly greater thanthat given by CMO, and this is due to the slip phenomenon on the walls in RK3. In Figure 14, weshow the distribution of u along the line x=0.99 and the inset demonstrates that the result givenby RK3 clearly exhibits the slip motion. The undesirable slip motion becomes more prominent ata larger �t . This implies that a decoupled method, such as RK3, is not suitable for the simulationof the micro-scale flows.

The body force f=− �e∇�, driving the fluid flow and/or contributing to the build-up of thepressure, is strongest near the edge of the bottom electrodes. Figure 15(a) and (b) show distributionof the body-force components near an edge of the electrode. The tangential component f u isconcentrated at the edge, whereas the normal component f v is spread over the whole range of theelectrode. The time sequence of the plot for f reveals that neither component vanishes but both

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1265

t

V

0.0064

0.0066

0.0068

t

V

0.050

0.002

0.004

0.006

0.1 0.15 0.2

0.005 0.01 0.015

Figure 13. Time history of the average velocity V of the unsteady electro-osmotic flow givennumerically at �0=0.2V from the method of CMO with uniform (solid line) and non-uniform(dash-dot line) grids along the x-direction, and the one given from the method of RK3 with

uniform grids along the x-direction (dashed line).

y

u

00

0.005

0.01

0.015

y

u

0

-0.005

0

0.005

0.01

0.015

0.02

0.2 0.4 0.6 0.8 1

0.005 0.01 0.015 0.02

Figure 14. Distribution of the velocity component u along the line x=0.99 obtained for �0=0.2V fromthe method of CMO (solid lines) and that from RK3 (dashed lines).

asymptotically approach to finite-valued fields. Considering the fact that the electro-osmotic flowunder dc ultimately decays, we can say that those steady finite-valued body-force fields shouldcontribute only to the osmotic pressure. For a more detailed investigation of the effect of the bodyforce, we consider the Helmholtz decomposition of f as follows:

f=g+∇�, (25)

where g and ∇� are rotational and irrotational parts, respectively. Take curl to Equation (24b) toobtain the vorticity transport equation as follows:

��

Sc�t=∇2�+h, (26)

where � is the vorticity, k is the unit vector normal to the (x, y) plane, and

h≡(∇×f) ·k=(∇×g) ·k=− (∇�e×∇�) ·k (27)

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1266 Y. K. SUH AND S. KANG

-160

-40

x

y

0.9 0.95 1 1.05 1.1 1.150

0.05

0.1

10

10

20

40

6090

xy

0.9 0.95 1 1.05 1.1 1.150

0.05

0.1

x

y

0.9 0.95 1 1.05 1.1 1.150

0.05

0.1

300

100

200

(a)

(b)

(c)

Figure 15. Distribution of (a) f u(x, y); (b) f v(x, y); and (c) (∇×f)·k at t=0.01 near the leading edgeof the bottom electrode located at x=1, obtained for �0=0.2V from CMO with non-uniform grids.

corresponds to the vorticity source. The second equality of Equation (27) is given by substituting(25) into (27) for f, and we can see that the last term on the RHS of Equation (25) vanishes.This means that the irrotational part of the body force does not directly contribute to the fluidflow but to the build-up of pressure. The third equality of Equation (27) is given by substitutingf=− �e∇� for f. Shown in Figure 15(c) is the distribution of the vorticity source h. Although themagnitude is highest at the leading edge, it is spread all over the electrode surface. This field nowdecreases in time after the peak at t∼=0.01 and tends to zero. This result clearly illustrates that inestimating the electro-osmotic flow, we must rely on the rotational part of the electric body-forceterm, which manifests itself through the curl operation.

The fundamental mechanism for the dc electro-osmotic flow can be understood with the help ofthe contour plots of the charge density �e and the potential �. Figure 16 shows superposition oftwo fields. In the initial acceleration stage, Figure 16(a), without significant charge developmentnear the electrode, the potential distribution shows the typical clustering of equi-potential linesnear the edge, making the vector ∇� more inclined to the LHS than in the deceleration stage. Onthe other hand, due to the stronger electric field near the edge, the cations are more attracted tothe edge than to the other area of the electrode. This results in the contour lines of �e in such waythat the direction of −∇�e tends to be inclined more to the RHS than in the deceleration stage.This then results in crossing of the two vectors −∇�e and ∇� as shown in Figure 16(a). Thevorticity source h is in turn positive and the flow must show counter-clockwise rotation on theLHS of the domain as shown in Figure 12. Even in the deceleration stage, cations still accumulatenear the electrode, but this results in the adjustment of the potential distribution. Eventually ∇�tends to be parallel to −∇�e or �e=�e(�), as shown in Figure 16(b), so that the vorticity source

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1267

x

y

0.95 1 1.05 1.10

0.05

0.1

0.15

0.2

0.4

0.3

3

1

2

x

y

0.95 1 1.05 1.10

0.05

0.1

0.15 0.6

0.5

0.3

0.4

64

2

(a)

(b)

Figure 16. Contour plots of the charge density �e (solid lines) and the potential � (dashedlines) near the LHS edge of the bottom electrode at (a) t=0.01 and (b) t=0.2 obtainedfor �0=0.2V from CMO with non-uniform grids. Arrows in (a) indicate the vectors −∇�e

(solid) and ∇� (dashed) at an arbitrary point.

tends to vanish. The final state, in which the Boltzmann distribution C± =Cref exp[±(�ref−�)]or �e=2Cref sinh[�ref−�] may hold, corresponds to this situation and both ion and fluid motionsare stopped; when �e and � do not follow the Boltzmann distribution but still satisfy �e=�e(�),then we can expect the ion transport but contributing only to the build-up of the pressure and notto the fluid motion.

5.3. Numerical results at a high external potential

When a higher external potential, �0=1V, is applied, unexpected results are obtained as shown inFigure 17. The average velocity first increases rapidly showing a first peak at t=0.0031 and thendecreases in the same way as the one for �0=0.2V. From t=0.017, however, it unexpectedlyincreases again rather slowly showing a second peak at t=0.039. Later on, it undergoes a typicaldeceleration process.

At the time when V starts to increase again, we can observe another pair of small vorticesnear the top electrode as shown in Figure 18(a). These vortices grow to size comparable to theoriginal vortex pair; see Figure 18(b) and (c). It turns out that those vortices contribute to thesecond increase of V shown in Figure 17. In the final deceleration stage, the streamline pattern ofFigure 18(c) is almost unchanged.

To investigate the cause for the generation of the top vortices, we plotted the contours of �eand � in Figure 19. This shows that in the upper left region the vector −∇�e is inclined towardthe LHS of the vector ∇� so that the vorticity source becomes negative, which should result in theclockwise rotational motion of fluid as shown in Figure 18(b) and (c). By comparing the contour

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1268 Y. K. SUH AND S. KANG

t

V

00

0.0005

0.001

0.0015

0.002

0.05 0.1 0.15 0.2 0.25

Figure 17. Time history of the average velocity V given at �0=1V from the method of CMO withnon-uniform grids along the x-direction.

x

y

0 0.5 1 1.5 2 2.50

0.5

1

x

y

0 0.5 1 1.5 2 2.50

0.5

1

x

y

0 0.5 1 1.5 2 2.50

0.5

1

(a)

(b)

(c)

Figure 18. Streamlines of the electro-osmotic flows obtained for �0=1V from the method of CMO withnon-uniform grids; given at (a) t=0.03; (b) t=0.05; and (c) t=0.2.

plots of �e and � for both the lower and the higher potential cases, we have found that, while theplots of � are not so different from each other, there is a significant difference in the plots of �e.Figure 20 shows contour plots of C± at two instants of time. Since the external potential �0=1Vis so high that the conduction effect is stronger than the diffusion effect, the central region whereions are depleted remains depleted for a long time and the diffusion from the regions near thelateral boundaries is relatively slow. We assume that such an unusual pattern in the distribution ofthe concentrations should result in the generation of a negative source of vorticity starting from the

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1269

x

y

0 0.5 1 1.5 2 2.5 30

0.5

1

0.6

0.8

0.7

0.4

-0.4

0

-0.2

0.2

Figure 19. Contour plots of the charge density �e (solid lines) and the potential � (dashed lines) at t=0.05obtained for �0=1V from CMO with non-uniform grids. Arrows indicate the vectors −∇�e (solid) and∇� (dashed) drawn at two points selected to represent the clockwise and the counterclockwise vorticalmotions, respectively. The contour lines of �e showing the magnitude larger than 1 near the electrodes

are not shown to avoid unnecessary complex view.

x

y

0 0.5 1 1.5 2 2.50

0.5

10.1

0.5

0.61

0.60.5

1

x

y

0 0.5 1 1.5 2 2.50

0.5

1

0.10.8

0.5 0.40.9

x

y

0 0.5 1 1.5 2 2.50

0.5

1

0.1

0.50.8

1

0.70.8

1

x

y

0 0.5 1 1.5 2 2.50

0.5

1

0.1 0.3

0.8

0.9

0.40.5

0.6

(a)

(b)

(c)

(d)

Figure 20. Contour plots of instantaneous field of (a) C+ at t=0.02; (b) C+ at t=0.06; (c) C− att=0.02; and (d) C− at t=0.06 obtained for �0=1V from CMO with non-uniform grids. Contours of

too-much-high values near the electrodes are not shown.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1270 Y. K. SUH AND S. KANG

top electrode surface. It seems, as a matter of fact, that the total amount of ions spread over thecavity is insufficient to achieve the complete fill-in of the EDL of the electrodes. Related to thisobservation, we note from Figure 17 that the magnitude of the average velocity is only one thirdof that calculated for a lower potential �0=0.2V, see Figure 13. Since the total amount of ionsis not sufficient, the continuous feed-in process stops and so does the fluid acceleration process.This observation can also explain the reason for the shorter period of the first acceleration stage at�0=1V than at �0=0.2V; measurement from Figures 13 and 17 indicates that the accelerationperiod of the former is less than one third of the latter.

5.4. Electro-osmotic flows caused by shut-off of the external potential

It is now understood that the transient charging process can produce a fluid flow. The transientcharging of course occurs right after a constant potential difference is applied across the electrodes,as has been the case up to now. On the other hand, we can also expect that the shut-off of theapplied potential may bring the reverse of the charging process, i.e. de-charging, which should bethe other kind of transient process. Therefore, it would be interesting to see if such de-chargingprocess can induce an electro-osmotic flow. We conducted simulation for the case when the appliedpotential �0=1V has been shut off at t=0.25. In this simulation, the shut-off process must bedone in a gradual way for numerical stability; in this study, we employed an exponential functionas the modulation of the external potential after t=0.25, i.e. �0=exp[−1000(t−0.25)]. The SORfactor of the linear solver for � must also be adjusted for the stability. In this study, we applied1.95 before the shut-off and abruptly decreased the magnitude down to 1.0 at t=0.25 and thenincreased it asymptotically up to 1.95 in the same way as �0 was decreased.

Figure 21 shows the time history of V . The magnitude of V is shown to increase abruptlyas soon as the potential is shut off, implying that the shut-off of the applied electricity indeedproduces a fluid flow. The average velocity reaches almost 70% of the second peak velocitymeasured at t=0.039. It also indicates that the decaying process after roughly t=0.35 is veryfast compared with that observed during the deceleration stage with the external potential beingapplied, i.e. during the time interval 0.1�t�0.25. Such difference can be explained in terms of theconduction effect. The conductive-flux term in Equation (24c), i.e. ±�∗

0C±∇�, indicates that the

conductive flux is proportional to the local concentration C±. When the external potential �0=1Vis applied, the bulk is devoid of ions during the deceleration stage, as shown in Figure 20, so thatthe ion transport toward the electrodes slows. On the other hand, when the electric potential isshut off, the accumulated ions near the electrodes return back to the bulk, causing an increase ofthe bulk concentration so that the conductive flux becomes increased and the bulk, in particular,is neutralized quickly.

t

V

00

0.0005

0.001

0.0015

0.002

0.1 0.2 0.3 0.4 0.5

Figure 21. Time history of the average velocity V given with �0=1V for the period 0�t�0.25 and with�0=0V for the period 0.25�t�0.5.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1271

x

y

0 0.5 1 1.5 2 2.50

0.5

1

x

y

0 0.5 1 1.5 2 2.50

0.5

1

-0.08

0.04

-0.06

0

-0.04

-0.08

0.4

-0.4

0-0.2

0.2

(a)

(b)

Figure 22. Numerical results of the electro-osmotic flow after the shut-off of the applied potential given att=0.27: (a) streamlines and (b) contour plots of �e (solid lines) and � (dashed lines). Lines with |�e|>1are not shown to avoid unnecessary complex view near the electrodes. Dash-dot lines in (b) divide the

LHS region into two depending on the sign of the vorticity source −(∇�e×∇�)·k.

We can confirm from the plot of the velocity field, Figure 22(a), that the direction of the primaryvortex flow is reversed right after t=0.25. The fundamental reason for such electro-osmotic flowcan be given in terms of the contour plots of �e and � as shown in Figure 22(b). The level of � issignificantly reduced but still non-zero, and the level lines of �e and � do not coincide with eachother, resulting in the non-zero values of h. Also, the LHS half region of the domain can be dividedinto two depending on the sign of h, as indicated by dash-dot lines in Figure 22(b); the region nearthe left-bottom corner is characterized by h>0, whereas the rest by h<0. This is why the primaryvortex is clockwise, whereas the smaller one near the left-bottom corner is counterclockwise, asshown in Figure 22(a). However, these vortices decay fast and disappear soon.

6. CONCLUSIONS

The main purpose of this study is to develop an efficient numerical scheme for solving incompress-ible flows at small Reynolds numbers for use in microfluidic applications such as electro-osmoticflows. General understanding in the CFD is that it is difficult and time-consuming to directly solvethe momentum and pressure (or continuity) equations for incompressible flows with fully coupledalgorithms [10, 20]. In this paper, however, we have shown numerically that a coupled scheme,with the use of SOR method, works better than the projection method in terms of both accuracyand CPU time.

To verify its superior performance, we applied our coupled schemes to a model flow whose exactsolution is available. Numerical tests were performed for various parameter sets and it was foundthat in every respect the coupled scheme in its original form, i.e. CMO, shows better performancethan other methods, i.e. CMV1, CMV2 and RK3. In employing the CMO method, however, wemust pay attention to setting the compatible pressure BC. On the other hand, the variant schemesCMV1 and CMV2 are more convenient in that we can apply the physical BC to the velocity andzero to the normal pressure gradient. As the solver for the momentum and pressure equations, SORis better than ICCG because the residual of the velocity equations is dependent on the solution ofthe pressure equation and vice versa.

We applied CMO to dc electro-osmotic flows. For this application, we employed coupledmethods separately to the fluid flow and the ion transport problems. For the fluid-flow equations,

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

1272 Y. K. SUH AND S. KANG

we simply followed the CMO process; the velocity components and the pressure are treated asunknowns simultaneously and the equations are solved iteratively by using the SOR method.Similarly for the ion transport equations, the ion concentrations and the potential are treatedunknown simultaneously and the equations are solved iteratively by using the SOR method. Weclarified, with the aid of numerical results, the fundamental mechanism of the electro-osmoticflows; that is, crossing of the level lines of the charge density and those of the potential plays therole of the vorticity source, which produces the fluid flow. When those two level lines are parallelto each other, they contribute only to the build-up of pressure (i.e. osmotic pressure). Numericalresults obtained at a low external potential (�0=0.2V) and those given at a high external potential(�0=1V) show strikingly different developments of the average velocity. In the former, a singlepeak is obtained conforming to the usual concept of transient dc electro-osmotic flows. The lattercase, however, reveals double peaks in the average velocity evolution. We have shown that thesecond peak is caused by a strong conduction effect compared with the diffusion effect at such ahigh external potential. In the deceleration stage, we intentionally shut off the external potential,and then, interestingly enough, there occurs a reverse flow with abruptly increased average velocityat the time of shut-off. While ions return back to the original neutral state, it happens that levellines of the charge density and those of the potential cross each other to make a non-zero vorticitysource, which in turn drives the observed reverse flow.

Understanding such mechanism for electro-osmotic flows may be useful in material handlingin microfluidic devices such as lab-on-chips. We plan to apply our code to a more practicalmicrofluidic application, i.e. mixing of fluid in the junction between the micro- and nano-channelsunder a dc electric field.

APPENDIX A

The original ICCG algorithm solving a symmetric linear system

Az=b (A1)

(all the variables used in this appendix have nothing to do with those in the text), the RHS beingfixed having the initial condition z=z0, can be given as follows [17]: We set from the beginningk=1, the scalar s0 at a large value and the vector u0=0. Get the initial residual q0=b−Az0.

(i) Solve the preconditioning matrix problem Mw=qk−1 for w, where D-ILU is used for thepreconditioning.

(ii) Get s1=qk−1 ·w and �=s1/s0.(iii) Obtain uk =w+�uk−1, w= Auk and �=s1/(uk ·w).(iv) Get zk =zk−1+�uk .(v) Get qk =qk−1−�w.(vi) Replace s0 with s1 and repeat (i)–(v) until the convergence is attained.

As mentioned in the text, in the coupled method, the RHS of Equation (A1) changes everysweep. Hence we cannot use the equation shown in the step (v) above for calculating the residualqk . In this study we use qk =bk−Azk .

We tested the above algorithm for a simple but non-linear problem

∇2�=−�2+(a sinhy sinx)2 (A2)

within a square cavity, 0�x�1, 0�y�1. BC is �=a sinhsinx at y=1 and �=0 at x=0,1 andy=1. Here the parameter a adjusts the system’s non-linearity. The exact solution of this problemis �=a sinhy sinx . We treat intentionally only the LHS of Equation (A2) unknown and updatethe RHS every sweep. After discretizing Equation (A2) by using the central difference scheme, weobtain Equation (A1). We used both the SOR and the modified ICCG methods to obtain convergedsolutions. First, we compared the performance of each solver for a linear equation

∇2�=0 (A3)

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 66:1248–1273DOI: 10.1002/fld

ALGORITHMS FOR SOLVING CREEPING FLOWS 1273

with the same BC as above so that the exact solution is unchanged. It was found, as expected, thatICCG is superior to SOR; as an example, for grids 101×101, SOR spent CPU time 1.1 s whereasICCG spent 0.72 s with the same accuracy, and for grids 201×201, CPU time is measured to be11.5 s for SOR and 4.9 s for ICCG; a laptop computer was used in this test. For the non-linearsystem, however, SOR is better than ICCG and its superior nature is more pronounced as the non-linearity increased. For instance, with a=1 no significant difference is observed between the twoas for the performance. For a=2, however, SOR becomes superior to ICCG; for grids 101×101,SOR’s CPU time is 1.61 s whereas ICCG’s is 33.4 s. With grids 201×201, CPU times of SORand ICCG are measured to be 26.6 and 138 s, respectively. The accuracy of SOR also turned outto be higher than ICCG by a factor of 17. Even for a=3, SOR gave a converged solution showingCPU time 4.8 s, but ICCG did not converge to a solution.

ACKNOWLEDGEMENTS

This work was supported by the National Research Foundation of Korea (NRF) through the NationalResearch Lab. Program funded by the Ministry of Education, Science and Technology (No. 2005-01091).This work was also supported by NRF grant No. 2009-0083510 through Multi-phenomena CFD Engi-neering Research Center.

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