J Eng Math (2011) 69:25–58DOI 10.1007/s10665-010-9364-1

ORIGINAL ARTICLE

Motion of paramagnetic particles in a viscous fluidunder a uniform magnetic field: benchmark solutions

Yong Kweon Suh · Sangmo Kang

Received: 17 July 2009 / Accepted: 5 February 2010 / Published online: 26 February 2010© Springer Science+Business Media B.V. 2010

Abstract Numerical simulations of the two-dimensional motion of multiple paramagnetic particles suspendedin a viscous fluid subjected to a uniform magnetic field are presented. Both the magnetic field and flow field canbe described efficiently with simple series in local coordinates attached to each particle. The coefficients of theseries can be obtained with fast convergence when only a few leading coefficients are treated implicitly. Numericalresults for the flow field are validated by comparing the data with those given by an asymptotic solution for a pair ofparticles separated by a small distance. The numerical results of the magnetic field are validated by comparison withthe solutions in bipolar coordinates. Simulations of the motion of multiple particles reveal interesting phenomenaand shed light on the fundamental mechanism of particles clustering into a straight chain. The data presented in thispaper can be used as a benchmark solution for verifying codes for simulating the motion of paramagnetic particlesin a magnetic field.

Keywords Asymptotic solution · Fluid flow · Magnetic force · Paramagnetic particles · Particle motion

1 Introduction

Understanding the motion of micro- and nano-particles in a viscous fluid is important in applications such as mag-netic tweezers [1], biological assays [2,3], drug targeting [4], and fluid mixing [5–10]. A prototype of magneticlab-on-a-chip for point-of-care sepsis diagnosis was recently reported [11]. Various applications of magnetism inmicrofluidic areas were introduced by Pamme [12].

Most previous theoretical and numerical investigations of the motion of paramagnetic particles in a viscous fluidhave been performed with simple models, i.e., dipole models [13, Chap. 1], [14–17]. In this model, a particle isconsidered as a dipole point responding to an external magnetic field moving in the field direction. Interactionsbetween neighboring particles are considered through the dipole–dipole model only. This simple model cannotprecisely capture the real dynamics of paramagnetic particles when the particle concentration is high or the particlesize is large, as is usual in micro- and nano-fluidics. Here paramagnetism means a form of magnetism occurringonly in the presence of an externally applied magnetic field [18].

Y. K. Suh (B) · S. KangDepartment of Mechanical Engineering, Dong-A University, Busan 604-714, Koreae-mail: yksuh@dau.ac.kr

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26 Y. K. Suh, S. Kang

An alternative to the investigation of particle motion is to perform direct numerical simulation of particles in asuspension. Kang et al. [19] recently reported simulation results for the two-dimensional case where particles withcylindrical shape are scattered within a circular container under a uniform magnetic field. Their numerical resultsfor a pair of particles are subject to validation because no solutions, numerical or analytical, have been reported todate for comparison.

This study has been motivated by the need to present accurate data for the motion of scattered paramagneticparticles subjected to a uniform magnetic field, so that these data can be used to verify new simulation methods forinvestigating particle motion. We propose a method of multiple coordinates for constructing semi-analytic solutionsfor the motion of multiple particles under a uniform magnetic field. The method will be validated by comparing thenumerical results with asymptotic solutions for a pair of particles in close proximity. We also verify the method bycomparing the numerical data with those calculated from solutions in bipolar coordinates.

In Sect. 2, we formulate the equations governing the magnetic and fluid-flow fields, together with the boundaryconditions. In Sect. 3, we present numerical methods for solving the governing equations. In Sect. 4, asymptoticsolutions of the flow field for a pair of particles with a small gap are derived to validate the multiple-coordinatemethod. Section 4 presents alternative solutions for the magnetic field for a pair of particles in bipolar coordinatesin order to validate the numerical methods. Numerical results are presented in Sect. 5 and our conclusions aresummarized in Sect. 6.

2 Formulation of the problem

We consider N paramagnetic particles with radius a∗ suspended inside a circular cylinder of radius a∗ R containinga viscous fluid, subject to a uniform magnetic field, as shown in Fig. 1. We study the two-dimensional motion ofparticles as well as the induced fluid flow by a semi-analytic method, i.e., a series solution, in the low-Reynolds-number regime. The fluid domain, �0, is enclosed by the boundary ∂�0. The interior domains of the particles,�1,�2, . . ., in contact with the fluid are enclosed by the interfaces, ∂�1, ∂�2, . . ., respectively. We will use globalcoordinates z0 = x + iy and local coordinates z1 = x1 + iy1, z2 = x2 + iy2, . . ., to represent the perturbed magneticand flow fields caused by the boundary ∂�0 and the particles�1,�2, . . .; see Fig. 1. The origin of each coordinatesystem is located at the center of the corresponding particle. Hereafter all the variables are dimensionless exceptfor those having an asterisk as the superscript and those separately indicated. The lengths and spatial coordinatesare scaled by a∗.

2.1 Formulation of the magnetic field

The Maxwell equations in the absence of current are

∇ × H = 0, ∇ · B = 0, (1a,b)

where B is the magnetic-flux intensity scaled by the magnitude of the external flux intensity B∗0 ,H = B/μ is the

magnetic-field intensity scaled by B∗0/μ

∗f , μ

∗f is the magnetic permeability of the fluid, and μ is the dimensionless

magnetic permeability of the fluid or particles scaled by μ∗f . In this study, both the fluid and the particles are linear

isotropic materials, and thus μ is taken to be constant within each material. As a consequence, B is irrotational andsolenoidal. The second property allows us to introduce a magnetic vector potential A (scaled by a∗ B∗

0 ) defined asB = ∇ × A, which converts Eq. 1a to ∇ × (∇ × A) = 0. For the two-dimensional problem, B is parallel to the(x, y) plane and A is orthogonal to this plane. This allows us to use a scalar field, A, defined as A = A(x, y)i × j,where i and j are unit vectors along the x- and y-axis, respectively. The governing equations reduce to the Laplaceequation,

∇2 A = 0. (2)

This equation must be solved in the fluid and particle regions.

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Motion of paramagnetic particles in a viscous fluid 27

x

y 1x

1y

1z

1bz

2bz

2z

0z

R

2x

2y

P

0BO

1

1

0Ω

2Ω

1Ω

2∂Ω0∂Ω

1∂Ω

Fig. 1 Multiple circular particles submerged in a viscous fluid confined within a domain of circular boundary under a uniform magneticfield

We will apply a constant flux density B0 = i at the surrounding boundary ∂�0. In terms of the magnetic vectorpotential, this corresponds to

A = y on ∂�0. (3)

The boundary conditions (in short BCs hereafter) at the interface between the fluid and a particle �i are

A f = Ap on ∂�i (i > 0), (4a)(∂A

∂n

)f

= 1

μp

(∂A

∂n

)p

on ∂�i (i > 0), (4b)

where the subscripts “f ” and “p” denote evaluation at the fluid and particle side, respectively, and n is a localcoordinate normal to the surface. The parameter μp is the dimensionless magnetic permeability of particles givenby μp = μ∗

p/μ∗f .

The fluid flow is solely driven by the motion of particles caused by the magnetic force. It can be shown that themagnetic-force density (force per unit volume) fm acting on each particle is given by

fm = −1

2H2∇μ, (5)

where H2 = H · H = H2x + H2

y . Note that μ is uniform over each domain�i but discontinuous across the interface∂�i . Integrating Eq. 5 over the whole region of the particle including the interface where μ sharply changes fromμp to 1, we obtain a formula for the total magnetic force of the particle exerted on the fluid in contact (see Appendixfor a detailed derivation),

123

28 Y. K. Suh, S. Kang

Fig. 2 An infinitesimalareal element on the particlesurface and thesurface-force componentsacting on the element

p

f

n dsσ ns dsσ s

ds

Fm = 1

2(1 − 1/μp)

∫∂�i

[(∂A

∂s

)2

p+ 1

μp

(∂A

∂n

)2

p

]n(s)ds, (6)

where the coordinate s is measured along the surface of the particle and n denotes the unit outward normal vector.In this study, forces are scaled by a∗2 B∗2

0 /μ∗f . Note that the integrand can also be evaluated at the fluid side by

applying the BCs (4a) and (4b). We note from (6) that the magnetic force contributes to the translational force butnot to the torque.

2.2 Formulation of the fluid-flow field

In the Stokes-flow regime, the fluid flow is governed by [20, Chap. 2]

∇ · u = 0, ∇ p = ∇2u, (7a,b)

where u is the fluid velocity vector scaled by a∗ B∗20 /(η

∗μ∗f ), p is the pressure scaled by B∗2

0 /μ∗f and η∗ is the fluid

viscosity.The magnetic force acting on a particle induces a particle translational motion. The motion of neighboring par-

ticles stirs the fluid, which again causes the particle to translate, rotate, or both via the fluid viscous effect. Thismeans that we must assume that, in general, the particles experience not only translation but also rotation. Let theparticle’s linear and angular velocities be up and ωp, respectively. The no-slip and impermeable BCs to be appliedat the walls of the container and the particles take the form

u = 0 on ∂�0, (8a)

u = up + ωps on ∂�i (i > 0), (8b)

where s is the unit tangential vector at the particle surface.The magnitudes of up and ωp to be used in Eq. 8b must be determined through a force balance for each particle.

Two forces participating in the force balance are the magnetic force given by Eq. 6 and the fluid force to be derivedin the following. We consider a small length element ds at the interface, as shown in Fig. 2. The infinitesimalfluid force acting on this surface element is dF f = (σnn + σss) ds, where σn and σs are the normal and tangentialcomponents of the stress acting on ds. Let Vn and Vs be the fluid velocity components in the normal and tangentialdirections, respectively. Then we have

F f =∫∂�i

[(−p + 2

∂Vn

∂n

)n +(∂Vs

∂n− Vs + ∂Vn

∂s

)s]

ds. (9)

2.3 Coupling the magnetic and flow fields

In the Stokes-flow regime and under the assumption of negligible particle inertia, the magnetic force Fm must beresisted by the fluid motion,

Fm = −F f . (10)

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Motion of paramagnetic particles in a viscous fluid 29

Substituting Eq. 9 for F f in (10), we obtain

Fm = −∫∂�i

[(−p + 2

∂Vn

∂n

)n +(∂Vs

∂n− Vs + ∂Vn

∂s

)s]

ds. (11)

Because the magnetic field does not exert a torque on the particle, we impose the torque-free condition∫∂�i

(∂Vs

∂n− Vs

)ds = 0. (12)

In summary, in order to determine up and ωp for a given Fm , Eqs. 11 and 12 must be solved in combination withthe other equations presented below.

To this end, we have two dimensionless parameters; μp, the particle’s dimensionless magnetic permeability, andR, the dimensionless radius of the outer boundary.

2.4 Motion of particles

The position of each particle, xp, is obtained from the time integration of the equation of motiondxp

dt= up. (13)

Since the particle position changes during the time evolution, all preceding equations must be solved at each timestep.

3 Method of solution

We introduce the conformal mapping zi = xi + iyi → wi = ξi + iηi for each local coordinate system, wherewi = log(zi ), so that the particle surface can be represented locally by ξi = constant. The relation between thelocal coordinates zi and the global coordinates z0 is

zi = z0 − zic, (14)

where zic denotes the coordinates of the ith particle center with the origin O. Then we arrive at the relationsdw j

dwi= exp(wi − w j ). (15)

We can derive the scale factors from these equations such as the real and imaginary parts of (15) corresponding to∂ξ j/∂ξi and ∂η j/∂ξi .

Methods of solution for the magnetic and flow fields are based on simple physical and mathematical reasoning.Suppose there are two particles marked as “A” and “B” in an unbounded domain, and we want to obtain the magneticfield caused by an external uniform field. As a first approximation, we can take the uniform field as the ‘primary’solution. Since the BCs on the surface of a particle, say particle “A”, may not be satisfied, it must generate anadditional (secondary) field to meet the BCs, which consequently corrects the primary one. However, the secondaryfield may not satisfy the BCs on the surface of particle “B”, which therefore must generate an additional field. Thisfield, when evaluated at the surface of particle “A”, must also generate a tertiary field, and so on. Although wehave described the scenario from the standpoint of particle “A”, the same procedure applies to particle “B”. Themost important property in this hierarchy is that the induced fields become increasingly weak as the number ofcorrections is increased. This implies that, in determining the correction functions for a particle, the effect of theother particles can be treated explicitly. As we will see below, the convergence is faster as the distance betweena pair of particles becomes larger. A similar scenario applies to the solution method for the flow field, but inthis case the various leading-order terms must be treated implicitly for numerical stability. Happel and Brenner[20, Chap. 6] proposed to use the so-called ‘method of reflections’ for solving low-Reynolds-number flow aroundtwo particles by using the coordinates attached to each particle. The present method is more general in that morethan two particles can be treated in a partially implicit manner.

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30 Y. K. Suh, S. Kang

3.1 Solution for the magnetic field

To solve the Laplace equation (2) in �0, we decompose A into three contributions,

A = y + A(0)(ξ0, η0)+N∑

i=1

A(i)(ξi , ηi ) for �0. (16)

The first term corresponds to the external uniform magnetic field applied to the system; the second term representsthe contribution from the outer boundary ∂�0; the third reflects the disturbance caused by all the particles. As thesolutions of (2) in �0, each of these terms can be expanded in a Fourier series,

A(0) =K∑

k=0

[T (0)1,k exp(kξ0) cos kη0 + T (0)2,k exp(kξ0) sin kη0

], (17a)

A(i) =K∑

k=0

[T (i)1,k exp(−kξi ) cos kηi + T (i)2,k exp(−kξi ) sin kηi

], (i > 0), (17b)

where T (0)1,k , T (0)2,k , T (i)1,k and T (i)2,k are coefficients to be determined.For the domain inside particle i , we assume the form

A = A(i)(ξi , ηi ) for �i (i > 0). (18)

These can also be expanded in a Fourier series,

A(i) =K∑

k=0

[T (i)1,k exp(kξi ) cos kηi + T (i)2,k exp(kξi ) sin kηi

], (i > 0), (19)

where T (i)1,k and T (i)2,k are coefficients to be determined. In what follows, the free index i runs over the range i =0, 1, 2, . . . , N , unless mentioned otherwise.

From the application of BC (3) we obtain a formula for determining the coefficients T (0)1,k and T (0)2,k . The BC (3)is equivalent to

A(0)(ξ0b, η0) = −N∑

i=1

[A(i)(ξi , ηi )

]∂�0, (20)

where ξ0b denotes the value of ξ0 at the outer boundary ∂�0. The RHS of (20) is a function of η0 and thus it canbe expanded in a Fourier series,

−N∑

i=1

[A(i)(ξi , ηi )

]∂�0

=K∑

k=0

[ν(0)1,k cos kη0 + ν

(0)2,k sin kη0

]≡ k

[ν(0)1,k, ν

(0)2,k; η0

],

wherek indicates the Fourier-series expansion written in terms of the index k with a pair of coefficients multiply-ing the cosine (i.e., ν(0)1,k) and sine (i.e., ν(0)2,k) functions of the variable shown after the semi-colon within the square

brackets (i.e., η0). Then, we can obtain the coefficients for A(0) from

T (0)1,k = ν(0)1,k exp(−kξ0b), T (0)2,k = ν

(0)2,k exp(−kξ0b). (21)

Henceforth, the free index k ranges over k = 0, 1, 2, . . . , K , unless stated otherwise. This means that, in obtainingthe coefficients for A(0), we treat the functions A(1)and A(2) etc. as known.

Next, we must apply the BCs (4a) and (4b) at each interface to obtain the coefficients for A(i) and A(i). In thebeginning of the study, we tried to apply these conditions separately. For instance, at the interface ∂�1 with twoparticles, we treated not only the coefficients for A(0) and A(2), but also those for A(1) to be known in applying BC(4b) to obtain the coefficients for A(1). This method was successful when the parameter μp was large enough, butdid not yield converged solutions when μp was close to 1, e.g. 1.3. To overcome this difficulty, we applied both

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Motion of paramagnetic particles in a viscous fluid 31

(4a) and (4b) to obtain the coefficients for A(1) and A(1) simultaneously. Similarly, we applied both BCs to obtainthe coefficients for A(2) and A(2) simultaneously.

We now explain the method for obtaining the coefficients T (i)1,k , T (i)2,k , T (i)1,k and T (i)2,k used in A(i) and A(i). Byapplying BC (4b) on ∂�i , we find[

−∂A(i)

∂n+ 1

μp

∂ A(i)

∂n

]∂�i

=⎡⎣∂y

∂n+

N∑j=0 �=i

∂A( j)

∂n

⎤⎦∂�i

, (i > 0), (22)

where∂

∂n= ∂

∂ξ j

(∂ξ j

∂ξi

)+ ∂

∂η j

(∂η j

∂ξi

),

for the second term on the RHS and ∂/∂n = ∂/∂ξi for all the others in Eq. 22. Each of the scale factors can beobtained from (15). We express the RHS of (22) as a Fourier series with coefficients ν(i)1,k and ν(i)2,k . Then we obtain

the following equations for the unknowns T (i)1,k , T (i)1,k , T (i)2,k and T (i)2,k :

T (i)1,k + 1

μpT (i)1,k = 1

kν(i)1,k, T (i)2,k + 1

μpT (i)2,k = 1

kν(i)2,k (k > 0, i > 0). (23)

Next, we consider BC (4a) equivalent to

[−A(i) + A(i)

]∂�i

=⎡⎣y +

N∑j=0 �=i

A( j)

⎤⎦∂�i

(i > 0). (24)

The RHS can also be expanded in a Fourier series with coefficients ν(i)1,k and ν(i)2,k . Then we have

− T (i)1,k + T (i)1,k = ν(i)1,k, −T (i)2,k + T (i)2,k = ν

(i)2,k (k > 0, i > 0). (25)

By coupling Eqs. 23 and 25, we derive the following formulas for the coefficients T (i)1,k , T (i)2,k , T (i)1,k and T (i)2,k :

T (i)1,k = ν(i)1,k/k − ν

(i)1,k/μp

1 + 1/μp, T (i)2,k = ν

(i)2,k/k − ν

(i)2,k/μp

1 + 1/μp(k > 0, i > 0), (26a)

T (i)1,k = ν(i)1,k/k + ν

(i)1,k

1 + 1/μp, T (i)2,k = ν

(i)2,k/k + ν

(i)2,k

1 + 1/μp(k > 0, i > 0). (26b)

For k = 0, the solution is indeterminate because the modes of k = 0 are not only contained in (17b) and (19), butalso in (17a) and they are not independent. In fact, we can impose an arbitrary value on either T (i)1,k or T (i)1,k and either

T (i)2,k or T (i)2,k . We take T (i)1,k = T (i)2,k = 0 so that

T (i)1,k = ν(i)1,k, T (i)2,k = ν

(i)2,k for k = 0 (i > 0). (27)

It turns out that this method provides us with solutions with much faster convergence.The fundamental reason for the success of such an explicit treatment of the contribution from all the other parti-

cles can be understood from the fact that all the modal functions of each mode show an exponential decrease withdistance from the particle center.

The magnetic force shown in (6) can be decomposed into two components,

Fmx = 1

2(1 − 1/μp)

2π∫0

[(∂ A

∂η

)2

p+ 1

μp

(∂ A

∂ξ

)2

p

]cos ηdη, (28a)

Fmy = 1

2(1 − 1/μp)

2π∫0

[(∂ A

∂η

)2

p+ 1

μp

(∂ A

∂ξ

)2

p

]sin ηdη, (28b)

123

32 Y. K. Suh, S. Kang

where the free index i has been omitted for simplicity. From Eq. 19 we derive (∂ A/∂ξ)p and (∂ A/∂η)p, andsubstitute the results in (28a) and (28b) to obtain

Fmx = 1

2(1 − 1/μp)

(G1x + 1

μpG2x

), Fmy = 1

2(1 − 1/μp)

(G1y + 1

μpG2y

), (29a,b)

where

G1x =K∑

k=0

πk

2

{T2,k[(k + 1)T2,k+1 + (k − 1)T2,k−1 + (1 − k)T2,1−k

]−T1,k

[−(k + 1)T1,k+1 − (k − 1)T1,k−1 + (1 − k)T1,1−k]},

G1y =K∑

k=0

πk

2

{T2,k[−(k + 1)T1,k+1 + (k − 1)T1,k−1 − (1 − k)T1,1−k

]−T1,k

[−(k + 1)T2,k+1 + (k − 1)T2,k−1 + (1 − k)T2,1−k]},

G2x =K∑

k=0

πk

2

{T1,k[(k + 1)T1,k+1 + (k − 1)T1,k−1 + (1 − k)T1,1−k

]+T2,k

[(k + 1)T2,k+1 + (k − 1)T2,k−1 − (1 − k)T2,1−k

]},

G2y =K∑

k=0

πk

2

{T1,k[(k + 1)T2,k+1 − (k − 1)T2,k−1 + (1 − k)T2,1−k

]+T2,k

[−(k + 1)T1,k+1 + (k − 1)T1,k−1 + (1 − k)T1,1−k]}.

Note that the coefficients T1,k , T2,k , T1,k and T2,k are zero for k < 0.

3.2 Solution for the flow field

Introducing the stream function ψ , we can write the governing equation for the flow field in biharmonic form:∇4ψ = 0. We seek a solution of this equation for N circular particles inside a cylindrical container in the form

ψ = C00 +N∑

i=0

ψ(i)(ξi , ηi ), (30)

where C00 is a constant to be determined and ψ(i)(ξi , ηi ) represents the stream function written in terms of thelocal coordinates centered on the particle i satisfying

∇2w

(exp(−2ξ)∇2

wψ)

= 0, (31)

where ∇2w = ∂2/∂ξ2 + ∂2/∂η2. The local function ψ(i)(ξi , ηi ) can be written in the form

ψ(i) =5∑

m=1

C (i)m q(i)m (ξi )pm(ηi )+ Q(i)(ξi , ηi ), (32)

where the function Q(i)(ξi , ηi ) is defined as follows:

Q(i)(ξi , ηi ) =K∑

k=2

⎧⎨⎩[

D(i)1,k g(i)1,k(ξi )+ D(i)

2,k g(i)2,k(ξi )]

cos kηi

+[

E (i)1,k g(i)1,k(ξi )+ E (i)2,k g(i)2,k(ξi )]

sin kηi

⎫⎬⎭. (33)

In Eqs. 32 and 33, the unknown coefficients to be determined are C (i)m , D(i)

1,k , D(i)2,k , E (i)1,k and E (i)2,k . The modal

functions q(i)m (ξ), g(i)1,k(ξ) and g(i)2,k(ξ) are given in Table 1. The modal functions pm(η) are defined as p1 = 1,

p2 = p3 = cos η, and p4 = p5 = sin η. Our target for the analysis is to obtain the unknown coefficients C00, C (i)m ,

D(i)1,k , D(i)

2,k , E (i)1,k and E (i)2,k from the given BCs to be specified on each of the walls of the cylinder and particles. Note

that BCs on the walls must be satisfied for the total stream function ψ and not for the local function ψ(i).

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Motion of paramagnetic particles in a viscous fluid 33

Table 1 Modal functionsused in the series solutionfor the stream-functionequation (32)

q(i)m , g(i)1,k and g(i)2,k i = 0 i ≥ 1

q(i)1 exp(2ξ) ξ

q(i)2 exp(ξ) ξ exp(ξ)

q(i)3 exp(3ξ) exp(−ξ)q(i)4 exp(ξ) ξ exp(ξ)

q(i)5 exp(3ξ) exp(−ξ)g(i)1,k exp(kξ) exp(−kξ)

g(i)2,k exp[(k + 2)ξ ] exp[(2 − k)ξ ]

The BCs on the ith particle are given in (8b). In terms of the normal and tangential components, we have∂ψ

∂ηi= u(i)px cos ηi + u(i)py sin ηi on ∂�i , (34a)

∂ψ

∂ξi= −u(i)py cos ηi + u(i)px sin ηi − ω(i)p on ∂�i , (34b)

where u(i)px and u(i)py denote the Cartesian components of the velocity vector u(i)p .We apply the BC (34a). We first need to evaluate on the ith particle wall the stream functions contributed by all

the other particles and write it as a Fourier series in terms of the local coordinate ηi . We do this separately for thelast term and each of the first terms on the RHS of (32). For each of the first terms, we set[

q( j)m (ξ j )pm(η j )

]∂�i

= k

[α( j,i)m,k , β

( j,i)m,k ; ηi

]. (35a)

As with the free index i , the free index j also varies over j = 0, 1, 2, . . . , N , unless specified otherwise. The freeindex m covers the range m = 1, 2, . . . , 5. For the last term of (32) we set[

Q( j)(ξ j , η j )]∂�i

= k

[κ( j,i)k , λ

( j,i)k ; ηi

]. (35b)

Then we have

[ψ]∂�i= C00 +

5∑m=1

[C (i)

m q(i)m (ξib)pm(ηi )]

+ Q(i)(ξib, ηi )+K∑

k=0

N∑j=0, j �=i

[κ( j,i)k cos kηi + λ

( j,i)k sin kηi

+5∑

m=1

(α( j,i)m,k cos kηi + β

( j,i)m,k sin kηi

)C ( j)

m

], (36)

where the last term on the RHS corresponds to the contribution from all the other stream functions and ξib denotesthe value of ξi at the surface of the particle i ; i.e., ξ0b = log(R) and ξib = 0 for i = 1, 2, . . .. The terms of the zerothmode in (36) will vanish if we take ∂/∂ηi in order to apply BC (34a). Simply applying (34a) to the zeroth modeterms is useless. Instead of applying (34a) to these terms, we utilize them in calculating ψ(i)w (the stream functionaveraged over the particle) as

ψ(i)w = C00 + C (i)1 q(i)1 (ξib)+

N∑j=0, j �=i

[κ( j,i)0 +

5∑m=1

α( j,i)m,0 C ( j)

m

]. (37)

The wall stream function is constant when the particle velocity is zero. On the other hand, we can set an arbitraryvalue for the stream function on the cylinder wall (i.e., the outer boundary ∂�0), where both the linear and rotatingvelocities are set to zero. So, we set ψ = 0 at ∂�0 and the resulting equation

C00 + q(0)1 (ξ0b)C(0)1 +

N∑j=1

5∑m=1

α( j,0)m,0 C ( j)

m = −N∑

j=1

κ( j,0)0 (38)

123

34 Y. K. Suh, S. Kang

is considered as one element of the linear system of equations to be solved for the unknown coefficients. We nowapply ∂/∂ηi to (36) and impose BC (34a). Collecting the terms having sin ηi and cos ηi , respectively, yields

q(i)2 (ξib)C(i)2 + q(i)3 (ξib)C

(i)3 +

N∑j=0, j �=i

5∑m=1

α( j,i)m,1 C ( j)

m + u(i)py = −N∑

j=0, j �=i

κ( j,i)1 , (39a)

q(i)4 (ξib)C(i)4 + q(i)5 (ξib)C

(i)5 +

N∑j=0, j �=i

5∑m=1

β( j,i)m,1 C ( j)

m − u(i)px = −N∑

j=0, j �=i

λ( j,i)1 . (39b)

Similarly, the terms of cos kηi and those of sin kηi give, respectively,

g(i)1,k(ξib)D(i)1,k + g(i)2,k(ξib)D

(i)2,k = −

N∑j=0, j �=i

(κ( j,i)k +

5∑m=1

α( j,i)m,k C ( j)

m

), (40a)

g(i)1,k(ξib)E(i)1,k + g(i)2,k(ξib)E

(i)2,k = −

N∑j=0, j �=i

(λ( j,i)k +

5∑m=1

β( j,i)m,k C ( j)

m

), (40b)

Next we consider BC (34b). First we write

∂ψ

∂ξi= ∂ψ(i)

∂ξi+

N∑j=0, j �=i

[∂ψ( j)

∂ξ j

(∂ξ j

∂ξi

)+ ∂ψ( j)

∂η j

(∂η j

∂ξi

)], (41)

where the metrics can be evaluated from(∂ξ j/∂ξi

) = exp(ξi − ξ j ) cos(ηi − η j ) and(∂η j/∂ξi

) = exp(ξi −ξ j ) sin(ηi −η j ). The term within the square brackets on the RHS of (41) (to be referred to as I I ( j)) can be obtainedfrom

I I ( j) =5∑

m=0

[q( j)′

m (ξ j )pm(η j )(∂ξ j/∂ξi )+ q( j)m (ξ j )p′

m(η j )(∂η j/∂ξi )]

C ( j)m

+(∂Q( j)/∂ξ j )(∂ξ j/∂ξi )+ (∂Q( j)/∂η j )(∂η j/∂ξi ).

We evaluate this on ∂�i separately for the first and the other terms on the RHS and expand the results in the Fourierseries to obtain

q( j)′m (ξ j )pm(η j )(∂ξ j/∂ξi )+ q( j)

m (ξ j )p′m(η j )(∂η j/∂ξi ) = k

[α( j,i)m,k , β

( j,i)m,k ; ηi

], (42a)

(∂Q( j)/∂ξ j )(∂ξ j/∂ξi )+ (∂Q( j)/∂η j )(∂η j/∂ξi ) = k

[κ( j,i)k , λ

( j,i)k ; ηi

]. (42b)

Now we apply BC (34b). For the zeroth mode, we have

q(i)′1 (ξ0b)C(i)1 +

N∑j=0, j �=i

5∑m=1

α( j,i)m,0 C ( j)

m + ω(i)p = −N∑

j=0, j �=i

κ( j,i)0 . (43)

Similarly, the modes of cos ηi and sin ηi yield

q(i)′2 (ξib)C(i)2 + q(i)′3 (ξib)C

(i)3 +

N∑j=0, j �=i

5∑m=1

α( j,i)m,1 C ( j)

m + u(i)py = −N∑

j=0, j �=i

κ( j,i)1 , (44a)

q(i)′4 (ξib)C(i)4 + q(i)′5 (ξib)C

(i)5 +

N∑j=0, j �=i

5∑m=1

β( j,i)m,1 C ( j)

m − u(i)px = −N∑

j=0, j �=i

λ( j,i)1 , (44b)

and those of cos kηi and sin kηi yield

g(i)′1,k (ξib)D(i)1,k + g(i)′2,k (ξib)D

(i)2,k = −

N∑j=0, j �=i

(κ( j,i)k +

5∑m=1

α( j,i)m,k C ( j)

m

), (45a)

g(i)′1,k (ξib)E(i)1,k + g(i)′2,k (ξib)E

(i)2,k = −

N∑j=0, j �=i

(λ( j,i)k +

5∑m=1

β( j,i)m,k C ( j)

m

). (45b)

123123

Motion of paramagnetic particles in a viscous fluid 35

3.3 Coupling the magnetic and flow fields

In terms of the coordinates (ξ, η), the components of the fluid force (9) can be written as

F f x =2π∫

0

[(−p + 2

∂Vn

∂ξ

)cos η −

(∂Vn

∂η+ ∂Vs

∂ξ− Vs

)sin η

]dη, (46a)

F f y =2π∫

0

[(−p + 2

∂Vn

∂ξ

)sin η +

(∂Vn

∂η+ ∂Vs

∂ξ− Vs

)cos η

]dη. (46b)

After some algebra, we arrive at the following simplified formula (see Appendix for the derivation).

F f x =2π∫

0

(∂ζ

∂ξ− ζ

)sin ηdη, F f y =

2π∫0

(∂ζ

∂ξ− ζ

)cos ηdη, (47a,b)

where ζ is the vorticity, ζ = − exp(−2ξ)∇2wψ . We can calculate the vorticity in terms of the coordinates (ξi , ηi )

as the sum of the three contributions,

ζ(ξi , ηi ) = ζ (i) + ζ (0) +N∑

j=1 �=i

ζ ( j), (48)

where the first term on the RHS is already defined in terms of the coordinates (ξi , ηi ). The other terms must beconsidered as a transformation from their respective ones (ξ j , η j ) for j = 0, 1, . . . , N ( j �= i). Then we can obtainthe following formulas as the force balance for the particle i .

F (i)mx/2π = −2C (i)4 + 4

(β(0,i)2,1 − β

(0,i)2,1

)C (0)

3 + 4(β(0,i)4,1 − β

(0,i)4,1

)C (0)

5

+N∑

j=1 �=i

[(β( j,i)3,1 − β

( j,i)3,1

)C ( j)

2 +(β( j,i)5,1 − β

( j,i)5,1

)C ( j)

4

]+ 2

N∑j=0 �=i

(δ( j,i)1 − δ

( j,i)1

)(i > 0), (49a)

F (i)my/2π = 2C (i)2 − 4

(α(0,i)2,1 − α

(0,i)2,1

)C (0)

3 − 4(α(0,i)4,1 − α

(0,i)4,1

)C (0)

5

−N∑

j=1 �=i

[(α( j,i)3,1 − α

( j,i)3,1

)C ( j)

2 +(α( j,i)5,1 − α

( j,i)5,1

)C ( j)

4

]− 2

N∑j=0 �=i

(γ( j,i)1 − γ

( j,i)1

)(i > 0). (49b)

Here γ ( j,i)k and δ( j,i)

k are the Fourier coefficients of the function P( j,i),

P( j,i) = k

[γ( j,i)k , δ

( j,i)k ; ηi

], (50)

where P( j,i) denotes the function

P( j) =

⎧⎪⎪⎨⎪⎪⎩

K∑k=2

(k + 1) exp(kξ j )(

D( j)2,k cos kη j + E ( j)

2,k sin kη j

)for j = 0

K∑k=2

(1 − k) exp(−kξ j )(

D( j)2,k cos kη j + E ( j)

2,k sin kη j

)for j ≥ 1

(51)

evaluated at the surface of the particle i . Similarly γ ( j,i)k and δ( j,i)

k are the Fourier coefficients of ∂P( j)/∂ξi evaluatedat the surface of the particle i , i.e.,[(∂P( j)/∂ξ j )(∂ξ j/∂ξi )+ (∂P( j)/∂η j )(∂η j/∂ξi )

]∂�i

= k[γ ( j,i)k , δ

( j,i)k ; ηi ]. (52)

The magnetic-force components Fmx and Fmy in (49a) and (49b) are given in (29a) and (29b), respectively. Thetorque-free condition on each particle, Eq. 12, can be written in the following form:

123

36 Y. K. Suh, S. Kang

2π∫0

(ζ + 2

∂ψ

∂ξ

)dη = 0.

We consider three contributions for the vorticity, and further replace the term ∂ψ/∂ξ by (34b) to obtain

2C (0)1 + 4α(0,i)2,0 C (0)

3 + 4α(0,i)4,0 C (0)5 +

N∑j=1 �=i

(α( j,i)3,0 C ( j)

2 + α( j,i)5,0 C ( j)

4

)+ ω(i)p + 2

N∑j=0 �=i

γ( j,i)0 = 0 (i > 0). (53)

3.4 Numerical methods

The motion of particles is calculated each step at a time with the following sequence. First, the magnetic field iscalculated; second, the magnetic force is obtained; next, the flow field as well as the particle velocity componentsare obtained; finally, the new positions of the particles are calculated. This sequence is repeated at every time stepuntil a destined time is reached.

When solving the magnetic flow field, the flow field is inconsequential. We use (26a) and (26b) to obtain thecoefficients T (i)1,k , T (i)2,k , T (i)1,k and T (i)2,k for each particle. Then the magnetic-force components are obtained from (29a)and (29b).

The linear system of equations for the flow field involving the unknown coefficients C00 and C (i)m and the

unknown velocity components u(i)px , u(i)py and ω(i) includes (38), (39a), (39b), (43), (44a), (44b), (49a), (49b) and(53); the total number of unknowns (and so the total number of equations) is 8N + 6. In the computer program,we include the trivial unknowns u(0)px , u(0)py and ω(0), which are obviously zero, just for convenience of the indexing.

The total number of unknowns is now 8N + 9. The unknown coefficients D(i)1,k and D(i)

2,k for each k are obtained

from (40a) and (45a), and E (i)1,k and E (i)2,k from (40b) and (45b). In this system, the coefficients C ( j)m on the RHS are

treated as known during the iteration process. Similarly, the coefficients D(i)1,k , D(i)

2,k , E (i)1,k and E (i)2,k are also treatedas known in solving the linear system of equations addressed above; that is, they are treated as known in obtainingthe coefficients α( j,i)

m,k , β( j,i)m,k , α( j,i)

m,k , β( j,i)m,k , κ( j,i)

k , λ( j,i)k , κ ( j,i)

k and λ( j,i)k . The numerical procedure for determining

the flow field at a time step can be summarized as follows:

(i) obtain the Fourier coefficients α( j,i)m,k , β( j,i)

m,k , α( j,i)m,k and β( j,i)

m,k from (35a) and (42a).(ii) find the coefficients of the linear system of equations.

(iii) obtain the Fourier coefficients κ( j,i)k , λ( j,i)

k , κ ( j,i)k and λ( j,i)

k from (35b) and (42b).(iv) solve the linear system of equations by Gauss elimination.(v) solve (40a) and (45a) for the unknowns D(i)

1,k and D(i)2,k , and (40b) and (45b) for the unknowns E (i)1,k and E (i)2,k .

(vi) check for convergence and repeat the process (iii)–(v) until the convergence is attained.

The convergence criterion in step (vi) is set as max∣∣∣ϕ(new)

k − ϕ(old)k

∣∣∣ < εlim, where ϕk stands for the unknown

coefficients determined in step (v). In this study, the error limit εlim varies in the range 10−7–10−5 depending onthe level of numerical accuracy required.

Aside from the physical parameters, such as R, μp, N , and initial positions of particles, we have simulationparameters, i.e. I , the number of panels used for the Fourier transformation of the function evaluated on the walls,and K , the number of the Fourier modes included in the simulation. From a preliminary study with a pair of parti-cles, we have confirmed that in most simulation cases, the parameter set I = 65 and K = 16 is enough to providereasonable accuracy. When particles come close together, we need to increase these parameter values to maintainthe level of accuracy. The convergence property of the series solutions will be addressed in Sect. 4.3.

123123

Motion of paramagnetic particles in a viscous fluid 37

4 Validation

In this section, we demonstrate the validation of the method described so far separately for the fluid-flow (Sect. 4.1)and the magnetic-field problem (Sect. 4.2).

4.1 Pair of particles with small gap—validation of flow solutions

To validate the method and confirm the formulas derived in this paper regarding the fluid-flow calculations, weconsider an asymptotic case where two cylinders aligned along the x-axis with a small distance 2d in the absenceof an external magnetic field advance toward each other with velocity u0, as shown in Fig. 3. For such a narrowgap, we can obtain an analytic solution for the ratio between the particle velocity and the fluid force. The key ideais to replace the circles with parabolas (Fig. 3). In this approximation, we assume that the pressure rise in the regionbetween the gap and the shear stress on the facing surfaces caused by the particles’ advancing motion is responsiblefor the fluid forces. Furthermore, it may be assumed that the effect of these factors should be more prevalent in thenarrower region than in the wider region in the upper or lower spaces. Therefore, approximating the circle with aparabola is asymptotically correct as long as the gap remains small. We assume that the vertical velocity componenttakes the form

v = v0(y)[1 − (x/xe)

2], (54)

where v0(y) is the velocity magnitude at x = 0 and

xe = d + y2/2

denotes the parabola surface. From the continuity equation, we derive

v0(y) = 3u0

2

y

d + y2/2. (55)

Now, we apply the lubrication approximation to the x-momentum equation

∂p

∂y= ∂2v

∂x2 . (56)

x

y 21

2x d y= +

d

0u0u

0u

v

0u

1

Fig. 3 Illustration of the approximation of circles by a pair of parabolas for deriving an analytic solution for the lubrication approximation

123

38 Y. K. Suh, S. Kang

Fig. 4 Comparisonbetween the asymptoticsolution (solid line) andresult obtained with thepresent method (symbols)for the ratio of the particlevelocity and the fluid forceacting on the particles. Thenumerical results areobtained with parametervalues R = 200,μp = 200, I = 129 andK = 32

log10(d)

log 10

(u0

/Ffx

)

-2 -1.5 -1 -0.5

-3.5

-3

-2.5

-2

-1.5

Substituting Eq. 54 in the RHS of (56), applying (55) for v0 and integrating the result over −∞ < y < ∞, we get

p = 3u0

2(d + y2/2)2,

where the pressure is taken zero at infinity. Integrating this once more, we get the pressure force

F f xp =∞∫

−∞pdy = 3π

√2u0

4d3/2 . (57)

The contribution of the wall shear stress τw = (−∂v/∂x) on the surface x = xe to the force is considered next.The x-component of the shear force reads

F f xs =∞∫

−∞τwdx . (58)

Substituting (54) and (55) again in (58) and integrating the result yields

F f xs = 3π√

2u0

d1/2 . (59)

Comparing (57) and (59), we see that the contribution of the pressure is dominant over that of the wall shear stress.The total force is then

F f x = F f xp + F f xs = 3π√

2u0

4d3/2 (1 + 4d). (60)

Finally we derive the ratio of the velocity u0 and the force F f x as follows:

u0/F f x = 4d3/2

3π√

2(1 + 4d). (61)

Figure 4 shows numerical results for the ratio u0/F f x obtained from the present multiple-coordinate method andthat given by the asymptotic solution, i.e., (61). The asymptotic solution is remarkably close to the numerical solu-tions and the agreement is better for narrower gaps, as expected. This not only validates our code for the flow-fieldcalculation, but also guarantees the accuracy of our numerical solutions, even for such small gaps.

123123

Motion of paramagnetic particles in a viscous fluid 39

4.2 Bipolar-coordinate method—validation of magnetic-field solutions

The first step for validating our code applies for the flow-field calculation. As a second step for the validation ofour full code, we now compare the magnetic force obtained from our code with that obtained from another methodfor a pair of particles subjected to horizontal or vertical magnetic field. For this we use the bipolar coordinatesshown in Fig. 5. First, we introduce the transformed plane (α, β) defined as x = c sinh α/(cosh α − cosβ) andy = c sin β/(cosh α − cosβ), where c is a constant controlling the gap between a pair of particles of a givensize. The mapping (x, y) → (α, β) is conformal, so that the Laplace equation (2) governing the magnetic vectorpotential is invariant and can be written in terms of the transformed coordinates (α, β). As the solution for the firstcase where the externally applied magnetic field is along the x-direction, we assume the form

Ao = y +∞∑

k=1

Cok cosh kα sin kβ (62)

for the magnetic vector potential in the fluid domain �o and

Ai =∞∑

k=1

Cik exp(−kα) sin kβ (63)

for the magnetic vector potential within the solid domain�i . Note that the superscripts ‘o’ and ‘i’ here indicate the‘outer’ and ‘inner’ sides, respectively. The boundary conditions are the same as before;

Ao = Ai at α = α0, (64a)

∂Ao

∂α= 1

μp

∂Ai

∂αat α = α0, (64b)

where α0 denotes the coordinate of α on the surface of the particle. The gap between the two particles is denotedby 2d, and α0 is given by

α0 = cosh−1(1 + d) = log(

1 + d +√

d2 + 2d).

x

y

0αα =0αα −=

0=α

0=βπβ =0=β

2/πβ =

Fig. 5 A bipolar coordinate system used for deriving a second analytic solution for a pair of particles

123

40 Y. K. Suh, S. Kang

The parameter c is determined from the condition that the particle radius should be unity in dimensionless variables,i.e., c = sinh α0.

From the BCs (64a) and (64b) we obtain the coefficients

Cok = −bk + μpbk/k

cosh kα0 + μp sinh kα0, (65a)

Cik =

μp exp(kα0)[bk sinh kα0 + (bk/k) cosh kα0

]cosh kα0 + μp sinh kα0

, (65b)

where bk and bk are the Fourier coefficients of y and ∂y/∂α, respectively, which are odd functions of β whenevaluated at the particle surface. It can be shown that

bk = 2 exp(−kα0) sinh α0, bk = kbk . (66)

Substituting (66) in (65a) and (65b) we obtain

Cok = 2(μp − 1) exp(−kα0)

cosh kα0 + μp sinh kα0, Ci

k = 2μp exp(kα0) sinh α0

cosh kα0 + μp sinh kα0.

The magnetic-force is then obtained from (6):

Fmx = 1

2c(1 − 1/μp)

2π∫0

[(∂Ai

∂β

)2

p+ 1

μp

(∂Ai

∂α

)2

p

](cosh α0 − cosβ) cos θdβ,

where θ measures the angle between the radial line of point (α0, β) on the particle surface and the x-axis. Since theparticles are aligned with the x-axis, the y-component of Fm is zero. By using the relation x = 1 + d + cos θ =c sinh α0/(cosh α0 − cosβ), we can write

Fmx = 1

2c(1 − 1/μp)

2π∫0

[(∂Ai

∂β

)2

p+ 1

μp

(∂Ai

∂α

)2

p

][c sinh α0 − (1 + d)(cosh α0 − cosβ)] dβ. (67)

Substitution of (63) in (67) and performing some algebraic manipulations leads to the following formula for themagnetic force,

Fmx = π

2c

(1 − 1/μ2

p

) ∞∑k=1

{k(k + 1)(1 + d)Ci

kCik+1 exp[−(2k + 1)α0] − [kCi

k exp(−kα0)]2}. (68)

It turns out that Fmx tends to a finite value for d → 0. It can be shown that the asymptotic formula of Fmx underthis limit takes the form (see Appendix for the derivation):

Fmx = π

⎡⎣(μ2

p − 1) ∞∫

0

f 2(x)dx − 1

3(μp − 1 + ν)

⎤⎦ , (69)

where f (x) = x/(cosh x + μp sinh x), ν = ∑∞k=1 λ

k/k2 and λ = (μp − 1)/(μp + 1). It is known that ν =π2/12 − (log 2)2/2 = 0.5822 . . . for λ = 1/2 and ν = π2/6 = 1.645 . . . for λ = 1.

In the independent limit μp → ∞, it is shown that f → x/(μp sinh x) and∫∞

0 f 2(x)dx → π2/6. Thus we canstate:

Fmx → −π3

(μp + π2

3− 1

). (70)

Similar work can be done for the case where the external magnetic field is applied in the y-direction. In that case,we set

Ao = x +∞∑

k=0

Dok sinh kα cos kβ (71)

for the magnetic vector potential within the fluid domain �o and

123123

Motion of paramagnetic particles in a viscous fluid 41

Ai =∞∑

k=0

Dik exp(−kα) cos kβ (72)

for the magnetic potential within the solid domain �i . Applying the BCs on the particle surface we obtain theformulas for the coefficients,

Dok = 2(μp − 1) exp(−kα0)

sinh kα0 + μp cosh kα0, Di

k = 2μp exp(kα0) sinh α0

sinh kα0 + μp cosh kα0.

The formula for the magnetic force also takes the form (68), except that the coefficients Cik must be replaced by Di

k .The asymptotic formula of the magnetic force in the limit d → 0 with the vertical magnetic field can be obtained

in a manner similar to that for the horizontal field. The result is

Fmx = π

⎡⎣(μ2

p − 1) ∞∫

0

g2(x)dx − (1 + γ )μp − 1

3μp

⎤⎦ , (73)

where g = x/(sinh x + μp cosh x) and γ = ∑∞k=1 (−1)k+1λk/k2. In the independent limit μp → ∞, we have

g → x/(μp cosh x), λ → 1 and∫∞

0 g2(x)dx → π2/(12μ2p), γ → π2/12. Thus we have

Fmx → π(π2 − 6)/18 = 0.67537 . . . . (74)

For the case when the external magnetic force acts in an arbitrary direction, the magnetic vector potential canbe obtained by the linear sum of AH , the potential solution for the purely horizontal field, and AV , the potentialfor the purely vertical field; A = mx AH + my AV , where mx = cos θm and my = sin θm are direction cosines ofthe external field actuation inclined by the angle θm with respect to the x-axis. The magnetic force is given by thenonlinear operator, Eq. 6. After some algebraic work, we obtain the magnetic-force components

Fmx = π

2c(1 − 1/μ2

p)

∞∑k=1

⎧⎨⎩

k(k + 1)(1 + d)(

m2x Ci

kCik+1 + m2

y Dik Di

k+1

)exp[−(2k + 1)α0]

−[k2(

m2x (C

ik)

2 + m2y(D

ik)

2)

exp(−2kα0)]

⎫⎬⎭, (75a)

Fmy = π

2(1 − 1/μ2

p)mx my

∞∑k=1

{k(k + 1)

(Ci

k+1 Dik − Ci

k Dik+1

)exp[−(2k + 1)α0]

}. (75b)

Note that the vertical component Fmy vanishes for either the horizontally acting external field, my = 0, or the ver-tically acting field, mx = 0. The vertical component Fmy becomes maximum at mx = my = 1/

√2, corresponding

to θm = π/4.Table 2 and Fig. 6 show data for the magnetic force for the horizontal and vertical alignments of the external

field obtained with the bipolar-coordinate method, compared with those given from the present multiple-coordi-nate method. We see that the agreement is almost perfect. This implies again that the present multiple-coordinate

Table 2 Magnitude of the magnetic-force component Fmx for a horizontal and vertical magnetic field applied to a pair of particlesaligned horizontally

d Fmx (horizontal;bipolar coord)

Fmx (horizontal;multiple-coord)

Fmx (vertical;bipolar coord)

Fmx (vertical;multiple-coord)

0.01 −16.663 −16.651 0.65916 0.65910

0.05 −5.9592 −5.9584 0.63116 0.63111

0.1 −3.4751 −3.4746 0.59604 0.59599

0.2 −1.8438 −1.8436 0.52578 0.52574

0.4 −0.85238 −0.85229 0.39541 0.39538

1 −0.22486 −0.22484 0.16644 0.16642

2 −0.06060 −0.06059 0.05376 0.05375

The data are obtained working in bipolar coordinates and from the multiple-coordinate method at μp = 200 and R = 200

123

42 Y. K. Suh, S. Kang

d

Fm

x

0 0.5 1 1.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

vertical magnetic field

horizontal magnetic field

Fig. 6 Numerical results of the magnetic-force componentsFmx obtained from the present multiple-coordinate method(symbols) in comparison with those given by the bi-polar coor-dinate method (solid lines) for horizontal and vertical arrange-ment of the external magnetic field for a pair of particles atR = 200 and μp = 200

log(d)

log(

-Fm

x)

-15 -10 -5 0

-4

-2

0

2

4

6asymptotic value

analyticalresult

Fig. 7 The magnetic force obtained for the horizontallyapplied magnetic field with R = 200 and μp = 200 for a pairof particles separated by a small gap. The asymptotic valueshown in this figure is calculated from Eq. 70

method is robust, accurate and reliable. In particular, we see that the maximum differences are less than 0.1%.We also see that the attracting force for the horizontal alignment apparently tends to be infinite as d → 0. How-ever, as predicted by the asymptotic analysis, i.e. (69) and (70), it should tend to a finite value. For μp = 200,Eq. 70 predicts Fmx = −212 in the limit d → 0. So we need to confirm the asymptotic formula at smaller valuesof d.

Figure 7 shows the magnetic force obtained with the horizontal magnetic field versus the gap distance on alog-log scale obtained by using the analytic formula (68) together with the asymptotic value −212 obtained from(70). The graph reveals that the magnetic force indeed approaches a finite value in the limit d → 0. However, thelimit value can be realized only when d remains below a critical value, which is as small as 5 × 10−6 for μp = 200.Note that both the critical value of d−1 and the asymptotic value of |Fmx | increase as μp increases.

Figure 8 presents data on the magnetic force versus the value of μp for the horizontal field with d = 10−8.The numerical result obtained from (68) approaches the asymptotic values predicted by formula (70) in the limitμp → ∞, which lends further support to the validity of the formula.

4.3 Convergence

Figure 9 illustrates the convergence of the series solution for the flow field given by Eqs. 32 and 33. Presented in

this figure are data of∣∣∣E (1)2,k

∣∣∣ obtained for the case of a pair of particles arranged horizontally as in Fig. 5. The coeffi-

cients show exponential decay for increasing values of k. The parameter μp exerts no influence on the convergenceproperty. As expected, when the gap d is decreased, convergence becomes slower, and K must be increased toensure a level of numerical accuracy.

123123

Motion of paramagnetic particles in a viscous fluid 43

log (μp)

log

(-F

mx

)

0 2 4 6 8-4

-2

0

2

4

6

8

formula formoderate μp

asymptotic theoryfor large μp

Fig. 8 Magnetic force obtained at d = 10−8 from the full equa-tion (68) (solid line) or given from the asymptotic formula (70)for d = 0 valid in the limit μp → ∞ (dashed line)

k

⎟E

2,k(1

)⎢

0 5 10 15 20 25 3010-5

10-4

10-3

10-2

10-1

d=0.02, μp=5

d=0.02, μp=200

d=0.1, μp=200

d=0.02, μp=2

d=0.5, μp=200

Fig. 9 Distribution of∣∣∣E (1)2,k

∣∣∣ representing the typical conver-

gence property of the series solution (32) obtained for the two-particle problem with R = 200 and various combinations of dand μp . The results are obtained with I = 129 and K = 32

5 Numerical results and discussion

We present numerical results for the magnetic field, the flow field and the particle motion, each obtained in differentsituations. The results of the magnetic field given in the following section are obtained for a pair of stationaryparticles in a point-symmetric configuration. The results of the flow field given in Sect. 5.2 are obtained for two orthree particles located in an arbitrary configuration with assigned linear and rotational motions in the absence ofany external magnetic field. The results of the particle motion given in Sect. 5.3 are obtained for multiple particlesin the presence of a uniform magnetic field.

5.1 Magnetic field

Simulations in a range of parameter values for a pair of motionless particles initially located at (x p, yp) =±(rp cos θp1, rp sin θp1), i.e., with point-symmetric configuration, have been performed to check the scheme’srobustness in terms of the magnetic-field calculation and to investigate the influence of the basic parameters. Hereθp1, in the range 0 ≤ θp1 ≤ π/2, is the angle of the radial line of the center of the particle at the first quadrantmeasured from the x-axis, and rp is half the gap distance, rp = d/2. As an example, when R = 10, μp = 2,d = 0.04 and θp1 = π/4, we obtain A = 0.75224 at the point (x, y) = (−0.5, 0.5), with K = 16 and I = 65.Halving K and I gives A = 0.75226 at the same point, four digits being unchanged. In most cases shown in thissection, we set I = 65 and K = 16 for the simulation, unless otherwise stated.

Figure 10 reveals the effect of the parameter μp on the magnetic-potential distribution. First, we note that thegradient of the magnetic potential is almost uniform within the particles regardless of μp. For larger values ofμp, the potential lines in the exterior of the particles tend to meet the particle interfaces orthogonally. This can beunderstood from the BC (4b). Suppose that the normal gradient ∂A/∂n takes a moderate value on the particle sideof the boundary. Then we can see from (4b) that the normal gradient must tend to zero in the limit μp → ∞ at thefluid side of the boundary.

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44 Y. K. Suh, S. Kang

x

y

-3 -2 -1 0 1 2 3

-2

0

2

x

y

-3 -2 -1 0 1 2 3

-2

0

2

(a) (b)

Fig. 10 Distribution of the magnetic vector potential for a pair of particles obtained for parameter set R = 10, rp = 1.5 and θp1 = π/4at two different values of μp; a μp = 2 and b μp = 200

Next, we investigate the effect of the particle position. As shown in Fig. 11, at lower values of μp (Fig. 11a, b)the field tends to be uniform not only within the particles but also in the fluid domain, and thus the effect of μp ismore prevalent for larger values of μp. We can see from the results with the horizontal arrangement at μp = 2000(Fig. 11c) that iso-potential lines are more clustered along the facing side than on the outer side. As will be shown,this causes an attractive force. We first note that (∂A/∂s) and (∂A/∂n) are of the same order of magnitude in theinterior of the particles. Therefore, in the limitμp → ∞, the second term within the square brackets in the integrandon the RHS of Eq. 6 is negligibly small compared with the first term. Then (6) can be approximated as

Fm = 1

2

∫∂�

(∂A

∂s

)2

n(s)ds.

Clustering of iso-potential lines at the interface indicates a stronger magnetic force acting along the normal direc-tion there. Therefore, the particles must experience an attracting force in the horizontal arrangement. In the verticalarrangement, the particles facing sides show a more uniform distribution of A than on the other sides, as shown inFig. 11d. Therefore, the particles must repel each other in the vertical arrangement.

The magnetic force field is next obtained for the various particle positions with a gap greater than 0.2. Figure 12shows the magnetic-force-field lines (lines tangentially connecting the magnetic-force vectors) obtained for differ-ent μp values. This reveals that the magnetic-force vector field is almost invariant of the parameter μp. Of course,the magnetic-force strength depends on μp; as μp approaches unity, the strength decreases, as it should.

5.2 Flow field

We present typical flow solutions in order to check the physical relevance of the results. In this calculation, we donot consider the magnetic field but a simple compulsory linear and rotating motion of the particles to be used asthe BCs on the interfaces. In most cases shown in this and next subsections, we set I = 129 and K = 32 for thesimulation, unless otherwise stated.

The first example is the case of two particles located at (2.5, 0) and (0, 5), the former particle having linearmotion and the latter (counter-clockwise) rotating motion. The streamlines shown in Fig. 13a are for the first parti-cle moving outward, while the ones in Fig. 13b are given with the first particle moving inward. Depending on thedirection of the linear motion, the streamline pattern takes a different shape; the first case shows two recirculatingflows while the second one shows a three-cell structure due to the counteractive effect of the two particles’ BCs inthe intermediate space between the two.

123123

Motion of paramagnetic particles in a viscous fluid 45

x

y

-3 -2 -1 0 1 2 3

-2

0

2

(a)

x

y

-3 -2 -1 0 1 2 3

-2

0

2

(b)

x

y

-3 -2 -1 0 1 2 3

-2

0

2

(c)

x

y

-3 -2 -1 0 1 2 3

-2

0

2

(d)

Fig. 11 Magnetic-potential distribution around a pair of particles with parameter set R = 10 and rp = 1.5; a θp1 = 0, μp = 2,b θp1 = π/2, μp = 2, c θp1 = 0, μp = 2000, d θp1 = π/2, μp = 2000

As a second example, we consider three particles; the first particle is restricted by a linear (upward) motion, thesecond by the (clockwise) rotating motion, and the third particle is stationary. The results in Fig. 14 show that thestreamline pattern for particles are separated from each other by rather long distances. Due to the upper stationaryparticle, the flow around that particle is weak.

When particles come closer, it takes many iterations to arrive at the steady state. The results shown in Fig. 15afor a close arrangement took 300 iterations to achieve the maximum residual 10−5. This clearly reveals the truephysics even for such a very close arrangement. The number of iterations is further increased when the radius ofthe outer cylinder is shrunk. The result shown in Fig. 15b for R = 3 is obtained after 500 iterations having 10−5

residual (increment of ψ in Fig. 15b is half of that in Fig. 15a).

5.3 Particle motion

In this section we report on the motional trajectories of particles caused by the magnetic forces determined from(29a) and (29b). Most results presented in this section are obtained for particles located initially on a circle withequal spacing in the circumferential direction; that is, the initial position of the particle i is set at

x (i)p = rp cos[θp1 + (i − 1)�θp

], y(i)p = rp sin

[θp1 + (i − 1)�θp

],

where rp is the radius of the circle, �θp = 2π/N is the angular spacing and θp1 represents the first particle’sangular position.

123

46 Y. K. Suh, S. Kang

x

y

1 2 30

1

2

3(a)

x

y

1 2 30

1

2

3(b)

x

y

1 2 30

1

2

3(c)

x

y

-4 -2 0 2 4

-4

-2

0

2

4(d)

Fig. 12 Magnetic-force-field lines for a pair of particles located anti-symmetrically with respect to the origin of the coordinate sys-tem with gap greater than 0.2 at a μp = 2, b μp = 20, c μp = ∞ and d μp = 2. Parameters used in the simulations are:R = 10, I = 129, K = 32

x

y

-10 -5 0 5 10-10

-5

0

5

10 (a)

x

y

-10 -5 0 5 10-10

-5

0

5

10(b)

Fig. 13 Streamline patterns around a pair of particles obtained at R = 10; the first particle is located at (2.5, 0) assigned with a linearmotion with unit velocity outward in a and inward in b; the second particle at (0, 5) is assigned with a counterclockwise rotating motionwith unit tangential velocity at the particle surface

123123

Motion of paramagnetic particles in a viscous fluid 47

Fig. 14 Streamline patternaround three particlesobtained at R = 10; theparticle at (2.5, 0) movesupward with unit linearvelocity; the particle at(−5/

√2,−5/

√2) rotates

clockwise with unittangential velocity at theparticle surface; the particleat (0, 5) is stationary

x

y

-10 -5 0 5 10-10

-5

0

5

10

x

y

-4 -2 0 2

-2

0

2

4 (a)

x

y

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3(b)

Fig. 15 Streamline patterns around three particles, each executing the same motion as in Fig. 14 but now located at (1.1, 0), (0, 1.8)and (−1.8/

√2,−1.8/

√2) with smaller gaps between each pair of particles: a R = 10, b R = 3

First, we show a typical magnetic potential line and streamline pattern in Fig. 16, obtained numerically for a pairof particles initially located point-symmetrically with θp1 = 0. It can be seen that the particles tend to attract eachother (Fig. 16a), because the magnetic potential lines are more clustered on the particle opposing sides than on theback sides. This results in a particle advancing motion that gives rise to the streamline pattern shown in Fig. 16b.Next, we show in Fig. 17 results for a pair of particles initially located point symmetrically with θp1 = π/4. Thestreamline pattern suggests that particles now instantaneously move along the clockwise revolving path rather thanattract or repel each other.

Particle trajectories obtained at μp = 2 and μp = 200 for various values of θp1 are shown in Fig. 18. Theyindicate that the particle trajectories are not so much affected by the parameter μp; for a larger value of μp, theparticle motion is faster, but trajectories are slightly closer to the circle rp = 2 where they started. The resultsfor μp = 2 are in good agreement with those obtained by Kang et al. [19]. It is found that the maximum radialdistance of the particle during the travel measured from rp = 2 is 0.582 for μp = 1.5, 0.584 for μp = 2.0, 0.511for μp = 20, and 0.505 for μp = 200.

123

48 Y. K. Suh, S. Kang

x

y

-10 -5 0 5 10-10

-5

0

5

10(a)

x

y

-10 -5 0 5 10-10

-5

0

5

10(b)

Fig. 16 Numerical result of a the magnetic potential line and b the streamline around a pair of particles located anti-symmetricallywith θp1 = 0. Parameters are set at R = 20 and μp = 200

x

y

-10 -5 0 5 10-10

-5

0

5

10(a)

x

y

-10 -5 0 5 10-10

-5

0

5

10 (b)

Fig. 17 Numerical result of a the magnetic potential line and b the streamline around a pair of particles located anti-symmetricallywith θp1 = π/4. Parameters are set at R = 10 and μp = 200

x

y

0 1 2 30

1

2

3

45

7060

80

85

(a)

x

y

0 1 2 30

1

2

3

45

7060

80

85

(b)

Fig. 18 Trajectories of the particle located in the first quadrant of the pair located anti-symmetrically with the initial gap set at 2. Thevalue of θp1 (in degree) is shown for each line. Parameters are set at R = 10 and �t = 2 with a μp = 2 and b μp = 200. An opencircle attached to one end of each trajectory indicates the initial position

123123

Motion of paramagnetic particles in a viscous fluid 49

x

y

0 1 2 30

1

2

3

(a)

x

y

0 1 2 30

1

2

3

(b)

Fig. 19 Numerical result of a the magnetic-force field acting on the particle in the first quadrant and b the particle velocity for a pairof anti-symmetrically positioned particles given at R = 10 and μp = 2

Table 3 Velocity components(u px , u py

)and magnetic-force components

(Fmx , Fmy

)for the pair-particle problem, at the first particle

starting point (rp, θp1) = (2, 80◦), for various values of μp

μp u px u py ωp Fmx Fmy

1.5 4.04 × 10−4 3.45 × 10−4 −4.44 × 10−5 3.85 × 10−3 6.50 × 10−3

2.0 1.11 × 10−3 9.34 × 10−4 −1.22 × 10−4 1.06 × 10−2 1.75 × 10−2

20 7.93 × 10−3 6.05 × 10−3 −8.84 × 10−4 7.43 × 10−2 1.15 × 10−1

200 9.45 × 10−3 7.11 × 10−3 −1.06 × 10−3 8.84 × 10−2 1.35 × 10−1

The force and velocity vector fields for a pair of particles located initially point symmetrically are shown inFig. 19. Both fields are similar, but the velocity-vector trajectories are slightly flatter than those of the force vector.Thus, we may use the magnetic force field to qualitatively estimate the translational motion.

For a pair of particles located point symmetrically, the particles turn around and finally tend to align on the x-axis,as seen in Fig. 18. In order to clarify the main reason for this phenomenon, we enlarged the size of the outer cylinderR. It was found that the phenomenon is more pronounced as R is increased, approaching an ultimate state wherethe magnetic forces and the particle velocities asymptote to finite values. We have broken the point-symmetricconfiguration for a large container size and found that the phenomenon is preserved, indicating that the returningmotion of the pair particles is purely the outcome of the mutual effect.

For use in validating a two-dimensional code calculating the motion of paramagnetic particles within a circulardomain, we present in Table 3 the data (u px , u py) and (Fmx , Fmy) of the particle located in the first quadrant of thedomain for various values of μp for the pair-particle problem.

Consider the case with three particles. First, we investigate the instantaneous magnetic potential and the stream-line pattern. For the initial configuration of � shown in Fig. 20 (symmetric about the y-axis), the particles attracteach other. Note that the increment of the stream function for the plot (�ψ = 0.0002) is 10 times smaller than thatfor the other cases (�ψ = 0.002) presented so far. The flow is weak because all particles concentrate at the centralpoint. For the initial configuration of � shown in Fig. 21 (symmetric about the x-axis) too, they attract each other,and concentrate at the origin.

Now we consider the motion of three particles. It has been found that, for the initial arrangement of the shape ∇ asshown in Fig. 20, two upper particles attract each other and move upward, while the lower particle moves downwardfaster. When we position the particles in the shape � (Fig. 22a), they first move inward slowly and then the upperone on the RHS moves faster than the lower one to make pairing with the one on the LHS. This pairing makesthe overall motion faster. The final shape is a chain aligned parallel to the x-axis, where the particles are almost

123

50 Y. K. Suh, S. Kang

x

y

-10 -5 0 5 10-10

-5

0

5

10(a)

x

y

-10 -5 0 5 10-10

-5

0

5

10(b)

Fig. 20 Numerical results for three particles arranged in the form of a triangle obtained at R = 20 and μp = 200: a magnetic potentiallines, b streamlines

x

y

-10 -5 0 5 10-10

-5

0

5

10(a)

x

y

-10 -5 0 5 10-10

-5

0

5

10(b)

Fig. 21 Numerical results for the case with three particles arranged in the form of a triangle obtained at R = 20 and μp = 200:a magnetic potential lines, b streamlines

motionless except for a very small movement caused by the outer boundary. Although the initial configuration ofthe particle arrangement is exactly of an equilateral triangle, a small perturbation in the particle motion can resultin such symmetric breaking. Indeed, when the particle at the bottom on the RHS is intentionally shifted initially tothe LHS over a small distance 0.01 (Fig. 22b), it moves faster than the upper one and pairs with the one on the LHSgiving rise to a sequence of particles, at the final linear alignment, different from Fig. 22a.

When the particles come very close to each other, the numerical scheme converges very slowly or diverges.The limit gap between a pair of particles for the numerical stability is 0.02 in the present method. Most numericalmethods reported in the literature avoid this situation by providing an artificial repelling force that increases withthe inverse of the gap distance. In this study, we employ the gap-readjustment technique; when the gap becomesless than the limit value, 0.05 in this study, two particles are forced to move away from each other so that the gapis not less than 0.05.

As the number of particles increases, the motion becomes more complex and diversified, as expected. The patternobviously depends on the initial setting of the particle positions. For the four-particle case, the simplest pattern occursfor θp1 = π/4. Pairing takes place between the particles at the first and second quadrants and between those at thethird and fourth quadrants. Then each of the paired group behaves like a single particle. Due to symmetric property,the particles move away traveling along the y-direction.

123123

Motion of paramagnetic particles in a viscous fluid 51

x

y

-3 -2 -1 0 1 2-3

-2

-1

0

1

2

3(a)

x

y

-3 -2 -1 0 1 2-3

-2

-1

0

1

2

3(b)

Fig. 22 Dramatic difference in the trajectories of three particles with a slight difference in the initial configuration of the form of atriangle obtained for parameter set R = 20 and μp = 200. In a, three particle initial positions constitute the equi-triangle exactly.In b, the lower particle on the RHS is moved slightly to the left by the distance 0.01

Fig. 23 Motion of fourparticles withrp = 5, θp1 = 0, R = 200and μp = 200. The openand closed circlescorrespond to the initial andfinal positions of particles

x

y

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

Figure 23 shows numerical results of four-particle motions for θp1 = 0. Initially, particles keep the initial sym-metric positions only moving along the axes. Later they escape from the axes and make pairing, after which theyrapidly approach the typical chain shape. Again, this implies that a small perturbation can cause the symmetry tobreak and lead to the chain-shape state, similar to the three-particle case. Figure 23 further shows that, even duringthe initial symmetric evolution, the particles on the y-axis move inward all the time, but those on the x-axis initiallymove outward and later inward. This peculiar motion can be understood from the variation of the magnetic forceacting on each particle with different initial positions of particles located at (±x p1, 0) and (0,±yp2). Shown inFig. 24 are the contour of the magnetic-force component Fmx acting on the first particle at the x-axis, (x p1, 0), andthat of Fmy acting on the second particle at the y-axis, (0, yp2). We can recognize three parameter spaces from thisresult. When the ratio x p1/yp2 is either much larger or smaller than 1, the first particle experiences an attractingforce while the second particle experiences a repelling force. However, when the ratio is close to 1, as in the caseof Fig. 23, the particles receive the reversed forces. The thick line segment drawn in Fig. 24 indicates the particleevolution while keeping the initial symmetry property. Due to the attracting magnetic force, the second particlemoves downward (inward) all the time, whereas the first one moves initially outward due to the repelling force.When the trajectory crosses the boundary in the contour for Fmx (i.e., the contour line of Fmx = 0 in Fig. 24a),

123

52 Y. K. Suh, S. Kang

xp1

y p2

2 4 6 8

2

4

6

8 (a)

xp1

y p2

2 4 6 8

2

4

6

8 (b)

Fig. 24 The contours of the magnetic-force components obtained for parameter set R = 200 and μp = 200 with various initialpositions of four particles located symmetrically at (±x p1, 0) and (0,±yp2): a contour of Fmx acting on the first particle at the x-axis,(x p1, 0), and b contour of Fmy acting on the second particle at the y-axis, (0, yp2). Contour level is 0.05 and the dashed lines indicatethe negative values. The thick line segment drawn in each plot is the locus (x p1, yp2) representing the trajectory of the first and secondparticles during the particle motion shown in Fig. 23; the open circle denotes the initial position; the closed circle denotes the timeinstant when those particles begin to break the symmetric configuration indicated by the maximum distance of the particle from theaxis reaching 0.01

x

y

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

123

4 5

(d)

x

y

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

1

2

3

45

(a)

x

y

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

1

23

4 5

(b)

x

y

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4123

4 5

(c)

Fig. 25 Motion of five particles with rp = 3, θp1 = 5o, θp1 = 5o and μp = 200: a t = 0, b t = 100, c t = 200, d t = 360

123123

Motion of paramagnetic particles in a viscous fluid 53

the first particle still moves outward, even though the magnetic-force component Fmx is negative there; this is,of course, caused by the viscous force of the surrounding fluid driven by the second particle’s downward motion.Finally, due to the attracting magnetic force acting on the first particle, the particle moves inward. The particlesmove faster after this because the magnetic forces become stronger as the particles come closer to each other, andthe instability brings the subsequent events, i.e., symmetry-breaking, pairing and line alignment.

In the five-particle case, we first simulate two kinds of initially symmetric configuration (one symmetric withrespect to the x-axis and the other with respect to the y-axis) and then investigate the asymmetric case. In the firstcase, with θp1 = 0, where the first particle is located on the x-axis and the initial configuration is symmetric withrespect to the x-axis, pairing occurs between the second and third particles and between the fourth and fifth parti-cles, respectively, leaving the first particle alone. Then the two pairs move outward without breaking the symmetricconfiguration. In this process the first particle never escapes from the x-axis, but it shows back-and-forth motion,i.e., outward–inward–outward movement. In the second case, with θp1 = 3π/10, the initial state is symmetric withrespect to the y-axis. Pairing occurs between two particles located at y > 0. The other three particles at y < 0also show clustering to become a three-particle chain aligned along the x-direction. Then those two groups movesteadily outward keeping the y-axis-symmetry configuration. The asymmetric case with θp1 = 5◦ is simulatedand the motion of particles is shown in Fig. 25. Pairing occurs between the second and third particle and betweenthe fourth and fifth particle (Fig. 25b). Later, the first particle is attracted to the right-hand side of the upper pair(Fig. 25c). Two groups of particles then undergo a turn-around motion to arrive at the final state, a single chain(Fig. 25d).

It should be stressed again that increasing the number of particles adds to the complexity in the pattern of particlemotion. For the randomly distributed initial positions, the motion is hardly predictable. For the six-particle case

x

y

-5 0 5 10

-10

-5

0

5 (b)

12

3

45

6

x

y

-5 0 5 10

-10

-5

0

5 (a)

12

3

4 5

6

x

y

-5 0 5 10

-10

-5

0

5 (c)

12

3

4 5

6

x

y

-5 0 5 10

-10

-5

0

5 (d)123

4 5

6

Fig. 26 Motion of six particles initially located at (5, 1), (2, 2), (−2, 4), (−3, 0), (1, 0) and (2,−3) in the sequence of numbers assignedto each particle, with the parameter set at R = 200 and μp = 200: a t = 0, b t = 11, c t = 35, d t = 120

123

54 Y. K. Suh, S. Kang

with arbitrary selection of initial positions, as shown in Fig. 26a, the first pairing occurs between particles 2 and 5.Then the particle 1 is attached to the right-hand side of the pair to make the chain 5-2-1 (Fig. 26b), and later particle4 sticks to the left-hand side of the chain to make it a longer one 4-5-2-1 (Fig. 26c). Particle 3 then undergoes aturn-around motion, finally adhering to the left-hand side of the chain to make a longer one 3-4-5-2-1 (Fig. 26d).During this evolution, particle 6 never gets a chance to become a part of the primary chain, but it just moves outward.We expect this particle must turn around following a long-distance trajectory and attach to the right-hand side ofthe chain after a long period of travel.

During the simulations, we have found that particle pairing occurs faster for the pair particles at a closer distance;for instance, in Fig. 26, particle 2 chooses particle 5 as the first partner for pairing and not particle 1, because theformer is located at a closer distance than the latter. We have also found that particle pairing occurs faster for thepair particles aligned in a more longitudinal direction; for instance, in Fig. 25, the initial distance between particles2 and 3 is the same with that between 2 and 1, but the first pairing occurs between the former pair, not between thelatter, for this reason.

In general, as R → ∞, we can expect that the final state should be composed of an almost stagnant long chainaligned in the longitudinal direction with some particles located on the line drawn normal to the chain at its centerpoint and moving slowly outward. For the initially asymmetric distribution of particles, it is almost impossible tofind those particles located exactly on the normal line. Most probably, all particles cluster with each other to becomea single longitudinal chain after a sufficiently long time.

6 Conclusion

We have presented an efficient numerical method for calculating the motion of paramagnetic particles under theinfluence of a uniform magnetic field. The key idea is to employ local coordinates attached to each particle. Thenumerical results presented should be used as benchmark solutions for validating the numerical method developedfor a more general purpose, such as finite volume or finite element methods for the motion of paramagnetic particles.

In calculating the magnetic field and forces acting on the particles, the fluid flow is decoupled. In determining theunknown coefficients of the series for the magnetic field perturbed by a particle, the contribution from all the otherparticles can be treated explicitly, and all modes in the series are decoupled. Such an explicit method is successfulbecause all modal functions in the series show an exponential decrease with distance from the particle center.

In calculating the flow field, the magnetic forces obtained after determining the magnetic field serve as the drivingterms. In this calculation, all five leading coefficients as well as the two linear-velocity components and the rotatingvelocity of the particles are treated as unknowns. The number of unknowns is then 8N + 9. Referring to Table 1,we see that we could have treated the coefficients C (i)

3 and C (i)5 explicitly without numerical instability because the

corresponding modal functions q(i)3 (ξ) and q(i)5 (ξ) show an exponential decrease with distance from the particlecenter. In this study, we treated these terms implicitly because this accordingly improves the convergence property.

In order to validate the numerical method for the flow field, we derived the asymptotic solutions of the viscousflow between a pair of particles separated by a small distance. We also developed a series-solution method by usingthe bipolar coordinates for a pair of particles in order to validate the numerical method for the magnetic field.Very good agreement was confirmed between the present numerical results and those provided from such separatecalculations, thus proving the validity of the present method.

The motion of multiple particles caused by the externally applied magnetic field was then simulated primarilywith the particle initial positions on a circle with equal spacing in the circumferential direction. For a pair of parti-cles, we obtained the particle trajectory typified as the ‘turn-around’ motion in almost the same form as that givenby Kang et al. [19]. We also provided the magnetic-force data as reference values for use in validation of codes.We demonstrated from simulations of three-particle cases that a small difference in the initial configuration of theparticle positions can result in a completely different sequence of particle locations in the final state, i.e., a straightchain. For the case of four particles with a diamond-shape distribution, a peculiar phenomenon was observed. Par-ticles on the x-axis show back-and-forth motion, while the other two reveal consistent inward motion. When they

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Motion of paramagnetic particles in a viscous fluid 55

move altogether inward, the motion becomes unstable and breaks the symmetry property, arriving ultimately at astraight-chain shape. Through simulations with five and six particles we could capture the fundamental mechanismin the development of particle clustering. Particles first show pairing between a pair of particles at a close distance tobecome a seed for the future chain structure. Then neighboring particles are attracted to the left-hand or right-handside of the pair after the turn-around motion to make the chain longer. Such local pairing and attaching effects giverise to groupings of particles, and those groups merge with each other to make them longer. If sufficient time isallowed, a single chain is formed encompassing all particles.

Acknowledgement This work was supported by the National Research Foundation of Korea through the NRL Program funded bythe Ministry of Education, Science and Technology (Grant No. 2005-1091).

Appendix

I. Derivation of Eq. 6

The magnetic force arises by integrating (5) over the whole region of the particle including the interface where μsharply changes from μp to 1,

Fm = −1

2

∫∫H2∇μdxdy. (A1)

We can write

H = 1

μB = 1

μ

[(∂A

∂y

)i −(∂A

∂x

)j].

In terms of the local coordinates (s, n),

H = 1

μ

[(∂A

∂s

)n −(∂A

∂n

)s]. (A2)

Note that the normal component Hn = (∂A/∂s) /μ is discontinuous while the tangential component Hs =− (∂A/∂n) /μ is continuous. Further, from H2 = H2

s + H2n we have

Fm = −1

2

∫∫H2∇μdxdy = −1

2

∫∫ [(1

μ

∂A

∂s

)2

+(

1

μ

∂A

∂n

)2]

∇μdxdy = −1

2[I1 + I2] .

The integrand of the above equation is zero over the whole interior region of the particle but infinite at the inter-face due to the discontinuous distribution of μ. We can take the very thin layer near the interface as the effectiveintegration domain. The first term within the square brackets on the RHS becomes

I1 =∫∫ (

∂A

∂s

)2 1

μ2 ∇μdnds = −∫∫ (

∂A

∂s

)2

∇(

1

μ

)dnds (A3)

over the thin layer. Note that ∂A/∂s is continuous across the interface because A is continuous; see BC (4a). Thismeans that we can integrate (A3) across this thin layer to obtain

I1 = −(1 − 1/μp)

∫∂�

(∂A

∂s

)2

n(s)ds. (A4)

Similarly, we can formulate I2 as follows:

I2 =∫∫ (

1

μ

∂A

∂n

)2

∇μdnds = (1 − μp)

∫∂�

(1

μ

∂A

∂n

)2

n(s)ds. (A5)

Note that in (A4) and (A5) the integrand can be evaluated either on the fluid or particle side. Then we can derive(6); the integrand is now to be evaluated on the particle side.

123

56 Y. K. Suh, S. Kang

II. Derivation of Eqs. 47a and 47b

For the derivation, we use the cylindrical coordinates (r, θ). From Eq. 7b we can derive

∂p

∂r= − ∂ς

r∂θ,

∂p

r∂θ= ∂ς

∂r, (A6a,b)

where ς is the vorticity defined as

ς = −∇22ψ,

where the Laplacian ∇22 is now ∇2

2 = ∂2/∂r2+∂/r∂r +∂2/r2∂θ2. In terms of (r, θ), Eqs. 46a and 46b are convertedinto

F f x =2π∫

0

[(−p + 2

∂Vr

∂r

)cos θ −

(∂Vr

r∂θ+ ∂Vθ

∂r− Vθ

r

)sin θ

]rdθ, (A7a)

F f y =2π∫

0

[(−p + 2

∂Vr

∂r

)sin θ +

(∂Vn

r∂θ+ ∂Vs

∂r− Vθ

r

)cos θ

]rdθ. (A7b)

The first terms within the square brackets in the above equations can be converted, by using the integration by part,into the forms2π∫

0

(−p + 2

∂Vr

∂r

)cos θrdθ =

2π∫0

(∂p

∂θ− 2

∂2Vr

∂θ∂r

)sin θrdθ,

2π∫0

(−p + 2

∂Vr

∂r

)sin θrdθ =

2π∫0

(−∂p

∂θ+ 2

∂2Vr

∂θ∂r

)cos θrdθ.

Next, we apply (A6b) to the above and substitute the results in (A7a) and (A7b) to obtain

F f x =2π∫

0

[r∂ς

∂r− ς − 2

r

∂

∂r

(ψ + ∂2ψ

∂θ2

)]r sin θdθ, F f y =

2π∫0

−[

r∂ς

∂r− ς − 2

r

∂

∂r

(ψ + ∂2ψ

∂θ2

)]r cos θdθ.

It can be shown that the contribution of the third terms within the square brackets in the above equations is zero.Evaluating at the particle surface, r = 1, leads to (47a) and (47b).

III. Derivation of Eq. 69

We derive the asymptotic formula of Fmx , given in Eq. 69, in the limit d → 0. The limit d → 0 is equivalent toα0 → 0. We can replace α0 by �x and k�x by xk , and write

a ∼= �x,Cik = 2μp exp(xk)

cosh xk + μp sinh xk, 1 + d ∼= 1 + �x2

2,

so that (68) becomes

Fx = 2π(μp − 1)(μp + 1)J, (A8)

where

J =∞∑

k=1

fk

[fk+1(1 +�x2/2)− fk

]/�x = 1

2

∞∑k=1

fk fk+1�x +∞∑

k=1

fk( fk+1 − fk)/�x (A9)

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Motion of paramagnetic particles in a viscous fluid 57

and the function f (x) is shown in (69). The first part on the RHS of (A9), J1, becomes

J1 = 1

2

∞∑k=1

fk fk+1�x ∼= 1

2

∞∫0

f 2(x)dx . (A10)

The second part on the RHS of (A9), i.e. J2, reads

J2 =∞∑

k=1

fk f ′k + 1

2

∞∑k=1

fk f ′′k �x .

It can be shown that the first term on the RHS vanishes in the limit �x → 0. For the second term, we performintegration by part to derive

J2 = 1

2

∞∫0

f (x) f ′′(x)dx = −1

2

∞∫0

[ f ′(x)]2dx .

Substituting f (x) in the above and performing some algebraic work, we arrive at

J2 = 4

3(1 + μp)2

∞∫0

(1 + x2) exp(−2x)

[1 − c exp(−2x)]2 dx,

where c = sinh α0 as defined in the text. Referring to the integral tables of Gradshteyn and Ryzhik [21, p. 330] weobtain

J2 = μp − 1 + ν

−6(μp − 1)(μp + 1), (A11)

where ν is another constant defined in (69). Substituting (A10) and (A11) in (A9) and the result into (A8), we obtain(69).

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