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Computational Electrokinetics

Roman Zeyde

Technion - Computer Science Department - M.Sc. Thesis MSC-2013-12 - 2013

Computational Electrokinetics

Research Thesis

Submitted in partial fulfillment of the requirements

for the degree of Master of Science in Computer Science

Roman Zeyde

Submitted to the Senate of

the Technion — Israel Institute of Technology

Adar 5773 Haifa March 2013


The research thesis was done under the supervision of Prof. Irad Yavneh in the Computer Science Depart-

ment.

Acknowledgment

I would like to thank my adviser, Prof. Irad Yavneh for his outstanding guidance, helpful discussions and

valuable advice. I am deeply grateful to my family, and especially to my wife, Galit, for their love and

support.

The generous financial support of the Technion is gratefully acknowledged.


Contents

Abstract 1

Abbreviations and Notations 2

1 Introduction 3

1.1 The Physical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Electrokinetic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Separation of scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 Bulk-scale effective boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.5 Steady-state velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Summary of achievements in this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Numerical Scheme 10

2.1 Numerical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Flux Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Ghost Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.5 Richardson Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

iTechnion - Computer Science Department - M.Sc. Thesis MSC-2013-12 - 2013

2.2 Algorithm Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Operator Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.4 System Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Solver Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Operator Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Steady-state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Solver Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Operator Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Handling the Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.3 Iterative Newton solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Source code overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.2 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.4 Sample source code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Results and Discussion 29

3.1 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 First-Order (linear in β) Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.2 Second-Order (quadratic in β) Solution . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.3 Third-Order (cubic in β) Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Ion-exchanger results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.2 Electrophoresis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.3 Electrophoresis for large β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

iiTechnion - Computer Science Department - M.Sc. Thesis MSC-2013-12 - 2013

3.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A Effective Boundary Conditions 54

A.1 Debye layer scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.2 Ion Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.3 Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.4 Radial Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.5 Tangential Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

B Differential operators in spherical coordinates 58

B.1 Scalar Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

B.2 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 62

Abstract in Hebrew `

iiiTechnion - Computer Science Department - M.Sc. Thesis MSC-2013-12 - 2013

List of Figures

1.1 Schematic structure of the double layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Spherical coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Computational grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Variable class source code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Grid class source code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Operator interface source code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Constant operator source code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Pointwise function source code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Linear operator source code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.9 Join operator source code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Sample problem source code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Ion exchanger steady-state velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Ion exchanger streamlines – numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Ion exchanger streamlines – analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Quadratic convergence of steady-state velocity . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Highly-charged surface particle steady-state velocity . . . . . . . . . . . . . . . . . . . . . . 38

3.6 Highly-charged surface particle steady-state velocity . . . . . . . . . . . . . . . . . . . . . . 39

3.7 Large β results for U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.8 Large β results for U – difference from linear regime . . . . . . . . . . . . . . . . . . . . . . 41

3.9 Rate of convergence estimation for U at large β . . . . . . . . . . . . . . . . . . . . . . . . 41

3.10 β = 0.1 results for C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.11 β = 0.8 results for C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

ivTechnion - Computer Science Department - M.Sc. Thesis MSC-2013-12 - 2013

3.12 β = 5.0 results for C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.13 β = 0.1 results for Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.14 β = 0.8 results for Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.15 β = 5.0 results for Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.16 Streamfunction results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.17 Streamfunction results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.18 Streamfunction asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.19 Electric field boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.20 Ionic concentration boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.21 The product of C(r = 1, θ = 0)Er(r = 1, θ = 0)/β as a function of β. . . . . . . . . . . . . . 52

vTechnion - Computer Science Department - M.Sc. Thesis MSC-2013-12 - 2013

viTechnion - Computer Science Department - M.Sc. Thesis MSC-2013-12 - 2013

Abstract

In this work we study the electrokinetic migration of particles in an electrolyte solution due to the appli-

cation of an external electric field, and propose a numerical framework for the iterative solution of such

problems. The electrokinetic transport process can be used for transporting and manipulating micro-

and nanoscale objects in many nanotechnology applications, nano-fluidic devices, packed-bed separation,

desalination processes and various electrophoresis applications. Due to strong electrostatic forces, a thin

boundary layer forms near the particle surface. This results in scale disparity of the boundary layer, which

makes a full numerical solution challenging. We employ nonlinear macroscale effective boundary condi-

tions that have been derived using the specific chemical properties of the particle (for an ion-exchanger

particle by Yariv in [1] and for a surface charged inert particle by Schnitzer and Yariv in [2]). The resulting

macroscale nonlinear partial differential system is discretized and an iterative Newton solver is constructed

automatically from the discrete equations. Numerical results are obtained for an ion-exchanger and for a

surface charged inert particle. An asymptotic analytical solution is used for the validation of the solver.

The numerical results are compared to the asymptotic solutions, and good correspondence is achieved.

Finally, the new solver is applied in the regime of moderately large values of the external field, where no

analytical results are known. The numerical results uncover the strong-field scaling behavior, including a

new boundary layer at the front of the particle that scales exponentially with respect to the external field,

and also a well-defined dependence of the particle steady-state velocity on this field.

1Technion - Computer Science Department - M.Sc. Thesis MSC-2013-12 - 2013

Abbreviations and Notations

Φ — Electric potential

C — Ionic concentration

V — Fluid velocity

P — Fluid pressure

T — Stress tensor

F — Force vector

n — Surface normal

O — Operator

β — Dimensionless electric field magnitude

δ — Dimensionless Debye layer width

U — Dimensionless steady-state velocity


Chapter 1

Introduction

1.1 The Physical Problem

1.1.1 Electrokinetic Phenomena

Electrokinetic theory describes the dynamics of charged particles in ionic fluids [3, 4]. When a particle

acquires surface charge, a layer of ions of opposite charge is attracted to the surface via electric forces,

creating a double-layer structure around the particle (see Figure 1.1). This structure, called “Debye layer”,

electrically screens the surface charge, creating a potential difference between the particle and the outer

layer of the fluid bulk. In cases where the layer width is much smaller than the particle size, an analytical

asymptotic solution for the Debye layer dynamics can be derived [5].

The variables of the electrokinetic flow equations are the electrostatic potential ϕ, fluid velocity v and its

pressure p, and ionic concentration c. The boundary conditions are determined by the specific problem

under consideration and defined by the particle’s geometry, chemical characteristics, and the fluid dynamics.

The partial differential equations that describe the system dynamics under an external electric field, are

coupled and nonlinear, and in general have no analytic solution. Moreover, any numerical solver must

handle the scale disparity caused by the Debye layer width being much smaller than the particle size. An

effective boundary condition outside the Debye layer has been developed for spherical particles, yielding a

macroscale model in [1, 2]. The model is solved for weak electric field in an axisymmetric setting but it

is hard to extend this analytic solution to more general systems. Once the electric field becomes stronger,

significant nonlinear phenomena emerge. This interesting regime has not yet been explored extensively.

1.1.2 Applications

Recently, it has been conceived that such type of phenomena can be very useful in microfluidics devices and

methods for controlling and manipulating fluid flows in micro- and nanoscale. Many different components

are used in microfluidics such as valves, pumps, sensors, mixers, filters, separators, heaters etc., and are

combined into lab-on-a-chip systems.


(1) (2) (3)

Figure 1.1: Schematic structure of the double layer: (1) charged particle surface, (2) Debye layer, (3)fluid bulk. If the particle surface has positive (black) surface charge, it attracts negative (white) ions fromthe fluid making the Debye layer negatively charged (as opposed to the rest of the fluid bulk, which iselectrically neutral). The zeta potential is defined as the voltage drop across the Debye layer (2).

The manipulation can be achieved by either applying the external field at inlets and outlets, or it can be

applied locally in the microchannel by integrated components. A typical microchannel is produced with

lithographic techniques. Normally the Reynolds number is very low (on the order of unity or smaller) which

means that viscosity dominates and the convective term of the Navier-Stokes equations is insignificant. This

also means that microflows most often are laminar.

In the last decade there has been an increase in microfluidics research, due to the spread of tools for

microfluidic systems fabrication, production of cheap and portable devices for fast medical analysis, and

fundamental research of physical, chemical and biological processes. Various applications are [6, 7, 8, 9]:

1. DNA separation and sequencing [10].

2. Cell analysis and forensic identification [11, 12, 13].

3. Integrated microchemical systems [14].

4. Packed-bed separation [15, 16].

5. Desalination processes [17].

6. Drug delivery and pathogen detection [18].

Because the nonlinear macroscale model for these electrokinetic phenomena has no closed-form analytic

solution, a numerical solver for this system is of importance.


1.2 Governing Equations

1.2.1 Nondimensionalization

We follow the derivation presented by Yariv in [5]. Spatial coordinates r are normalized by a∗ (the

characteristic size of the particle). Positive and negative ionic concentrations are denoted by c+ and c−,

respectively, and are normalized by the ambient ionic concentration c∗. The ionic valences are ±z. The

dimensional ionic fluxes, j±, are given by the Nernst-Plank equation (using the Einstein relation for the

electric mobility, µ∗q = D∗/k∗BT∗, where D∗ is the ionic diffusivity):

j∗± = −D∗∇c∗± + v∗c∗± ∓ ze∗D∗

k∗BT∗ c

∗±∇ϕ∗, (1.1)

The electric potential ϕ is normalized by the thermal voltage: ϕ∗ = k∗BT∗/ze∗ = R∗T ∗/zF ∗, where

R∗ = k∗BNA, F∗ = e∗NA is the Faraday number and NA is the Avogadro constant.

The dimensional Stokes equations with electrostatic forces are given by:

∇ · v = 0, −∇p∗ + µ∗∇2v∗ + ǫ∗∇2ϕ∗∇ϕ∗ = 0. (1.2)

The velocity v is normalized by v∗ = ǫ∗(ϕ∗)2/a∗µ∗, and the pressure p is normalized by p∗ = µ∗v∗/a∗ =

ǫ∗(ϕ∗)2/(a∗)2. The fluxes j∗± are normalized by j∗ = D∗c∗/a∗.

The dimensional Poisson equation can be written as

(c∗+ − c∗−)ze∗ = −ǫ∗∇2ϕ∗, (1.3)

After nondimensionalization, the Nernst-Plank, Stokes and Poisson equations read:

j± = −∇c± + αvc± ∓ c±∇ϕ, (1.4)

∇ · v = 0, −∇p+∇2v +∇2ϕ∇ϕ = 0, (1.5)

c+ − c− = −2δ2∇2ϕ, (1.6)

where δ = δ∗/a∗ is the dimensionless Debye thickness (δ∗ =√

ǫ∗ϕ∗/2c∗e∗z) and α = v∗a∗/D∗ =

ǫ∗(ϕ∗)2/µ∗D∗ is the dimensionless Peclet number. The force on the particle is denoted by F normal-

ized by µ∗v∗a∗ = ǫ∗(ϕ∗)2. The stress tensor is denoted by T and normalized by p∗ = µ∗v∗/a∗. The electric

field is denoted by E and normalized by E∗ = ϕ∗/a∗. The surface charge is denoted by σ and normalized

by σ∗ = ǫ∗ϕ∗/δ∗.

The ionic fluxes are given by ion diffusion, electrostatic forces and ion advection by the fluid in (1.4). Due

to conservation of ion concentrations, the fluxes are divergence-free:

∇ · j± = 0.


These equations may be re-written by using salt c = (c+ + c−)/2 and charge q = (c+ − c−)/2. Thus, salt

and charge fluxes are defined by:

j =j++j

−

2 = −∇c− q∇ϕ+ αvc, i =j+−j

−

2 = −∇q − c∇ϕ+ αvq, (1.7)

and are divergence-free as well:

∇ · j = 0, ∇ · i = 0. (1.8)

In addition, the charge concentration is governed by the Poisson equation (1.6)

q = −δ2∇2ϕ. (1.9)

1.2.2 Boundary conditions

We assume that a constant electric field of nondimensional magnitude β, is applied to the system, in the

direction denoted by the unit vector ı. As a result, the particle will equilibrate at a non-zero steady-state

velocity U ı, which is a function of the applied field, and is not known a priori.

We employ the reference frame of the particle. Far away from the particle, we have a uniform applied

electrostatic field, uniform flow, and ambient ionic concentrations:

v → −U ı, ∇ϕ→ −βı, c± → 1. (1.10)

We consider a spherical particle. The chemical properties of the particle’s surface S (defined at r = 1 with

normal n = r), determine the boundary conditions on S. We consider the following scenarios:

Ion-exchanger

We follow the microscale model presented at [1] and assume high conductance of the particle (yielding a

uniform electric potential), zero slip, anion impermeability, cation selectivity (with fast Butler-Volmer[19]

kinetics n · j+ = k(1 − c+/γ), where k → ∞):

ϕ = V , v = 0, n · j− = 0, c+ = γ, (1.11)

where γ is the ratio between the cationic concentration on the surface and the cationic concentration in

the fluid bulk and V is an unknown constant electric potential at the ion-exchanger surface.

Inert particle

We follow the microscale model presented at [2] and assume constant surface charge zero slip and ion

impermeability:

n · (∇ϕ− χ∇ϕs) = δ−1σ, v = 0, n · j± = 0, (1.12)

where χ is the ratio between the dielectric constants of the particle and the fluid.


1.2.3 Separation of scales

When δ ≪ 1, the fluid remains electroneutral (q ≈ 0, thus c+ ≈ c−), except inside a thin boundary layer

of width O(δ) surrounding the particle (the “Debye layer”). This boundary layer makes it very hard to

construct a direct numerical solver for the full electrokinetic problem. However, the scale disparity at the

Debye layer can be exploited (as shown by Yariv in [5]), which allows the PDE system to be rewritten in

the “inner” region as an ODE system using the asymptotic limit of δ → 0. This system can be integrated

and solved analytically, yielding “effective” boundary conditions near the particle surface for the “outer”

bulk region equations.

Bulk-scale equations

The Nernst-Planck equations above (1.7) for the fluid bulk outside the boundary layer [5], using bulk

variables denoted by uppercase letters, read:

Q = 0, C = C+ = C−, J = −∇C + αV C, I = −C∇Φ. (1.13)

Because the flow is incompressible (∇ · V = 0), ion conservation results in the following salt and charge

conservation equations – the salt and charge fluxes above (1.13) are divergence-free (1.8):

∇2C − αV ·∇C = 0, ∇ · (C∇Φ) = 0. (1.14)

The macroscale Stokes flow equations have the same form as before (1.5):

∇ · V = 0, ∇2V −∇P +∇Φ∇2Φ = 0. (1.15)

1.2.4 Bulk-scale effective boundary conditions

By integrating the equations across the Debye layer, the effective boundary conditions at the particle

surface are:

Ion-exchanger

See [1] for full derivation of the nonlinear effective boundary conditions (and also Appendix A):

0 = Φ+ logC,

0 =∂

∂n(Φ− logC) , (1.16)

V = ζ ·∇SΦ+ 2 log

(

1− tanh2ζ

4

)

·∇S logC,

where ζ = V − Φ = log(C/γ) is the zeta potential.


Highly charged inert particle

See [2] for full derivation of the nonlinear effective boundary conditions. Note that due to surface conduc-

tion, the resulting boundary conditions are:

0 =∂C

∂n−Du∇2

s (Φ− logC),

0 =∂C

∂n+ C

∂Φ

∂n, (1.17)

V = ζ ·∇S (Φ− logC) + 4 log 2 ·∇S logC,

where ζ = ζ − logC is the zeta potential and ζ = 2 log σ.

1.2.5 Steady-state velocity

The stress tensor in the fluid is composed of Newtonian and Maxwel stresses:

T = ∇V + (∇V )† − P I+∇Φ∇Φ− 1

2(∇Φ ·∇Φ). (1.18)

In steady state (constant particle drift velocity U ), the total force acting on the particle (computed by

the following surface integral) must vanish:

F (β,U ) =

∮

ST · ndA = 0. (1.19)

The total force F (β,U ) is a function of the applied electric field β and the velocity U . For any given

electric field β, the force-free constraint above can be solved numerically for U to yield the corresponding

steady-state velocity.

1.3 Summary of achievements in this work

In this work, we develop and implement a novel software framework, that enables the generation of iterative

numerical solvers for nonlinear macroscale electrokinetic problems. A novel feature is the combination of

a numerical framework with an analytical solution of the thin Debye layer, that is available from the

literature and provides effective boundary conditions. Therefore, the Debye layer does not need to be

resolved by the computational grid, saving a tremendous amount of numerical resources.

We apply this framework to the study of systems that have no closed-form solutions, such as ion-exchanger

migration [1] and the electrophoresis problem [2] for a spherical particle in an infinite domain. The

numerical results can be used to gain insight into the chemical and physical behavior in far more general

regimes than are currently well-understood, and also to possibly lead to further analytical developments,

based on new scaling behavior that is discovered numerically. We have also discovered interesting physical

phenomena that may be observed experimentally. One such phenomenon is nonlinear electrokinetic vortex

flow, which is useful for microfluidic mixing applications [20, 21].


In addition, we develop and present an asymptotic derivation for the ion-exchanger migration problem up

to third order terms. This new analytical result is also used to verify the numerical framework and the

numerical results.


Chapter 2

Numerical Scheme

2.1 Numerical Framework

We start by presenting the numerical framework that is used for the construction of the nonlinear solver.

2.1.1 Finite Volume Method

Finite Volume methods [22] are used to define a discretization of the conservation equations’ integral forms.

The simulation domain is divided into a set of discrete non-overlapping control volumes. Each conservation

equation is discretized over the control volumes, such that the integral form of the conservation law is

satisfied using an approximation of the fluxes over the control volume faces. This method ensures that the

discretization is conservative in the discrete sense.

2.1.2 Flux Discretization

In order to define the fluxes on the boundaries of each control volume, we employ the conservative for-

mulation of the governing equations above. The flux terms are defined using finite differences and linear

interpolation of the discrete variables (defined at the centroid of each cell), except for the advection flux

which should not be discretized using linear interpolation – due to a numerical instability for large cell

Peclet numbers [23].

2.1.3 Ghost Points

In order to discretize boundary condtions for Φ, C and V , “ghost” points are employed. The grid is

extended to include points outside the domain interior, and the variables at the “ghost” points are set

to satisfy the discretized boundary conditions on the grid boundaries. Note that P has no boundary

conditions – so the pressure variable is defined only in the domain interior.


2.1.4 Newton’s Method

Newton’s method is used to solve a nonlinear system of equations of the form O(x) = 0. For small step

∆x, we can linearize the operator by

O(x+∆x) ≈ O(x) +∇O(x)∆x (2.1)

Thus, we use the following iteration, starting from an initial solution x0:

∆xn = − [∇O(xn)]−1 · O(xn) (2.2)

xn+1 = xn +∆xn (2.3)

Then, a sparse linear system A∆x = b needs to be solved to yield the update ∆x. When the method

converges, xn → x∞, the asymptotic convergence is in general quadratic:

‖xn+1 − x∞‖ ≤ k‖xn − x∞‖2. (2.4)

Thus, when the initial guess is close enough to the solution, the solver will typically converge in very few

steps.

2.1.5 Richardson Extrapolation

In order to achieve higher numerical accuracy, we employ Richardson extrapolation [24] on the resulting

steady-state velocity U (h), as a scalar function of the grid spacing h. This method enables us to detect

and analyze nonlinear effects for β ≪ 1, where a higher-order discretization is required because otherwise

the numerical error becomes comparable to the difference between the linear analytical regime and the

numerical results.

2.2 Algorithm Description

The numerical solver first requires constructing a discrete approximation for the nonlinear PDE system,

for a given electric field β and assumed velocity U . The resulting nonlinear discrete system is solved

numerically using Newton’s Method, so the total force on the particle can be computed. We seek the

steady-state velocity U , which corresponds to zero total force, F (β,U ) = 0, which we solve by the

standard secant method.

2.2.1 Spherical Coordinates

Because the system is axisymmetric and the particle is a sphere, we employ spherical coordinates (r, θ, φ),

as shown in Figure 2.1.


The scalar gradient and the divergence can be written as:

∇f =∂f

∂rr +

1

r

∂f

∂θθ, (2.5)

∇ · F =1

r2∂

∂r

(

Frr2)

+1

r sin θ

∂

∂θ(Fθ sin θ) . (2.6)

The scalar Laplacian can be written as:

∇2f = ∇ · (∇f) =1

r2∂

∂r

(

r2∂f

∂r

)

+1

r2 sin θ

∂

∂θ

(

sin θ∂f

∂θ

)

. (2.7)

The vector gradient and vector Laplacian can be written as:

∇F =∂Fr∂r

rr +∂Fθ∂r

rθ +1

r

(

∂Fr∂θ

− Fθ

)

θr +1

r

(

∂Fθ∂θ

+ Fr

)

θθ,

r ·∇2F = ∇2Fr −2Frr2

− 2

r2 sin θ

∂ (Fθ sin θ)

∂θ(2.8)

θ ·∇2F = ∇2Fθ −Fθ

r2 sin2 θ+

2

r2∂Fr∂θ

.

Note that in the spherical coordinate system, the vector Laplacian operator cannot be computed in a de-

coupled manner as in the Cartesian system, because the unit vectors r, θ, φ are functions of the coordinates

in a curvilinear coordinate system. See Appendix B for the full derivation of the identities above.

2.2.2 Computational Grid

A regular grid of size nr × nθ is defined, with:

(ri, θj) ∈ [1,∞)× [0, π],

ri = (1 + ∆r)i, (2.9)

θj = ∆θ · j,

where a logarithmic grid spacing has been chosen for r and a uniform grid for θ. Note that r0 = 1 and

Rmax = (1 + ∆r)nr , where ∆r = (Rmax)

1/nr − 1 and ∆θ = π/nθ.

Because the grid is finite, the choice of Rmax must be large enough so as to have a negligible effect on the

solution. On the other hand, in order to have a high resolution grid near the particle surface, ∆r should

be minimized. It should be noted that for a given grid size nr, these requirements cannot be satisfied

simultaneously – thus, a compromise is required between large Rmax and small ∆r.

This grid induces a disjoint subdivision of the domain Ω = [1, Rmax]× [0, π] =⋃

ij Ωij into cells. A specific

cell Ωij and its center (ri, θj) are defined by (2.9):

Ωij = [ri−1, ri]× [θj−1, θj ],

ri = (ri−1 + ri)/2, (2.10)

θj = (θj−1 + θj)/2.


Each discrete variable is located with respect to its cell as shown at Figure 2.2. Φ, C and P are represented

by their value at the center of each cell, using an all-centered grid, whereas V is represented by its values

at cell boundaries, using a staggered grid:

Φh[i,j] = Φ(ri, θj),

Ch[i,j] = C(ri, θj),

P h[i,j] = P (ri, θj), (2.11)

V hr [i,j] = Vr(ri, θj),

V hθ [i,j] = Vθ(ri, θj).

2.2.3 Operator Discretization

A finite-volume method with linear interpolation is used for flux discretization. Define the following discrete

central difference operators:

Dr(fh)[i+ 1

2,j] =

fh[i+1,j] − fh[i,j]

r[i+1,j] − r[i,j], (2.12)

Dθ(fh)[i,j+ 1

2 ]=fh[i,j+1] − fh[i,j]

θ[i,j+1] − θ[i,j]. (2.13)

Define the following linear interpolation operators:

Ir(fh)[i+ 1

2,j] =

r[i+ 1

2,j] − r[i,j]

r[i+1,j] − r[i,j]fh[i+1,j] +

r[i+1,j] − r[i+ 1

2,j]

r[i+1,j] − r[i,j]fh[i,j], (2.14)

Iθ(fh)[i,j+ 1

2 ]=r[i,j+ 1

2 ]− r[i,j]

r[i,j+1] − r[i,j]fh[i,j+1] +

r[i,j+1] − r[i,j+ 1

2 ]

r[i,j+1] − r[i,j]fh[i,j]. (2.15)

If the total flux of a cell is equal to zero (∇ · f = 0), then using (2.12), we have

1

r2Dr

(

fhr r2)

+1

r sin θDθ

(

fhθ sin θ)

= 0. (2.16)


x

r

θ

φ

ı

Figure 2.1: Spherical coordinate system is employed for computational grid construction.

Φ, C, P

Φ, C, P

Φ, C, P

Φ, C, P

Φ, C, P

Vr

Vr

Vr

Vr

Vr

Vr

VrVr

Vr

Vr

Vr

Vr

VθVθVθ

Vθ

Vθ

Vθ

Vθ

Vθ

Vθ

Vθ

Vθ

VθΦ, C, P

Φ, C, P

Φ, C, P

Φ, C, P

Figure 2.2: The computational grid is defined using a spherical coordinate system, due to axisymmetry ofthe physical problem and boundary conditions discretization on the particle surface.


2.2.4 System Equations

Charge and salt fluxes (1.13) are discretized on grid cell boundaries (using an upwind scheme U for

numerical stability at large cell Peclet number) using (2.12) and (2.14):

Ihr = −Ir(Ch) · Dr(Φh),

Ihθ = −Iθ(Ch) ·Dθ(Φ

h)

r,

Jhr = −Dr(Ch) + αV h

r · UV h

r (Ch),

Jhθ = −Dθ(Ch)

r+ αV h

θ · UV h

θ (Ch), (2.17)

UV h

r (Ch)[i, j] = Ch[

i− sign(V hr )

2, j

]

,

UV h

θ (Ch)[i, j] = Ch[

i, j − sign(V hθ )

2

]

.

Mass flux V is discretized on grid cells boundaries, using the staggered velocity grid. Force components

(F = −∇P + ∇2V + ∇Φ∇2Φ) are discretized on the staggered velocity grid using the finite difference

method, where linear interpolation is used for the Coulomb force and for vector Laplacian components.

F hr = −Dr(Ph) + L(V h

r )−2

r2V hr − 2

r2 sin θIr(Dθ(V

hθ sin θ)) +Dr(Φ

h) · Ir(L(Φh)),

F hθ = −Dθ(Ph)

r+ L(V h

θ )−F hθ

r2 sin2 θ+

2

r2Iθ(Dθ(F

hr )) +

Dθ(Φh)

r· Iθ(L(Φh)),

L(fh) = 1

r2Dr

(

Dr(fh)r2

)

+1

r2 sin θDθ

(

Dθ(fh) · sin θ

)

. (2.18)

2.2.5 Boundary Conditions

Far away from the particle boundary (1.10), we have:

Dr

(

Φh)

= −β cos θ, Ir(

Ch)

= 1 V hr = −U cos θ, V h

θ = U sin θ. (2.19)

For θ = 0 and θ = π, the “ghost” points are defined by axial symmetry considerations:

Dθ

(

Φh)

= 0, Dθ

(

Ch)

= 0, Dθ

(

V hr

)

= 0, V hθ = 0 (2.20)

The specific boundary conditions at r = 1 depend on the chemical properties of particle surface.


Ion-exchanger

The boundary conditions (1.16) are implemented by:

0 = Ir(Φh + logCh),

0 = Dr(Φh − logCh),

V hr = 0,

V hθ = 4 log

(

12

(

1 + exp

Iθ(ζh)/2))

· Dθ(ζh),

ζh = − log γ − Ir(Φh).

(2.21)

Highly charged inert particle

The boundary conditions (1.17) are implemented by:

0 = Dr(Ch)−Du ·

(

Dθ

(

sin θ · DθIr(

Φh − logC)))

/sin θ,

0 = Dr(Ch) + Ir(Ch)Dr(Φ

h),

V hr = 0,

V hθ = Iθ(ζh) · Dθ(IrΦh − log IrCh) + 4 log 2 · Dθ(log IrCh),ζh = ζ − log IrC.

(2.22)

2.3 Solver Design

Let O : RN → RN be a nonlinear operator. In order to solve the equation O(x) = 0, Newton’s method is

applied, given an initial solution x0. Each step requires the computation of the residual vector, r = O(x),

the gradient matrix of the operator, G = ∇O(x) (given the current solution x) and the solution of the

sparse linear system G∆x = −r.

2.3.1 Operator Representation

The system variable x is defined as the concatenation x = [Φ,C, Vr , Vθ, P ]. Thus, each problem variable

can be computed by applying an appropriate projection operator on x. The system can be written as a

nonlinear operator O(x) = 0 acting on the system variable x, where O is the system equations operator,

derived by concatenation of the system equations (described in Subsection 2.2.4) and the boundary con-

ditions operators (described in Subsection 2.2.5). This way, the same numerical solver handles both the

nonlinear system equations and nonlinear boundary conditions.

2.3.2 Steady-state

Let β be a given nondimensional electric field. Assume that U ı is the drift velocity of the particle (given

that the fluid is at rest far away from it). After the convergence of the solver, once the residual norm

is below a prescribed threshold ‖O(xn)‖ < ǫ, the total force acting on the particle can be computed by

integrating T · n over the particle surface.


Due to symmetry considerations, the total force F is aligned with ı and it vanishes iff ı · F = 0.

Fı = ı · F =

∮

S(ı · T · n) dA =

∫ π

0fı(θ)2π sin θdθ,

fı = ı · T · r =

(

−P + 2Dr(Vr) +1

2(Dr(Φ))

2 − 1

2r2(Dθ(Φ))

2

)

cos θ (2.23)

−(

Dr(Vθ)−Vθr

+1

rDr(Φ)Dθ(Φ)

)

sin θ.

The integral above is approximated by 1D numerical quadrature (using the mid-point rule)

F hı (β,Uh) =

nθ∑

i=1

fh(θi) · 2π sin θi ·∆θi, (2.24)

to yield the total force F hı (β,Uh) as a function of the drift velocity U h. Since the goal is to find the steady-

state solution, F hı is required to be zero – and the appropriate U h is found by a simple 1D root-finding

algorithm, applied as an outer loop:

F hı (β,Uh(β)) = 0 (2.25)

2.3.3 Continuation

The iterative solver above can in principle be used to compute the steady-state solution for any given β.

For β ≪ 1, the linear terms are the dominant ones, so the solution is approximately linear in β (as derived

in [1]), and the solver convergence is fast. However, this is no longer true for β & 1, because the nonlinear

terms become dominant and the iterative solver may not converge at all if we start with an arbitrary initial

guess. In order to find a solution for such β, a continuation method is used: the solver is applied to a

sequence of βini=0 such that β0 = 0, βn = β and the solution xi for βi is used as the solver initializer for

the problem of βi+1.

2.4 Solver Implementation

Object-Oriented Design methodology is used for implementation. The MATLAB programming language

is chosen due to strong numerical capabilities and high-level language features. The source code of the

solver is provided in [25].

MATLAB direct solver is applied to solve the sparse linear system for each Newton step, using the backslash

operator. The library used for solving the linear system is UMFPACK 5.0 [26] (Unsymmetric MultiFrontal

Sparse LU Factorization Package), as reported by MATLAB function call, spparms(’spumoni’, 2).

2.4.1 Operator Interface

The base Operator interface is defined by supporting the computation of a residual vector Operator.res()

and a gradient matrix Operator.grad(), given an input vector x. This interface is implemented by specific


operator classes, which are used to construct the system operator O, so that the Newton solver (that

requires computing the residual and the gradient) can be applied automatically. Each such operator may

have other operators as its inputs, so the residual and the gradient are computed recursively, using the

chain rule (applied to operators).

Linear operator

If O is linear, it can be represented by a matrix L, such that:

O(x) = Lx, ∇O = L. (2.26)

Note that finite difference D, interpolation I and upwind selection U operators can be implemented as

linear sparse operators, by constructing an appropriate sparse matrix L, having O(dimx) non-zeroes. This

way, the computation of the residual is very fast, taking O(dimx) time, and the gradient does not depend

on x.

Pointwise scalar function

Let f : R → R be a differentiable 1D function, whose derivative is denoted by f ′ : R → R. The operator

F can be defined to represent a pointwise application of f :

F(x) = [f(x1); . . . ; f(xn)], ∇F(x) = diagf ′(x1), . . . , f ′(xn). (2.27)

f is defined and differentiated automatically using the MATLAB symbolic toolbox.

Constant value

A constant operator C is defined by the constant vector c (which does not depend on x):

C(x) = c, ∇C = 0. (2.28)

Binary operators

Let O1 and O2 be two operators, so that their pointwise addition, subtraction, and multiplication, are

defined as follows:

(O1 ±O2)(x) = O1(x)±O2(x), (2.29)

∇(O1 ±O2)(x) = ∇O1(x)±∇O2(x),

(O1 · O2)(x) = O1(x) · O2(x), (2.30)

∇(O1 · O2)(x) = diag(O2(x))∇O1(x) + diag(O1(x))∇O2(x).


N-ary operators

Given N operators OiNi=1, their concatenation O is defined as:

O(x) = [O1(x); . . . ;ON (x)], (2.31)

(∇O)(x) = [(∇O1)(x); . . . ; (∇ON )(x)].

2.4.2 Handling the Singularity

The electric potential Φ (in electrophoresis case) and the pressure P (in both cases) are defined up to an

additive constant, since the system equations and the boundary conditions contain only their derivatives.

In order to overcome the singularity of the linear system G∆x = −r, we set the variable at a specific grid

point to be 0. In order to retain a square system, we remove one conservation equation at this grid point

too. In the case of pressure P , we remove one mass conservation equation, ∇ · V = 0, at this grid point.

In the case of electric potential, we remove the charge conservation equation, ∇ · (C∇Φ) = 0, at this grid

point. For r = O(x) and G = ∇O(x), the new square linear system reads:

RGP · δx = −Rr, (2.32)

where P matrix is used to remove the columns and R matrix is used to remove the rows of G, where the

update step is ∆x = P · δx.

2.4.3 Iterative Newton solver

Given an initial solution x0 and prescribed values for β and U , we apply the following algorithm until

convergence (‖rn‖ < ǫ):

1. Update the system operator: op = System(xn)

2. Compute the gradient: Gn = op.grad()

3. Compute the right-hand side: rn = op.res()

4. Solve sparse linear system: RGnP∆xn = −Rrn

5. Update solution: xn+1 = xn + P∆xn

2.5 Source code overview

In the following section we present the source code of the numerical framework and demonstrate its usage

for solving a boundary-value problem.


2.5.1 Variable

We define the variable vector x as Variable operator (Figure 2.3).

1 classdef Variable < handle

2 properties

3 value;

4 end

5 methods

6 function self = Variable(varargin)

7 self.value = col(varargin:);

8 end

9 function r = res(self)

10 r = self.value(:);

11 end

12 function G = grad(self)

13 G = speye(numel(self.value));

14 end

15 function update(self, dv)

16 self.value = self.value + dv;

17 end

18 end

Figure 2.3: Variable class source code.

Then the initial solution for the solver is set using x = Variable(Φ,C, Vr, Vt, P ) from the initial conditions

of the solver.

2.5.2 Grid

Most of the operators are using a Grid object (Figure 2.4), enabling validation of operators compatibility

and convenient interpolation between grids. Note that each operator must define its res() and grad()

functions, for residual and gradient computation, respectively. An Operator interface is used (Figure 2.5)

to support operator overloading MATLAB syntax.

Thus, various computations can be applied on simple operators, creating more complex operators, such as

op4 = op1 * exp(op2 + tanh(op3)). Since each operator can compute its residual and gradient, we can

compute automatically the residual and gradient of op4 from these of op1, op2, op3 using the derivation

rules described at 2.4.1.

2.5.3 Operators

An operator can be created with n existing operators as input. Const is an operator that corresponds to

n = 0, whose residual value is constant and gradient matrix is zero (see Figure 2.6).


1 % Regular structured grid class

2 classdef Grid < handle

3 properties

4 x, y;

5 X, Y;

6 size;

7 numel;

8 end

9 methods

10 function self = Grid(x, y)

11 self.x = x(:);

12 self.y = y(:);

13 [self.X, self.Y] = ndgrid(self.x, self.y);

14 self.size = [numel(self.x), numel(self.y)];

15 self.numel = prod(self.size);

16 end

17 end

Figure 2.4: Grid class source code.

Unary operators

Unary operators have one existing operator as input. Func operator implements point-wise analytic func-

tion application (whose derivative is computed using the symbolic differentiation toolbox) on the same grid

of the input operator (Figure 2.7). Linear operator is implemented using multiplication by a constant

sparse matrix L, which is also the gradient of the operator (Figure 2.8). Deriv and Interp operators

(implementing discrete differentiation and grid interpolation) derive from Linear base class operator and

specify L sparse matrix accordingly.

Binary operators

Binary operators such as addition, subtraction and point-wise multiplication are implemented as operators

whose output grid is the same as the grid of their inputs. The implementation is used by operator

overloading in the base class Operator.

N-ary operators

All equations and boundary condition operators are defined as operators op1, op2, ... opN. Then,

they are combined using the N -ary operator op = Join(op1, op2, ... opN), creating the system oper-

ator op (Figure 2.9).


2.5.4 Sample source code

The following example shows how to define the following boundary-value problem:

∇2F (x, y) = 1 (x, y) ∈ [0, 1]2 (2.33)

F (0, y) = F (1, y) = 0 y ∈ [0, 1] (2.34)

∂F

∂y(x, 0) =

∂F

∂y(x, 1) = 0 x ∈ [0, 1] (2.35)

and solve it using one Newton method step. The source code is presented at Figure 2.10.


1 classdef Operator < handle

2 properties

3 grid; % a grid object for the output

4 end

5 methods

6 function op = Operator(grid)

7 op.grid = grid;

8 end

9 function op = plus(op1, op2)

10 op = Sum(op1, op2);

11 end

12 function op = minus(op1, op2)

13 op = Minus(op1, op2);

14 end

15 function op = uminus(op)

16 op = Product(-1, op);

17 end

18 function op = mtimes(op1, op2)

19 op = Product(op1, op2);

20 end

21 function op = log(op)

22 op = Func(op, ’log(x)’);

23 end

24 function op = exp(op)

25 op = Func(op, ’exp(x)’);

26 end

27 function op = sinh(op)

28 op = Func(op, ’sinh(x)’);

29 end

30 function op = cosh(op)

31 op = Func(op, ’cosh(x)’);

32 end

33 function op = tanh(op)

34 op = Func(op, ’tanh(x)’);

35 end

36 end

Figure 2.5: Operator interface source code.


1 classdef Const < Operator

2 properties

3 val;

4 end

5 methods

6 function self = Const(grid, val)

7 self = self@Operator(grid);

8 if isa(val, ’float’)

9 if numel(val) == 1

10 val = repmat(val, grid.size);

11 end

12 self.val = val;

13 else % function handle evaluated point-wise

14 if isa(val, ’char’)

15 val = str2func([’@(r,t) ’ val]);

16 end

17 self.val = arrayfun(val, grid.R, grid.T);

18 end

19 end


21 r = self.val(:);

22 end


24 G = sparse(0); % [0]

25 end

26 end

Figure 2.6: Constant operator source code.


1 classdef Func < Operator

2 properties

3 op; % Input operator

4 func; % y = f(x)

5 deriv; % dy/dx = f’(x)

6 end

7 methods

8 function self = Func(op, func)

9 grid = op.grid; % Reuse operator’s grid


11 self.op = op;

12 if isa(func, ’char’)

13 func = sym(func);

14 end

15 self.func = matlabFunction(func);

16 self.deriv = matlabFunction(diff(func));

17 end


19 r = self.func( self.op.res() );

20 end


22 r = self.op.res();

23 G = spdiag( self.deriv(r) ) * self.op.grad();

24 end

25 end

Figure 2.7: Pointwise function source code.


1 classdef Linear < Operator

2 properties

3 L;

4 op;

5 end

6 methods

7 function self = Linear(grid, op, L)


9 self.op = op;

10 self.L = L;

11 end


13 r = self.L * self.op.res();

14 end


16 G = self.L * self.op.grad();

17 end

18 end

Figure 2.8: Linear operator source code.


1 classdef Join < handle

2 properties

3 ops;

4 end

5 methods

6 function self = Join(operators)

7 self.ops = operators;

8 for k = 1:numel(self.ops)

9 op = self.opsk;

10 end

11 end


13 r = cell(numel(self.ops), 1);


15 rk = self.opsk.res();

16 end

17 r = cat(1, r:);

18 end


20 G = cell(numel(self.ops), 1);


22 Gk = self.opsk.grad();

23 end

24 G = cat(1, G:);

25 end

26 end

Figure 2.9: Join operator source code.


1 x = linspace(0, 1);

2 y = linspace(0, 1);

3 g = Grid(x, y); % Grid definition

4 x_ = (x(1:end-1) + x(2:end)) * 0.5;

5 y_ = (y(1:end-1) + y(2:end)) * 0.5;

6 xi = x(2:end-1);

7 yi = y(2:end-1);

8 v = Variable(zeros(g.size));

9 F = Linear(g, v, speye(g.numel)); % Variable Definition

10 Fx = Deriv(Grid(x_, y), F, 1);

11 Fy = Deriv(Grid(x, y_), F, 2);

12 Fxx = Deriv(Grid(xi, yi), Fx, 1);

13 Fyy = Deriv(Grid(xi, yi), Fy, 2);

14 eq = (Fxx + Fyy) - Const(1); % Laplacian(F) = 1

15 bnd = Interp(Grid(0.0, y(2:end-1)), F); ... % F(0, y) = 0

16 Interp(Grid(1.0, y(2:end-1)), F); ... % F(1, y) = 0

17 Interp(Grid(x, 0.0), Fy); ... % dF/dy(x, 0) = 0

18 Interp(Grid(x, 1.0), Fy) % dF/dy(x, 1) = 0

19 op = Join([eq; bnd]); % System operator

20 r = op.res()

21 G = op.grad()

22 v.update(-(G\r)) % Solve by Newton iteration

Figure 2.10: Sample problem source code.


Chapter 3

Results and Discussion

3.1 Asymptotic Analysis

In order to validate the numerical results, an asymptotic analysis has been applied to the nonlinear systems

of ion-exchanger migration and highly charged particle electrophoresis. We present here the asymptotic

analysis for the ion-exchanger case [1], while the asymptotic analysis for the electrophoresis case [2] is

derived by Schnitzer and Yariv in [27].

The derivation is performed as follows. The nonlinear system (composed of the governing equations and

the boundary conditions) is written as O(x;β) = 0. For small β, the solution to the nonlinear system can

be expanded in an asymptotic series in β:

x = x(β) ≈∑

n

xnβn. (3.1)

The nonlinear terms are expanded around β = 0:

0 = O(x) =∑

i

Oi(x0, . . .xi)βi. (3.2)

Thus, the O(βk) term xk can be found recursively by solving Ok(x0,x1, . . . ,xk) = 0, given the previous

solutions for x0, . . . ,xk−1. The derivation is validated using the MATLAB symbolic toolbox by the code

at symbolic/asymp.m [25].

3.1.1 First-Order (linear in β) Solution

For β = 0 (no electric field is applied), the steady-state solution is U = 0:

Φ0(r, θ) = 0, C0(r, θ) = 1, V 0(r, θ) = 0, P0(r, θ) = 0. (3.3)


Assuming β ≪ 1, the linearized equations and boundary conditions are

∇2Φ1 = 0, ∇2C1 = 0, ∇2V 1 −∇P1 = 0. (3.4)

The first-order terms x1 are given by (see [1] for full derivation)

Φ1 =(

14r2

− r)

cos θ, C1 =3

4r2cos θ, Ψ1 = U1

(

1r − r2

)

sin2 θ2 , P1 = 0, (3.5)

and the linear velocity term is:

V 1 = U1

(

r

(

−1 +1

r3

)

cos θ + θ

(

1 +1

2r3

)

sin θ

)

(3.6)

U1(γ) = 2 log

(

1 + γ−1

2

2

)

. (3.7)

For γ > 1, the Debye layer has a positive charge, corresponding to a negative particle charge – yielding

negative drift velocity, U < 0. For γ < 1, we have U > 0.

3.1.2 Second-Order (quadratic in β) Solution

The quadratic terms x2 are computed by solving a linear PDE system with a right-hand side determined

by the linear terms x1, yielding

Φ2 =

(

U1 α

32− 1

16

)

3 cos2 θ − 1

r3− 3 + 3 sin2 θ

(

4 r3 − 1)

32 r4− 3U1 α− 6

32 r, (3.8)

C2 =

(

5U1 α

32+

1

16

)

3 cos2 θ − 1

r3− 3U1 α− 6

32 r+

3U1 α(

2 r3 sin2 θ + sin2 θ − 1)

16 r4, (3.9)

Ψ2 =

(

9

16(√γ + 1)

− 3

16U1(U1α+ 1)

)(

1

r2− 1

)

sin2 θ cos θ. (3.10)

Note that the quadratic term does not contribute to the total force and steady-state velocity terms, but it

does change the fluid flow, the electric potential and the ionic concentration.

3.1.3 Third-Order (cubic in β) Solution

Note that, due to symmetry considerations, U (β) is an anti-symmetric function. Therefore, U2 = 0 and

the next velocity term is the cubic one:

U (β) ≈ βU1 + β3U3 +O(β5). (3.11)

The cubic terms x3 are computed by solving a linear PDE system with a right-hand side determined by

the linear terms x1 and the quadratic terms x2.


Φ3 =15U1α+ 6

64cos θ − 3U1α

32cos3θ +

183U12α2 − 839U1α− 470

2560r2cos θ

+3cos θ

(

5U1α− 6cos2 θ − 4U1αcos2 θ + 4

)

64r3+

(2U1α− 1)(

cos θ − 3cos3θ)

64r5

+ cos3θ

(

15U1α

64+

3

32

)

r−2 −(

3 cos θ − 5cos3θ)

(

−19U12α2

5120+

97U1α

5120+

3U2α

320+

21

1280

)

r−4

+ cos3θ

(

U1α

64+

3

64

)

r−6,

C3 =3U1

2α2cos3 θ

32+

cos θ(

797U12α2 + 419U1α− 512U2α+ 174

)

2560r2− 3U1α cos θ (5U1α+ 2)

64

−(

3 cos θ − 5cos3 θ)

(

59U12α2

5120+

103U1α

5120+

69U2α

320+

3

1280

)

r−4

+α(

cos θ − 3cos3 θ) (

5αU12 + 2U1 + 16U2

)

128r5

− 3α cos θ(

−8αU12cos2 θ + 7αU1

2 + 2U1 + 32U2cos2 θ − 32U2

)

128r3

+U1

2α2cos3 θ

32r6− 3U1αcos

3 θ (5U1α+ 2)

64r2,

Ψ3 = r2sin2 θ

(

W2

15− W1

3+

209

3360

)

− sin2 θ

(

5W2

3− W1

3+

35

264

)

r−1 + sin4 θ

(

2W2 +761

5632

)

r−1

− sin2 θ(

1848r6sin2 θ − 2079r3sin2 θ + 924r3 − 7sin2 θ + 10)

19712r54sin2 θ − 5sin4 θ

r3

+

(

U1α

16− 1

8

)

2r3 − 3r2 + 1

4rsin2 θ,

W1 =69

512(√

γ + 1) −

1407α2log(√

γ+12√γ

)3

1280−

123 log(√

γ+12√γ

)

1280− 27

512(√γ + 1

)2

−1899αlog

(√γ+12√γ

)2

2560−α log

(√γ+12√γ

)

(

8116(

√γ+1)

− log(√

γ+12√γ

)

(

27α log(√

γ+1

2√

γ

)

4 + 278

))

320

+21α log

(√γ+12√γ

)

128(√γ + 1

) ,


W2 =9 log

(√γ+12√γ

)

256+

21α2log(√

γ+12√γ

)3+ 15αlog

(√γ+12√γ

)2

64− 27

512(√

γ + 1)2

−297α log

(√γ+12√γ

)

+ 54

1024(√

γ + 1) .

The cubic solution satisfies the boundary conditions only when αU1 = 0, due to boundary conditions

mismatch for C at r → ∞, when the diffusion and the advection terms become of the same order of

magnitude. The cubic correction velocity terms (for α = 0) turn out to be:

U3(γ) =31

320(√γ + 1)

− 9

320(√γ + 1)2

+1

1680− 11

160log

(

1 + γ−1

2

2

)

. (3.12)

The cubic term becomes dominant for small β when γ ≈ 1. Note that, for γ = 1, the linear term vanishes

and the cubic term of the velocity is U (β) = 112926880β

3.

Observe that a U = 0 solution may exist for a critical βc > 0 and α = 0, if it satisfies:

β2c = −U1(γ)

U3(γ). (3.13)

Specifically, for γ = 1 + ǫ, where 0 < ǫ ≪ 1, βc = O (√ǫ):

βc ≈√

−U ′1 (1)ǫ/U3(1) =

√

13440ǫ/1129 ≈ 3.45√ǫ. (3.14)

That is, there may be a specific nonzero external field for which the particle remains at rest. The physical

viability of this solution has yet to be explored.

3.2 Numerical Results

The numerical solver is applied to the nonlinear problem of an ion-exchanger migration driven by an electric

field [1] and the electrophoresis of highly charged surface inert particle [2]. The solver is applied to a series

of values of the electric field β, ranging from β ≪ 1 to β ∼ 1. For each β value, the Newton solver is

initialized with the linear solution and is iterated K times. Secant method iteration is applied M times

to find the steady-state velocity by finding U such that Fı(U , β) = 0, thus computing the steady-state

velocity U (β). The resulting U (β) is plotted on a log-log scale, so that the asymptotic behaviour of the

solution may be examined for a wide range of β values.

In order to validate the numerical solver, we employ the results of the asymptotic analysis of the nonlinear

system. The numerical results are compared to the asymptotic solutions, and a good correspondence is

observed.


3.2.1 Ion-exchanger results

For β ≪ 1, the linear regime [5] is dominant. However, when γ ≈ 1, the cubic regime, where U = O(β3),

dominates the linear one, even for weak electric fields. This phenomenon is confirmed both by the numerical

results and by asymptotic expansion of the nonlinear system (for α = 0). The switch from the linear regime

to the cubic regime occurs at an electric field value of β = O(√

|γ − 1|). The numerical results U h, as well

as the analytical ones (3.11), are given in Figure 3.1. The numerical solver uses Nr × Nθ = (400 × 50),

Rmax = 107 (∆r ≈ 0.041), K = 3, M = 5 and β = 10t for t = −2 : 0.1 : 0.5.

The linear solution for γ ≈ 1, yields a sign change of the zeta potential ζ = log(C/γ), when | log γ| ≈ 3β/4.

This results in a sign change in the slip velocity, which can be approximated for small |ζ| ≪ 1 as Vθ ≈ζ ·∇SΦ. This sign change creates a vortex at the downstream end of the ion-exchanger, as demonstrated

by the numerical solution and the asymptotic analysis results in Figures 3.2 and 3.3. Note that there is a

good correspondence between the numerical results and the analytical solution for β < 1, validating the

vortex formation.

3.2.2 Electrophoresis results

For β ≪ 1, the linear regime [2] is dominant. In order to verify the solver, the numerical steady-state

velocity is compared to the analytical solution [27], derived by Schnitzer and Yariv. Since the cubic

correction is much smaller than the linear solution for β ≪ 1, the difference between numerical results and

the linear solution is compared to the cubic correction term.

Because the leading order of the solution U (β) is linear for β ≪ 1, the numerical discretization error is

O(β) as well. In order to reduce the discretization error of the numerical solution (2.25) and after verifying

that the discretization error converges as O(h2) (Figure 3.4), we employ Richardson extrapolation to U h

and U 2h:

U =4

3U

h − 1

3U

2h. (3.15)

The results of the extrapolation U and the analytical results are given at Figures 3.5 and 3.6. The

numerical solver uses Rmax = 102, K = 3, M = 5 and β = 0.1 · 2t for t = −2 : 0.5 : 5. The solver is applied

to 2 grids of Nr ×Nθ = (128 × 128) as the coarse grid (with ∆r ≈ 0.037), and Nr ×Nθ = (256 × 256) as

the fine grid (with ∆r ≈ 0.018), in order to use Richardson extrapolation for U .

3.2.3 Electrophoresis for large β

When the electric field is large, nonlinear effects dominate the solution. In order to investigate the solution

for strong electric fields, we have run the solver on the TAMNUN HPC cluster [28]. Large grid sizes are used

to verify that the numerical solution converges when the grid becomes finer and that no numerical artifacts

are present in the solution. Moreover, several Rmax ∈ 10, 30, 100 values are used, to make sure it has no

major impact on the solution. The results are shown in the following figures. The steady-state velocity

results are shown for several grid resolutions and several Rmax values in Figure 3.7, and the differences

between the numerical results for steady-state velocity and the linear regime are shown in Figure 3.8. The

estimated rate of convergence is shown in Figure 3.9.


10−2

10−1

100

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Electric field magnitude (β)

Ste

ady−

stat

e ve

loci

ty m

agni

tude

γ = 0.9999γ = 0.9990γ = 0.9900γ = 0.9000

10−2

10−1

100

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101


Ste

ady−

stat

e ve

loci

ty m

agni

tude

γ = 1.0001γ = 1.0010γ = 1.0100γ = 1.1000

Figure 3.1: Steady-state velocity U as a function of the electric field magnitude β, for γ < 1 (top) andγ > 1 (bottom). The analytic cubic solution (3.11) is represented by curves (solid for U > 0, dashed forU < 0), and the numerical results are represented by symbols (circles for U > 0, squares for U < 0).Note that linear and cubic regimes match well the numerical results, including the critical β behaviour(3.13).


Numerical: β = 0.10




Figure 3.2: Streamlines of the flow for various electric field β values: numerical results for γ = 0.9. Thesteady-state velocity U is positive, so the ion-exchanger drifts to the right.


Analytical: β = 0.10




Figure 3.3: Streamlines of the flow for various electric field β values: analytical approximation for γ = 0.9(see Section 3.1). The steady-state velocity U is positive, so the ion-exchanger drifts to the right.


10−1

10−5

10−4

10−3

10−2

10−1


Dis

cret

izat

ion

erro

r

Quadratic convergence of steady−state velocity

32x3264x64128x128256x256

Figure 3.4: The discretization error U h − U1 between the numerical results and the linear asymptoticsolutions for α = 0, Du = 0.5, ζ = 6 and Rmax = 100. The symbols represent the discretization errors forvarious grid sizes, and the dotted lines correspond to linear functions of β, scaled by 4n. The graph showsthat, as the number of grid points in each dimension increases by a factor of 2, the discretization errordecreases by a factor of 4.


10−1

100

10−1

100

101

α = 0.0, Du = 0.5, ζ = 6.0


Ste

ady−

stat

e ve

loci

ty

AnalyticalNumerical

10−1

100

10−4

10−2

100

α = 0.0, Du = 0.5, ζ = 6.0


Dep

artu

re fr

om li

near

ana

lytic

sol

utio

n

AnalyticalNumerical

Figure 3.5: Steady-state velocity U as a function of the electric field magnitude β, for highly-chargedsurface particle electrophoresis (with α = 0,Du = 0.5, ζ = 6). The analytic solution velocity is representedby solid lines, and the numerical results are represented by symbols. The top figure shows the full numericalsolution compared to the analytical one (3.11), and the bottom figure compares the analytic cubic correction[27] to the difference between the numerical results and the linear regime.


10−1

100

10−1

100

101

α = 0.5, Du = 1.0, ζ = 10.0


Ste

ady−

stat

e ve

loci

ty

AnalyticalNumerical

10−1

100

10−4

10−2

100

α = 0.5, Du = 1.0, ζ = 10.0


Dep

artu

re fr

om li

near

ana

lytic

sol

utio

n

AnalyticalNumerical

Figure 3.6: Steady-state velocity U as a function of the electric field magnitude β, for highly-chargedsurface particle electrophoresis (with α = 0.5,Du = 1, ζ = 10). The analytic solution velocity is representedby solid lines, and the numerical results are represented by symbols. The top figure shows the full numericalsolution compared to the analytical one (3.11), and the bottom figure compares the analytic cubic correction[27] to the difference between the numerical results and the linear regime. When β is large, the analyticalcubic solution loses its validity because higher-order terms become significant.


0 1 2 3 4 5 60

5

10

15

20

25

30

β

Ste

ady−

stat

e ve

loci

ty

[128x128] R

max=10

[256x256] Rmax

=10

[512x512] Rmax

=10

Linear Solution (small β): 4.258 βLinear Fit (moderate β): 5.193 β −0.850

0 1 2 3 4 5 60

5

10

15

20

25

30

β

Ste

ady−

stat

e ve

loci

ty

[512x512] R

max=100

[512x512] Rmax

=30

[512x512] Rmax

=10

Linear Solution (small β): 4.258 βLinear Fit (moderate β): 5.193 β −0.850

Figure 3.7: Large β results for U (β) (Du = 1, ζ = 10, α = 0.5). The left figure shows the numericalresults of U (β) for Rmax = 10 and the following grid sizes: 128 × 128, 256 × 256, 512 × 512. The rightfigure shows the numerical results of U (β) for 512 × 512 grid for Rmax = 100, 30, 10. For small values ofβ, the linear regime of U (β) is dominant. For moderate β values, the steady state velocity is very close toan affine function of β. However, for larger values of β, the numerical solution deviates significantly fromthe linear regime, due to convergence problems at β > 6, and for β ∼ 7, the solver fails to converge. Thenumerical convergence depends on the grid resolution near the particle surface – as it becomes finer (whenNr increases or Rmax decreases), the solver convergence improves.

The numerical solver converges well for β < 5 and the resulting steady-state velocity for moderate β values

is very close to being an affine function of β. When β becomes large, the salt concentration C near the

front of the particle (at θ ≈ 0) approaches zero, while creating a wake behind the particle (at θ ≈ π) as

shown in Figures 3.10, 3.11 and 3.12. This results in high gradients in logC near the front of the particle

and, therefore, a strong electric field E forms near the particle surface (Figures 3.13, 3.14 and 3.15) due

to effective boundary conditions in (1.17). From Figs. 3.7-3.8 we conclude that even our most accurate

numerical solution is valid only up to β = 5 approximately, and therefore we only further examine solutions

in this range.

The streamline pattern does not change drastically for different electric field magnitudes. The resulting

fluid flow is nearly proportional to that of the Stokes creeping flow, as shown in Figure 3.16 and 3.17. The

difference between Ψ(r, θ) and Ψ(r, π − θ) is shown in Figure 3.18, demonstrating that the flow is almost

symmetric (with relative difference up to approximately 0.5%) for β < 5.

The numerical results show that a boundary layer starts to form near the front of the particle for strong

electric fields (see Figures 3.19 and 3.20). The width of the boundary layer with respect to the electric

field scales, to a good approximation, as: ρ(β) ≈ 0.2e−0.5β . This causes numerical problems for the current

solver at large β, due to insufficient grid resolution at the boundary layer of C and E.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

βDiff

eren

ce o

f ste

ady−

stat

e ve

loci

ty fr

om th

e lin

ear

regi

me

[128x128] R

max=10

[256x256] Rmax

=10

[512x512] Rmax

=10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

βDiff

eren

ce o

f ste

ady−

stat

e ve

loci

ty fr

om th

e lin

ear

regi

me

[512x512] R

max=100

[512x512] Rmax

=30

[512x512] Rmax

=10

Figure 3.8: Large β results for U (β)− βU1 (Du = 1, ζ = 10, α = 0.5 and U1 ≈ 4.258 is the linear regimecoefficient). The left figure shows the numerical results of U (β) for Rmax = 10 and the following grid sizes:128 × 128, 256 × 256, 512 × 512. The right figure shows the numerical results of U (β) for 512 × 512 gridfor Rmax = 100, 30, 10. It is seen that the difference is indeed nearly an affine function of β in the large-βregime, so long as the numerical solution remains sufficiently accurate.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

β

(V12

8x12

8 − V

256x

256)

/ (V

256x

256 −

V51

2x51

2)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1

−0.5

0

0.5

1

1.5

2

2.5

β

Con

verg

ence

ord

er

Figure 3.9: Rate of convergence estimation for U at large β (Du = 1, ζ = 10, α = 0.5), based on thenumerical results of U (β), for the grid sizes of 128 × 128, 256 × 256, 512 × 512. The rate of convergencer(β) is estimated by r(β) ,

(

U 4h(β) − U 2h(β))

/(

U 2h(β)− U h(β))

(left). The order of convergence is

estimated by p(β) , log2 r(β) (right). For small β, p(β) → 2, showing quadratic convergence for the linearregime. For moderate β, the convergence rate slows down, and it deteriorates for large β values.


−3 −2 −1 0 1 2 3−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5Salt concentration (C): β = 0.10

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

Figure 3.10: β = 0.1 results for C: grid size 512 × 512, Rmax = 10, Du = 1, ζ = 10, α = 0.5. Here thelinear regime dominates the solution for Cβ(r, θ) ≈ 1− β cos θ

2r2 .


−3 −2 −1 0 1 2 3−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


0.7

0.8

0.9

1

1.1

1.2

1.3

Figure 3.11: β = 0.8 results for C: grid size 512× 512, Rmax = 10, Du = 1, ζ = 10, α = 0.5. For moderateβ values, the salt concentration starts to form a wake behind the particle (θ ≈ π).


−3 −2 −1 0 1 2 3−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


0.5

1

1.5

2

2.5

3

Figure 3.12: β = 5.0 results for C: grid size 512 × 512, Rmax = 10, Du = 1, ζ = 10, α = 0.5. For largevalues of β, there is a significant drop in salt concentration at the front of the particle – C approaches zeroat θ ≈ 0, such that ∂C

∂r becomes large (since the change in C happens on a small length scale). Due to theboundary conditions of the electrophoresis problem (given by (1.17)), the electric field radial componentgiven by Er = −∂Φ

∂r = − 1C∂C∂r , becomes singular where C → 0.


−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Electric potential (φ) and field at the surface (E): β = 0.10

Figure 3.13: β = 0.1 results for Φ: grid size 512 × 512, Rmax = 10, Du = 1, ζ = 10, α = 0.5. The electricpotential and the electric field results are multiplied by β−1, so the linear regime appears unchanged fordifferent values of β. Here the linear regime dominates the solution for Φβ(r, θ) ≈ −βr cos θ, hence theelectric field is constant for weak electric fields: E ≈ βı.


−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

1

1.5


Figure 3.14: β = 0.8 results for Φ: grid size 512 × 512, Rmax = 10, Du = 1, ζ = 10, α = 0.5. The electricpotential and the electric field results are multiplied by β−1, so the linear regime appears unchanged fordifferent values of β. For moderate values of β, the symmetry breaks and the electric field becomes strongernear the front of the particle than at the back.


−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

1

1.5


Figure 3.15: β = 5.0 results for Φ: grid size 512 × 512, Rmax = 10, Du = 1, ζ = 10, α = 0.5. The electricpotential and the electric field results are multiplied by β−1, so the linear regime appears unchanged fordifferent values of β. For large values of β, the electric field radial component Er = −∂Φ

∂r at the particlefront becomes much larger than the electric field at the particle back.


−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3Streamlines (Ψ): β = 0.10

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2


−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2


−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2


Figure 3.16: β = 0.1, 0.8, 5.0 and Stokes streamlines results: grid size 512 × 512, Rmax = 30, Du = 1,ζ = 10, α = 0.5. The numerical results for Ψ are scaled by β−1, so the linear regime will appear thesame for different β values. Note that the numerical results show a flow pattern very similar to that of theanalytical Stokes flow.


−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5Streamlines (Ψ): β = 0.10

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Figure 3.17: β = 0.1, 0.8, 5.0 and Stokes streamlines results after subtraction of the uniform flow stream-function (U

2 r2 sin2 θ): grid size 512 × 512, Rmax = 30, Du = 1, ζ = 10, α = 0.5. The numerical results for

Ψ are scaled by β−1, so the linear regime will appear the same for different β values. Even for the largestbeta we could compute, the velocity remains fairly symmetric and “uneventful” also in the framework ofthe “laboratory” rather than that of the particle.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

β

Rel

ativ

e st

ream

lines

asy

mm

etry

Figure 3.18: Streamline results: grid size 512 × 512, Rmax = 10, Du = 1, ζ = 10, α = 0.5. The relativestreamline asymmetry is computed by ‖Ψ(r, θ) − Ψ(r, π − θ)‖/‖Ψ(r, θ)‖ as a function of β. The resultsshow that the streamline pattern is almost symmetric, even for moderately large β values.


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Radial distance from particle surface

Nor

mal

ized

ele

ctric

fiel

d ra

dial

com

pone

nt

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Radial distance from particle surface (scaled by eβ/2)N

orm

aliz

ed e

lect

ric fi

eld

radi

al c

ompo

nent

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

−2

10−1

β

Er b

ouda

ry la

yer

wid

th

Numerical resultsPoints to fitExponential fit

0.192 e−0.498 β

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

0

101

β

Er(r

=1,

θ=

0) /

β


1.237 e0.417 β

Figure 3.19: Electric field boundary layer formation: grid size 512× 512, Rmax = 30, Du = 1, ζ = 10, α =0.5. The upper left graph shows normalized electric field radial component Er near the particle front (θ =0). The numerical results for e(ρ) = Er(r = 1+ ρ, θ = 0) are normalized by e(ρ) = (e(ρ)− β) / (e(0)− β),where e(ρ = ∞) = β. The normalized electric field satisfies e(ρ) ∈ [0, 1] and is shown as function ofρ = r − 1 at the upper left graph and as function of ρe0.5β at the upper right graph. In order to estimatethe width of the boundary layer, for each β, we compute the value of ρ(β) = minρρ : e(ρ) < 0.5. Eachsuch ρ(β) is plotted at the left graph as a solid dot on the corresponding line. In addition, ρ(β) is plottedas a function of β using logarithmic y axis at the lower left graph to show that ρ(β) ≈ 0.2e−0.5β . Moreover,Er(β; r = 1, θ = 0)/β grows exponentially as a function of β, as shown at the lower right graph. Thus, aboundary layer is indeed forming near the front of the particle.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Radial distance from particle surface

Ion

conc

entr

atio

n (C

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

−1

100


C(r

=1,

θ=

0)


0.937 e−0.445 β

Figure 3.20: Ionic concentration boundary layer formation: grid size 512 × 512, Rmax = 30, Du = 1,ζ = 10, α = 0.5. The left graph shows the ionic concentration C near the particle front (θ = 0). Inaddition, Cβ(r = 1, θ = 0) is plotted as a function of β using logarithmic y axis at the right graph to showthat Cβ(r = 1, θ = 0) ≈ 0.94e−0.45β . The ionic concentration becomes very close to zero for β ≈ 7 at r = 1,creating a singularity near the particle front.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

β

C ⋅

Er /

β at

r=

1, θ

=0

Figure 3.21: The product of C(r = 1, θ = 0)Er(r = 1, θ = 0)/β as a function of β.


3.3 Discussion

A numerical framework for a nonlinear electrokinetic transport system is developed. The framework is

used to construct numerical solvers for two electrokinetic problems: the migration of ion-exchangers and

surface-charged inert particle electrophoresis. The numerical results for weak electric field exhibit a good

correspondence to the asymptotic models as shown above. Two interesting phenomena (cubic regime and

downstream vortex) were predicted for the ion-exchanger using the numerical results, and were verified

using an asymptotic expansion of the nonlinear system. The electrophoresis problem was solved on a

high-resolution grid to confirm the asymptotic cubic correction of the steady-state velocity, as well as to

provide insights into the solution behavior for moderately strong electric fields. For strong electric field,

the nonlinear effects become dominant and a boundary-layer structure begins to form. Due to diminishing

ion concentration C at the front of the particle, a boundary layer of thickness approximately 0.2e−0.5β is

formed with respect to the electric field E at the front of the particle. The value of Er/β at (r = 1, θ = 0) is

given approximately by eβ/2, while the value of C(r = 1, θ = 0) is given approximately by e−β/2, and their

product remains close to 1 for all β that we could compute (as shown in Figure 3.21). However, the velocity

of the particle remains nearly linear with β and the velocity field is very close to being symmetric. Finally,

we emphasize the importance of the analytical asymptotic derivation. This approach provides effective

boundary conditions for the macroscale problem and thus eliminates the inherent physical scale disparity.

This elimination enables the construction of the numerical scheme. Furthermore, the analytical asymptotic

solutions for the weak field regime enable important mutual verification of the numerical results.

3.4 Future Work

Future work may include using the numerical results described in this work to develop an analytical

solution for the strong field regime of the electrophoresis problem, which is of interest. The solver may

be used to investigate how the solution (and the evolving boundary layer, in particular) depends on the

problem parameters. Optimizing the direct sparse linear solver can be done by using the ordering data from

previous calls, or replacing the full Newton step by an iterative linear solver (e.g., using preconditioned

Krylov subspace methods [29] or Multigrid), allowing finer grids to be used. In order to achieve high

numerical accuracy, the grid discretization should be made finer near the particle surface (at r = 1), while

Rmax can be made smaller. Moreover, a different discretization scheme may be used for the r coordinate,

to have higher grid resolution near the particles. In addition, the continuation of the solution between

different β values should advance in finer steps, to ensure Newton method convergence. The discretization

of boundary conditions far away from the particle may be improved by including the linear correction from

the asymptotic solution, or reformulation of far-away boundary condition according to the asymptotic

behavior of the analytic solution far away from the particle. Moreover, the framework may be used for

automatic construction of Newton solvers for other nonlinear PDE systems, for example, for particles with

different surface properties, or different particle shape.


Appendix A

Effective Boundary Conditions

A.1 Debye layer scaling

We define the scaled coordinate ρ (normal to ion-exchanger surface), where δ ≪ 1:

ρ =r − 1

δ. (A.1)

Thus, the electric potential and the ionic concentrations are O(1) at the Debye layer (r ≈ 1):

ϕ(ρ, θ) ∼ ϕ, (A.2)

c±(ρ, θ) ∼ c±, (A.3)

c(ρ, θ) ∼ c, (A.4)

q(ρ, θ) ∼ q. (A.5)

The radial fluxes at the Debye layer are O(δ−1), due to coordinate rescaling:

j±(ρ, θ) ∼ δ−1j±. (A.6)

The pressure is O(δ−2), due to momentum balance equations:

p(ρ, θ) ∼ δ−2p. (A.7)

The tangential velocity component is O(1) and the radial component is O(δ).


A.2 Ion Fluxes

In order to match bulk O(1) outer radial ionic flux, the O(δ−1) term of the inner ionic radial flux, j±(ρ, θ)

must vanish:

−∂c±∂ρ

∓ c±∂ϕ

∂ρ= 0,

(Φ− ϕ)± (logC± − log c±) = 0, (A.8)

Φ− ϕ± logC±c±

= 0.

This relation can be rewritten as a Boltzmann distribution of c±, where limρ→∞ c± = C,

c± = C exp [±(Φ− ϕ)] . (A.9)

A.3 Electric Potential

The leading-order Laplace equation in the Debye layer reads

∂2ϕ

∂ρ2= −q = −c+ − c−

2= −C

2(exp [+(Φ− ϕ)] − exp [−(Φ− ϕ)]) = C sinh(ϕ− Φ). (A.10)

Denote ψ = ϕ− Φ, so that ψ(0) = V − Φ = ζ and ψ(ρ→ ∞) = 0:

∂2ψ

∂ρ2= C sinh(ψ) = 2C sinh

ψ

2cosh

ψ

2,

2∂ψ

∂ρ

∂2ψ

∂ρ2= 4C sinh

ψ

2cosh

ψ

2

∂ψ

∂ρ, (A.11)

∂

∂ρ

(

∂ψ

∂ρ

)2

=∂

∂ρ

(

2√C sinh

ψ

2

)2

.

Integrating with respect to boundary condtions yields:

∂ψ

∂ρ= −2

√C sinh

ψ

2= −4 sinh

ψ

4cosh

ψ

4

√C = −4 tanh

ψ

4cosh2

ψ

4

√C,

1/tanh ψ4

4 cosh2 ψ4

∂ψ

∂ρ= −

√C, (A.12)

∂

∂ρ

(

log tanhψ

4

)

=∂

∂ρ

(

−ρ√C)

.


Integrating with respect to boundary condtions yields:

log tanhψ

4− log tanh

ζ

4= −ρ

√C,

ψ = ϕ− Φ = 4 tanh−1

(

tanhζ

4· e−ρ

√C

)

. (A.13)

A.4 Radial Momentum

Leading-order O(δ−3) radial momentum balance in the Debye layer yields:

−∂p∂ρ

+∂ϕ

∂ρ

∂2ϕ

∂ρ2= 0. (A.14)

One integration yields:

p =1

2

(

∂ϕ

∂ρ

)2

=1

2

(

∂ψ

∂ρ

)2

. (A.15)

A.5 Tangential Momentum

Leading-order O(δ−2) tangential momentum balance in the Debye layer yields:

0 =∂2vθ∂ρ2

− ∂p

∂θ+∂ϕ

∂θ

∂2ϕ

∂ρ2,

∂2vθ∂ρ2

=1

2

∂

∂θ

(

∂ψ

∂ρ

)2

− ∂(ψ + Φ)

∂θ

∂2ψ

∂ρ2=

∂

∂θ

(

2C sinh2ψ

2

)

− ∂ψ

∂θC sinhψ − ∂Φ

∂θ

∂2ψ

∂ρ2

=∂

∂θ(C (coshψ − 1))− C sinhψ

∂ψ

∂θ− ∂Φ

∂θ

∂2ψ

∂ρ2(A.16)

=∂C

∂θ(coshψ − 1)− ∂Φ

∂θ

∂2ψ

∂ρ2=∂C

∂θ

(

2 sinh2ψ

2

)

− ∂Φ

∂θ

∂2ψ

∂ρ2

= −∂C∂θ

2√C

(

1

2sinh

ψ

2

∂ψ

∂ρ

)

− ∂Φ

∂θ

∂2ψ

∂ρ2= −∂C

∂θ

2√C

∂

∂ρ

(

coshψ

2

)

− ∂Φ

∂θ

∂2ψ

∂ρ2.

By integrating from specific ρ to ρ→ ∞ (where ψ = 0):

∂vθ∂ρ

= −∂C∂θ

2√C

(

coshψ

2− 1

)

− ∂Φ

∂θ

∂ψ

∂ρ= −∂C

∂θ

4√C

sinh2ψ

4− ∂Φ

∂θ

∂ψ

∂ρ. (A.17)

Note that:

∂ψ

∂ρ= −2

√C sinh

ψ

2= −4

√C sinh

ψ

4cosh

ψ

4, (A.18)

−4 sinhψ

4=

1/ cosh ψ4√

C

∂ψ

∂ρ. (A.19)


Thus:

∂vθ∂ρ

=

(

∂C

∂θ

1

C

)

·(

sinh ψ4

cosh ψ4

∂ψ

∂ρ

)

− ∂Φ

∂θ

∂ψ

∂ρ= 4

∂

∂ρ

(

log coshψ

4

)

∂

∂θlogC − ∂ψ

∂ρ

∂Φ

∂θ, (A.20)

∂vθ∂ρ

=∂

∂ρ

(

4 log coshψ

4

∂

∂θlogC − ψ

∂Φ

∂θ

)

. (A.21)

By integrating from ρ = 0 to ρ → ∞, we have Dukhin-Derjaguin slip formula for tangential velocity

component Vθ (where the radial component is Vr = 0) for ζ = V − Φ:

Vθ = ζ · ∂Φ∂θ

− 4 log coshζ

4· ∂∂θ

logC = ζ · ∂Φ∂θ

+ 2 log

(

1− tanh2ζ

4

)

· ∂∂θ

logC. (A.22)


Appendix B

Differential operators in spherical

coordinates

The following spherical coordinate system is used:

x = r sin θ cosφ

y = r sin θ sinφ (B.1)

z = r cos θ

Assume full symmetry around z axis – there is no dependence on φ:

∇f =∂f

∂rr+

1

r

∂f

∂θθ +

1

r sin θ

∂f

∂φφ (B.2)

∇·F =1

r2∂

∂r

(

r2 · Fr)

+1

r sin θ

∂

∂θ(sin θ · Fθ) +

1

r sin θ

∂Fφ∂φ

(B.3)

B.1 Scalar Operators

Scalar Laplacian derivation:

∇2f = ∇·∇f =1

r2∂

∂r

(

r2∂f

∂r

)

+1

r2 sin θ

∂

∂θ

(

sin θ · ∂f∂θ

)

+1

r2 sin2 θ

∂2f

∂φ2(B.4)

In conservative form:

r2 sin θ ·∇2f =∂

∂r

(

r2 sin θ∂f

∂r

)

+∂

∂θ

(

sin θ · ∂f∂θ

)

+∂

∂φ

(

1

sin θ· ∂f∂φ

)

(B.5)


The unit vectors in spherical coordinate system are:

r =∂r

∂r=

sin θ cosφ

sin θ sinφ

cos θ

, θ =

1

r

∂r

∂θ=

cos θ cosφ

cos θ sinφ

− sin θ

, φ =

1

r sin θ

∂r

∂φ=

− sinφ

cosφ

0

(B.6)

Note that:

∂r

∂r= 0,

∂θ

∂r= 0,

∂φ

∂r= 0,

∂r

∂θ= θ,

∂θ

∂θ= −r,

∂φ

∂θ= 0, (B.7)

∂r

∂φ= sin θ · φ, ∂θ

∂φ= cos θ · φ, ∂φ

∂φ= − sin θ · r − cos θ · θ.

B.2 Vector Operators

Vector gradient operator can be written by:

∇V =

r∂

∂r+ θ

1

r

∂

∂θ+ φ

1

r sin θ

∂

∂φ

V =

r∂

∂r+ θ

1

r

∂

∂θ+ φ

1

r sin θ

∂

∂φ

(

Vrr + Vθθ + Vφφ)

∂V

∂r=∂Vr∂r

r +∂Vθ∂r

θ +∂Vφ∂r

φ

∂V

∂θ=∂Vr∂θ

r +∂Vθ∂θ

θ +∂Vφ∂θ

φ+ Vrθ − Vθr (B.8)

∂V

∂φ=∂Vr∂φ

r +∂Vθ∂φ

θ +∂Vφ∂φ

φ+ Vr sin θφ+ Vθ cos θφ− Vφ

(

r sin θ + θ cos θ)

∇V =

1 0 0

0 1r 0

0 0 1r sin θ

∂Vr∂r

∂Vθ∂r

∂Vφ∂r

∂Vr∂θ − Vθ

∂Vθ∂θ + Vr

∂Vφ∂θ

∂Vr∂φ − Vφ sin θ

∂Vθ∂φ − Vφ cos θ

∂Vφ∂φ + Vr sin θ + Vθ cos θ

Vector Laplacian components can be written as:

∇2V = ∇ ·∇V =1

r2∂

∂r

(

r2∂V

∂r

)

+1

r2 sin θ

∂

∂θ

(

sin θ∂V

∂θ

)

+1

r2 sin2 θ

∂

∂φ

(

∂V

∂φ

)

(B.9)


r component:

∇2(Vrr) =1

r2∂

∂r

(

r2∂Vr∂r

)

r

+1

r2 sin θ

∂

∂θ

(

sin θ

(

∂Vr∂θ

r + Vrθ

))

+1

r2 sin2 θ

∂

∂φ

(

∂Vr∂φ

r + Vr sin θφ

)

∇2(Vrr) =

(

2

r

∂Vr∂r

+∂2Vr∂r2

)

r (B.10)

+1

r2 sin θ

(

cos θ

(

∂Vr∂θ

r + Vrθ

)

+ sin θ

(

∂2Vr∂θ2

r +∂Vr∂θ

θ +∂Vr∂θ

θ − Vrr

))

+1

r2 sin2 θ

(

∂2Vr∂φ2

r +∂Vr∂φ

sin θφ+∂Vr∂φ

φ sin θ − Vr sin θ(


)

∇2(Vrr) =

(

2

r

∂Vr∂r

+∂2Vr∂r2

− 2Vrr2

+1

r2∂2Vr∂θ2

+cot θ

r2∂Vr∂θ

+1

r2 sin2 θ

∂2Vr∂φ2

)

r

+2

r2∂Vr∂θ

θ +2

r2 sin θ

∂Vr∂φ

φ

θ component:

∇2(Vθθ) =1

r2∂

∂r

(

r2∂Vθ∂r

)

θ

+1

r2 sin θ

∂

∂θ

(

sin θ

(

∂Vθ∂θ

θ − Vθr

))

+1

r2 sin2 θ

∂

∂φ

(

∂Vθ∂φ

θ + Vθ cos θφ

)

∇2(Vθθ) =

(

2

r

∂Vθ∂r

+∂2Vθ∂r2

)

θ (B.11)

+1

r2 sin θ

(

cos θ

(

∂Vθ∂θ

θ − Vθr

)

+ sin θ

(

∂2Vθ∂θ2

θ − ∂Vθ∂θ

r − ∂Vθ∂θ

r − Vθθ

))

+1

r2 sin2 θ

(

∂2Vθ∂φ2

θ +∂Vθ∂φ

cos θφ+∂Vθ∂φ

cos θφ− Vθ cos θ(


)

∇2(Vθθ) =

(

−2 cot θ

r2Vθ − 2

∂Vθ∂θ

)

r +

(

2

r

∂Vθ∂r

+∂2Vθ∂r2

+cot θ

r2∂Vθ∂θ

+1

r2∂2Vθ∂θ2

)

θ

+1

r2 sin2 θ

(

∂2Vθ∂φ2

− Vθ

)

θ +2cot θ

r2 sin θ

∂Vθ∂φ

φ


φ component:

∇2(Vφφ) =1

r2∂

∂r

(

r2∂Vφ∂r

)

φ+1

r2 sin θ

∂

∂θ

(

sin θ∂Vφ∂θ

φ

)

+1

r2 sin2 θ

∂

∂φ

(

∂Vφ∂φ

φ− Vφ

(


)

∇2(Vφφ) =1

r2∂

∂r

(

r2∂Vφ∂r

)

φ+1

r2 sin θ

∂

∂θ

(

sin θ∂Vφ∂θ

φ

)

(B.12)

+1

r2 sin2 θ

∂

∂φ

(

∂Vφ∂φ

φ− Vφ

(


)

∇2(Vφφ) =

(

2

r

∂Vφ∂r

+∂2Vφ∂r2

)

φ+1

r2 sin θ

(

cos θ∂Vφ∂θ

+ sin θ∂2Vφ∂θ2

)

φ

+1

r2 sin2 θ

(

∂2Vφ∂φ2

φ− 2∂Vφ∂φ

(


− Vφφ

)

In summary:

∇2V =

(

∇2Vr −2Vrr2

)

r +2

r2∂Vr∂θ

θ +2

r2 sin θ

∂Vr∂φ

φ

− 2

r2 sin θ

∂ (Vθ sin θ)

∂θr +

(

∇2Vθ −Vθ

r2 sin2 θ

)

θ +2cot θ

r2 sin θ

∂Vθ∂φ

φ (B.13)

− 2

r2 sin θ

∂Vφ∂φ

r − 2 cot θ

r2 sin θ

∂Vφ∂φ

θ +

(

∇2Vφ −Vφ

r2 sin2 θ

)

φ

Since Vφ = 0 and ∂∂φ(·) = 0, we get:

∇2V =

(

∇2Vr −2Vrr2

− 2

r2 sin θ

∂ (Vθ sin θ)

∂θ

)

r +

(

∇2Vθ −Vθ

r2 sin2 θ+

2

r2∂Vr∂θ

)

θ

∇2V =

(

1

r2∂

∂r

(

r2∂Vr∂r

)

+1

r2 sin θ

∂

∂θ

(

sin θ · ∂Vr∂θ

)

− 2Vrr2

− 2

r2 sin θ

∂ (Vθ sin θ)

∂θ

)

r (B.14)

+

(

1

r2∂

∂r

(

r2∂Vθ∂r

)

+1

r2 sin θ

∂

∂θ

(

sin θ · ∂Vθ∂θ

)

− Vθ

r2 sin2 θ+

2

r2∂Vr∂θ

)

θ


Bibliography

[1] E. Yariv. Migration of ion-exchange particles driven by a uniform electric field. Journal of Fluid

Mechanics, 655:1–17, 2010.

[2] O. Schnitzer and E. Yariv. Macroscale description of electrokinetic flows at large zeta potentials:

Nonlinear surface conduction. Phys. Rev. E, 86:021503, Aug 2012.

[3] J. H. Masliyah and S. Bhattacharjee. Electrokinetic and Colloid Transport Phenomena. John

Wiley & Sons, Inc., 2005.

[4] B. Kirby. Micro- and Nanoscale Fluid Mechanics – Transport in Microfluidic Devices. Cambridge

Univ Press, 2010.

[5] E. Yariv. An asymptotic derivation of the thin-Debye-layer limit for electrokinetic phenomena.

Chemical Engineering Communications, 197(1):3–17, 2010.

[6] G.M. Whitesides. The origins and the future of microfluidics. Nature, 442(7101):368–373, 2006.

[7] D. Erickson and D. Li. Integrated microfluidic devices. Analytica Chimica Acta, 507(1):11–26,

2004.

[8] T.M. Squires and S.R. Quake. Microfluidics: Fluid physics at the nanoliter scale. Reviews of

modern physics, 77(3):977, 2005.

[9] S. Hardt and F. Schonfeld. Microfluidic technologies for miniaturized analysis systems, vol-

ume 18. Springer, 2007.

[10] A. Wainright, U.T. Nguyen, T.L. Bjornson, and T.D. Boone. Preconcentration and separation of

double-stranded dna fragments by electrophoresis in plastic microfluidic devices. Electrophoresis,

24(21):3784–3792, 2003.

[11] H. Wu, A. Wheeler, and R.N. Zare. Chemical cytometry on a picoliter-scale integrated microflu-

idic chip. Proceedings of the National Academy of Sciences of the United States of America,

101(35):12809–12813, 2004.

[12] R. Weinberger. Practical capillary electrophoresis. Academic Press San Diego, CA:, 1993.

[13] K.M. Horsman, J.M. Bienvenue, K.R. Blasier, and J.P. Landers. Forensic dna analysis on

microfluidic devices: a review. Journal of forensic sciences, 52(4):784–799, 2007.


[14] MV Kothare. Dynamics and control of integrated microchemical systems with application to

micro-scale fuel processing. Computers & chemical engineering, 30(10):1725–1734, 2006.

[15] F. C. Leinweber and U. Tallarek. Nonequilibrium electrokinetic effects in beds of ion-

permselective particles. Langmuir, 20(26):11637–11648, 2004. PMID: 15595793.

[16] M.W. Losey, M.A. Schmidt, and K.F. Jensen. Microfabricated multiphase packed-bed reactors:

Characterization of mass transfer and reactions. Industrial & engineering chemistry research,

40(12):2555–2562, 2001.

[17] S.J. Kim, S.H. Ko, K.H. Kang, and J. Han. Direct seawater desalination by ion concentration

polarization. Nature Nanotechnology, 5(4):297–301, 2010.

[18] J.T. Santini Jr, A.C. Richards, R. Scheidt, M.J. Cima, and R. Langer. Microchips as controlled

drug-delivery devices. Angewandte Chemie International Edition, 39(14):2396–2407, 2000.

[19] A. Bard and L Faulkner. Electrochemical Methods - Fundamentals and Applications. Wiley,

2001.

[20] Shau-Chun Wang, Yi-Wen Lai, Yuxing Ben, and Hsueh-Chia Chang. Microfluidic mixing by

dc and ac nonlinear electrokinetic vortex flows. Industrial & Engineering Chemistry Research,

43(12):2902–2911, 2004.

[21] Y. Ben and H.-C. Chang. Nonlinear smoluchowski slip velocity and micro-vortex generation.

Journal of Fluid Mechanics, 461:229–238, 2002.

[22] J.H. Ferziger and J.H.F.M. Peric. Computational Methods for Fluid Dynamics. Springer-Verlag

GmbH, 2002.

[23] J. Strikwerda. Finite difference schemes and partial differential equations. Society for Industrial

Mathematics, 2004.

[24] A. Sidi. Practical extrapolation methods: theory and applications, volume 10. Cambridge Uni-

versity Press, 2003.

[25] R. Zeyde. Electrokinetic Solver Source Code. http://github.com/romanz/thesis.

[26] Timothy A. Davis. Algorithm 832: Umfpack v4.3—an unsymmetric-pattern multifrontal

method. ACM Trans. Math. Softw., 30(2):196–199, June 2004.

[27] O. Schnitzer, R. Zeyde, I. Yavneh, and E. Yariv. Non-linear electrophoresis of a highly charged

colloidal particle. Submitted to Phyiscs of fluids, 2013.

[28] Technion Taub Computer Center. TAMNUN HPC cluster. http://hpc.technion.ac.il/.

[29] Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, 2003.


http://github.com/romanz/thesis

http://hpc.technion.ac.il/

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