A boundary integral method for motion of particles in unsteady

Stokes and linear viscoelastic flows

Hualong Fenga,b,c, Andres Cordobaa,b, Francisco Hernandezd, Tsutomu Indeia,b, ShuwangLid, Xiaofan Lid,∗, Jay D. Schiebera,b,e

aDepartment of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, IL 60616,

USAbCenter for Molecular Study of Condensed Soft Matter, Illinois Institute of Technology, Chicago, IL 60616,

USAcSchool of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China

dDepartment of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA.eDepartment of Physics, Illinois Institute of Technology, Chicago, IL 60616, USA

Abstract

In this paper, we develop a boundary integral method (BIM) for interacting particles inunsteady Stokes flow or linear viscoelastic (LVE) flow. The idea is to exploit a correspon-dence principle between them so a BIM can be established in the Fourier domain. Since theunsteady Stokes equation in the frequency domain is analogous to Brinkman equation in thetime domain, our method can also be used for flow through porous media. In addition todimension reduction vested in a boundary integral method, our formulation further reducesthe computational cost by eliminating double-layer integrals. To evaluate the single-layerintegrals more efficiently, we develop a hybrid numerical integration scheme based on kerneldecompositions. The resulting method is third-order accurate. We first compare our numer-ical results with a known analytic solution for motion of one particle, and then apply themethod to motion of two particles. Accurately capturing the hydrodynamic interaction oftwo particles in purely viscous or viscoelastic flow is of fundamental importance for studyingtwo-particle microrheology and binding kinetics of particles in those flows. We compare ournumerical results with an existing asymptotic solution for motion of two particles, which wasderived for the case of large separation of the particles, and document its accuracy when thetwo particles come close to each other.

Keywords:

particulate flow, linear viscoelastic fluid, unsteady Stokes, boundary integral method

1. Introduction

Micron- or nano-sized particles are ubiquitous in advanced materials, and provide en-hanced mechanical properties and conductivity, or inhibited mass transport [1, 2, 3, 4, 5, 6, 7].Because of a surrounding viscoelastic matrix, the diffusion dynamics of these particles is notgoverned by the Stokes-law drag force [8, 9, 10, 11, 12, 13, 14, 15]. Therefore, typical efficient

∗Email: lix@iit.edu

Preprint submitted to Journal of Computational Physics December 19, 2014

numerical methods based on Stokesian dynamics for hydrodynamically interacting Brownianparticles [16], which work for relatively long time scales or slow motions with Newtonian ap-proximations, cannot be implemented here. Since the fluid flow around a Brownian particleis by definition near equilibrium, it is described completely by linear viscoelasticity [17, 18].All non-Newtonian fluids are described by linear viscoelasticity in this regime.The dynamicsof these Brownian particles should be described by a generalized Langevin equation (GLE)[19, 20]. In unsteady Stokes or linear viscoelastic flow, however, the form of such a GLEis not currently known, except in the limiting case of large separation of the particles [21].Understanding the hydrodynamic interaction of particles in such flows is a first step towardsderiving a GLE for the particles.

There is a long history of numerically solving the linear viscoelastic equations [22, 23, 24,25, 26, 27, 28]. For moving boundary problems, there are a variety of computational methodsthat can be applied to complex fluids, such as the boundary integral methods [29, 30],the volume of fluid methods [31], and the immersed boundary/interface methods [32, 33,34], among others. When applicable, boundary integral/element methods are typically themost accurate methods because there are well-developed accurate quadratures and stablediscretizations of the boundary integral equations. Khayat et al. [35] presented a boundaryelement method in the time domain and studied a mixing flow of Newtonian fluid and linearviscoelastic (LVE) fluid. Their method is second-order accurate in space and first-orderaccurate in time. While a higher-order integration scheme may be possible for the timeevolution, the integral formulation would need significant revision because their methodtracks the sum of extra stress tensor and its time derivative. To achieve high accuracy, oneoften needs to track the extra stress tensor explicitly. Very recently, Jiang et al. [29] presentedan integral equation formulation for the unsteady Stokes equations in two dimensions, whichallows for a high-order accurate numerical scheme. Feng et al. [36] developed a boundaryelement method for the motion of a sphere near planar boundaries in an axisymmetric porousmedium by solving Brinkman equation in the time domain (which is identical to the LVEequation in the frequency domain), and their method is second order accurate in space.

As a part of effort to derive the GLE governing motion of Brownian particles in LVEfluid, we develop in this paper a high-order boundary integral method for computing forceson interacting particles in unsteady Stokes flow or linear viscoelastic flow. Using a correspon-dence principle between the unsteady Stokes flow and linear viscoelasticity [37, 38, 39], weformulate the problem as Fredholm boundary integral equations in the Fourier domain. Toreduce the computational cost, we consider a modified surface force density to eliminate thedouble-layer integrals in our formulation, following [40, 41]. Note that only a single-particlecase was studied in [40, 41]. Here we show that this elimination approach can also be donein problems with multiple particles provided that one modifies the surface density correctionappropriately. To guarantee the uniqueness of the solution, we remove the eigen-solutionfrom the null space of the kernel. The formulation reduces the computational cost whilemaintaining the mathematical structure and original physics.

Most of the computational cost in boundary integral methods lies in evaluating theGreen’s functions. We develop a hybrid integration scheme based on a kernel decompositiontechnique, which dramatically reduces the cost of evaluating the Green’s functions. Whenthe target and source points are close to each other, using the same amount of computa-tional time, it evaluates the axisymmetric Green’s functions seven orders of magnitude more

2

accurate than the trapezoidal rule, even though the trapezoidal rule is spectrally accuratefor evaluating integrals with periodic and smooth integrands. When applying a boundaryintegral method to a closed surface in an axisymmetric setting, it occurs that the quadra-ture errors are much larger near the axis of symmetry. Following the previous work byNitsche [42], we develop a similar correction scheme to improve the accuracy for evaluat-ing the boundary integrals at the points close to the poles in axisymmetric setting. It willbe demonstrated that this pole correction is critical in achieving the uniform convergenceof order three. Furthermore, we show that the accuracy order is optimal with the currentformulation. Our idea of achieving the optimal accuracy order can also be used for otherkernels of logarithmic nature.

Our numerical tests on a single particle in a linear viscoelastic fluid are in excellentagreement with the analytical solution, and demonstrate that the method is convergent,efficient, and highly accurate. Simulations of two particles immersed in an unsteady Stokesflow show that the differences between the boundary integral solution and the asymptoticsolution decay as ǫ3 as expected, where the nondimensional parameter ǫ measures the ratioof the particle size and the distance between the two particles. The boundary integral result,which serves as a nearly exact solution for computing the actual force acting on the particle,can be used to evaluate the accuracy of asymptotic solution. Indeed, as ǫ increases, theaccuracy of the asymptotic solution decreases.The asymptotic solution was derived for thecase of large separation of the particles, and it has been exploited to study two-particlemicrorheology in viscoelasticity [8, 43, 21, 44, 10, 9], via the correspondence principle. Fortwo-particle microrheology [11, 13, 45, 20], the two particles need to be sufficiently separated,while study of binding kinetics requires considering approaching particles. We examinethe validity of the asymptotic solution for studies of binding kinetics. By comparing theasymptotic solution with our numerical results, we conclude that the asymptotic solutionshould be used with caution when particles are close to each other. Consequently, accuratenumerical results are necessary in those studies.

This paper is organized as follows: in Section 2, we review the governing equations forthe linear viscoelastic problem and present its boundary integral formulation in the Fourierdomain and explain the canonical correspondence principle between unsteady Stokes flowand LVE flow in the Fourier domain; in Section 3, we present an integration scheme for thesingle-layer integral and derive our quadrature methods; in Section 4, we discuss numericalresults; and in Section 5, we give conclusions.

2. Governing equations

2.1. Differential form

We consider the motion of solid particles suspended in a Stokes or linear viscoelasticfluid. First, we recall the linearized incompressible Cauchy equation without body force

ρ∂v

∂t= ∇ · τ −∇p, ∇ · v = 0, (1)

where ρ is the fluid density, v is the fluid velocity, p is the pressure, and τ is the extra stresstensor. Note that if τ = 2µD, where µ is the fluid viscosity and D = (1/2)(∇v +∇vT ) is

3

the symmetrized strain rate, equation (1) reduces to the unsteady Stokes equation

ρ∂v

∂t= ∇2v −∇p, ∇ · v = 0. (2)

For a linear viscoelastic fluid, the stress tensor is given by

τ (x, t) = 2

∫ t

−∞G(t− t′)D(x, t′)dt′, (3)

where G(τ) is the relaxation modulus. For an N -mode Maxwell fluid, the relaxation modulusG takes the form of

G(t) =N−1∑

j=0

Gje−t/λj , (4)

where λj and Gj are the relaxation time and spectral strength of mode j, respectively. Phys-ically speaking, a larger λj signifies a longer memory of the fluid. Note that in Eq. (4), anyviscoelastic fluid near equilibrium can be approximated to an arbitrary accuracy using suffi-ciently many modes, i.e. by increasing N . Using a one-mode Maxwell fluid with relaxationmodulus G(t) = G0e

−t/λ0 , we can simplify the constitutive equation (3) as

τ (x, t) =

∫ t

−∞e(t−t′)/λ0G0(∇v +∇vT )dt′, (5)

or in differential form,

λ0∂τ (x, t)

∂t+ τ (x, t) = G0(∇v +∇vT ). (6)

The no-slip boundary condition is imposed on the particle surfaces. We note that thelinear viscoelastic equation is naturally linked to the unsteady Stokes equation in the Fourierdomain by a correspondence principle [38]. Specifically, we take the time Fourier transformof Eq. (1) to obtain

ρiωv = G(ω)∇2v −∇p, ∇ · v = 0, (7)

where v and p depend on space variable x and frequency ω. For a Stokes flow, G(ω) = µis independent of ω; while for a viscoelastic flow, G(ω) is the Fourier transform of G(t) intime. Let l be the characteristic length scale, then one can non-dimensionalize Eq. (7) witha single parameter λ2 = iωl2ρ/G(ω)

(∇2 − λ2)v = ∇p, ∇ · v = 0, (8)

where we have dropped all the hats for simplicity. This should not cause confusion sincewe will work in the Fourier domain hereafter. The first equation in Eq. (8) is known asBrinkman equation. It is essentially the constitutive equation for the unsteady Stokes or theviscoelastic equation in the Fourier domain. In later sections, we may refer to Eq. (8) eitheras the unsteady Stokes equation or the linear viscoelastic equation, and the meaning shouldbe clear from the context since λ2 is purely imaginary for unsteady Stokes and fully complexfor LVE. We next discuss the Green’s functions and appropriate boundary value problemsfor this equation.

4

LR R

V1 V2

∂Ω1

∂Ω2

n n

Figure 1: Sketch of two spherical particles suspended in a viscoelastic fluid. The particles occupy the regionsΩ1 and Ω2 with the surfaces of the particles denoted by ∂Ω1 and ∂Ω2 respectively. The entirely interface∂Ω is the union ∂Ω = ∂Ω1 ∪ ∂Ω2 as used in the boundary integral formulation.

2.2. Boundary integral formulation

Following [40, 41], we solve the non-dimensionalized governing equations for a linearviscoelastic fluid in the Fourier domain, Eq. (8), via its boundary integral formulation.Referring to the schematic shown in Fig. 1, the velocity veiωt at a point x0 on the surface∂Ω satisfies the following boundary integral equation (BIE)

vj(x0) = − 1

4πµP.V.

∫

∂Ω

fi(x)Gij(x,x0) dS(x) +1

4πP.V.

∫

∂Ω

vi(x)Tijk(x,x0)nk(x) dS(x), (9)

where feiωt is the traction on the surface, the normal vector n points into the fluid, and P.V.stands for principal value integral. For clarity, we use the notation x and x0 for a sourcepoint and a target point respectively. The expressions of the Green’s functions Gij and Tijkfor Brinkman equation (8) are given in [40, 41],

Gij(x,x0) = A(X)δijr

+ B(X)xixjr3

, (10)

Tijk(x,x0) = − 2

r3(δijxk + δkjxi)

[

e−X(X + 1)−B(X)]

− 2

r3δikxj(1−B(X))

−2xixjxkr5

[

5B(X)− 2e−X(X + 1)]

, (11)

where x = x− x0, r = |x|, X = λr, and we use the auxiliary functions

A(X) = 2e−X

(

1 +1

X+

1

X2

)

− 2

X2, B(X) = −2e−X

(

1 +3

X+

3

X2

)

+6

X2. (12)

The parameter λ in Brinkman equation (8) should be chosen with a positive real part suchthat A(X) and B(X) will tend to zero as r goes to infinity, and it is also clear that theyreduce to the Green’s functions for Stokes flow as λ (hence X) tends to 0.

The velocity and the traction of the particles are related to each other according to theBIE (9), so one can solve for the traction if the velocity is given and vice versa. Once boththe velocities of and the traction on the particles are known, one can find all flow quantitiessuch as velocity v and the pressure p anywhere in space by evaluating boundary integrals.

For single-particle cases, Pozrikidis [40, 41] showed that, to reduce the computationalcost, one can eliminate the double-layer integral in Eq. (9) as follows. For a target point x0

5

inside the fluid, the boundary integral equation (9) holds if both coefficients on its right-handside are halved,

vj(x0) = − 1

8πµ

∫

∂Ω

fi(x)Gij(x,x0) dS(x) +1

8π

∫

∂Ω

vi(x)Tijk(x,x0)nk(x) dS(x). (13)

We introduce an imaginary flow v′ inside the solid particle such that v′ agrees with the realflow v on the particle surface. For x0 in the interior of the fluid, we have [41]

0 = −∫

∂Ω

f ′i(x)Gij(x,x0) dS(x) + µ

∫

∂Ω

v′i(x)Tijk(x,x0)nk(x) dS(x). (14)

Replacing the double-layer integral of Eq. (13) using Eq. (14), we obtain the boundaryintegral equation involving only the single-layer potential

vj(x0) = − 1

8πµ

∫

∂Ω

qi(x)Gij(x,x0) dS(x), (15)

where q := f − f ′ is an unknown surface density. The equation (15) holds when we take thelimit as the point x0 approaches the boundary ∂Ω. Thus, the boundary integral equation(15) is valid when x0 is inside the fluid or on the interface of particles. f ′ is the tractioncorresponding to an imaginary flow v′ inside the particle that agrees with v on the surface.While Eq. (15) is simpler than Eq. (9) after the double-layer integral is eliminated, thequantity q can not be used to find the traction f on the surface unless we know the solutionpair (v′, f ′), which fortunately is available in simple cases. For instance, for a single sphericalparticle undergoing a single frequency oscillation veiωt (v is a constant), we know [46]

v′ = v, f ′ = λ2(x · v)n, (16)

where n is the unit outward normal to the surface.We now extend the result to show the simplified boundary integral equation (15) holds

with two or multiple particles. In this case, for a point x0 inside the fluid, Eq. (13) holds forthe union of the particle surfaces ∂Ω = ∂Ω1 ∪ ∂Ω2. However, since the point x0 is outsideboth particles, the equation (14) is valid when the interface ∂Ω is replaced by ∂Ω1 or ∂Ω2

separately, namely,

0 = −∫

∂Ωp

f ′i(x)Gij(x,x0) dS(x) + µ

∫

∂Ωp

v′i(x)Tijk(x,x0)nk(x) dS(x), p = 1, 2. (17)

Note that the pair (v′, f ′) could be different for particle 1 and particle 2. Using Eq. (17), wecan eliminate the double-layer integrals in (13) and, consequently show that (15) is true fortwo- or multiple-particle cases.

2.3. Axisymmetric flow

We study the motion of a single solid spherical particle or two moving along their centerline, so the flow is axisymmetric. In this case, we can integrate along the azimuthal direction

6

and reduce the surface integral in Eq. (15) to a line integral along the contour C (semi-circle)of each particle in an azimuthal plane,

vi(x0) = − 1

8πµ

∫

C

Mij(x,x0)qj(x) ds(x), (18)

where i, j = 1, 2 indicate the axial and radial directions respectively, and M is the Green’sfunction tensor. For convenience and brevity, we abuse the notations x and x0 that wereused for points on particle contours to denote points in 3D space also. Suppose x = (x, σ, ψ)and x0 = (x0, σ0, ψ0) in cylindrical coordinates, one can calculate

r = |x− x0| =√

(x− x0)2 + σ2 + σ20 − 2σσ0 cosφ, where φ = ψ − ψ0. (19)

Following [40] and letting x = x−x0, the axisymmetric Green’s function tensorM in Eq. (18)is

M = σ

(

A10 + x2B30 x(σB30 − σ0B31)x(σB31 − σ0B30) A11 + (σ2 + σ2

0)B31 − σσ0(B30 + B32)

)

, (20)

where

Amn =

∫ 2π

0

cosn φ

rmA(λr) dφ, Bmn =

∫ 2π

0

cosn φ

rmB(λr) dφ. (21)

These two sets of integral Amn, Bmn are related to complete elliptic integrals, and can beevaluated accurately using the proposed hybrid scheme explained in § 3.1. The axisymmetricboundary integral formulation (18) also works for the case of two or multiple particles if thedefinition of the boundaries C includes all interfaces. Once the integration with regard tothe azimuthal angle φ is completed, as shown in Eq. (21), each entry of the Green’s functiontensor M in Eq. (20) is a function of (α, α0), the polar angles corresponding to the two pointsx and x0 in the meridional plane.

2.4. Uniqueness of solution

There is an eigen-solution ni(x) to the integral equation (15) because of the identity∫

∂Ω

ni(x)Gij(x,x0) dS(x) = 0. (22)

On the other hand, physical considerations dictate that the solution q to Eq. (15) be orthog-onal to n,

∫

∂Ω

q · n dS(x) = 0. (23)

To impose the constraint in Eq. (23), we form the following combination of Eq. (15) withEq. (23) to ensure the uniqueness of the solution

vj(x0) = − 1

8πµ

∫

∂Ω

qi(x)Gij(x,x0) dS(x)−1

8πµnj(x0)

∫

∂Ω

qi(x)ni(x) dS(x). (24)

7

Next, we show that Eq. (24) is equivalent to Eq. (15) and Eq. (23). We multiply Eq. (24)by nj(x0) and integrate over the particle surface and obtain

∫

∂Ω

nj(x0)vj(x0) dS(x0) =− 1

8πµ

∫

∂Ω

qi(x)

[∫

∂Ω

nj(x0)Gij(x,x0) dS(x0)

]

dS(x)

− 1

8πµ

∫

∂Ω

nj(x0)nj(x0) dS(x0)

∫

∂Ω

qi(x)ni(x) dS(x). (25)

The integral on the left hand side of Eq. (25) vanishes because of the incompressibilitycondition. The first term on the right-hand side also vanishes because of the identity∫

∂Ω

nj(x0)Gij(x,x0) dS(x0) = 0. As a result, the second term on the right hand side has to be

0, which implies Eq. (23) is true, because the integral

∫

∂Ω

nj(x0)nj(x0) dS(x0) =

∫

∂Ω

dS(x0)

is the surface area of ∂Ω. Therefore, we can deduce Eq. (15) from Eq. (24). In practice, weonly need to solve the equation (24) and the solution will be unique. We rewrite Eq. (24) as

vj(x0) = − 1

8πµ

∫

∂Ω

qi(x) [Gij(x,x0) + nj(x0)ni(x)] dS(x). (26)

For an axisymmetric flow, it reads

vi(x0) = − 1

8πµ

∫

C

[Mij(x,x0) + ni(x0)nj(x)] qj(x) dS(x), (27)

where i, j now indicate axial and radial directions. Thus, if the velocity v is given, Eq. (27)provides a Fredholm integral equation of the first kind for the unknown surface density q.

3. Numerical methods

In this section, we describe the numerical schemes used for computing the forces onspherical particles in a viscoelastic or unsteady Stokes flow in the Fourier domain when thevelocities of the particles are provided.

The most computationally expensive part of a boundary integral method is solving thedense linear system resulting from the discretization. When we compute the matrix-vectormultiplication directly without fast multipole or other fast algorithms, much of the compu-tational time is spent on evaluating of the Green’s functions. In Section 3.1, we present anaccurate and efficient method for evaluating the Green’s tensor M.

3.1. Evaluation of the Green’s function tensor

To compute the Green’s function tensorM in Eq. (20), one needs to evaluate the integralsAmn and Bmn in Eq. (21). Note that both integrands are 2π-periodic functions of φ, for whicha trapezoid rule should achieve spectral accuracy. However, a direct implementation of thetrapezoid rule will not suffice in practice if two points x and x0 on the particle contour areclose. It can be seen from Eq. (19) that r is close to 0 at φ = 0. Thus, the integrandsbehave like having a singularity there with a sharp spike. We take A32 as an example, andlet λ2 = i, so λ = (1+ i)/

√2. In Fig. 2(a), we plot the integrand in A32 for x = (0.2, 0.4) and

8

−2 0 2 4 6−50

0

50

100

150

200

250

300

350

400

Real partImaginary part

−0.2 0 0.20

200

400

−0.2 0 0.2−60

−40

−20

(a)

−2 0 2 4 60

0.002

0.004

0.006

0.008

0.01

Real, closeImag, closeReal, farImag, far

(b)

Figure 2: (a) The real and imaginary parts of the integrand for Amn in Eq. (21) are plotted for m = 3, n =2, λ2 = i,x = (0.2, 0.4),x0 = (0.3, 0.5), and φ ∈ [−π, π]. Close-ups near φ = 0 are plotted in the insets,showing smoother features. (b) The smoother part, i.e., the integrand in the second integral of Eq. (32), thatcorresponds to the integrand as shown in (a) is plotted. The smoother part is also plotted for an integrandsimilar to the one shown in (a), except that x is brought closer to x0 : x = (0.29, 0.49),x0 = (0.3, 0.5).

x0 = (0.3, 0.5) against φ ∈ [0, 2π]. The real part of the integrand does not appear smoothon the large scale, but it is smooth if we zoom in around φ = 0, as shown in the insets.The insets show the integrand on the interval [−0.2, 0.2]. This large curvature signifies alarge second derivative, causing difficulty for trapezoidal rule. If we applied the trapezoidalrule straightforwardly for the numerical integration, it would take costly large number ofquadrature points to achieve spectral convergence, as we will show later.

A common method to deal with the nearly singular integrand is to do a variable transfor-mation such that the quadrature points are more clustered around the ”pseudo”-singularity[36, 47]. To use this strategy, one needs to try and test different transformations. If it isimplemented with the right transformation, the computational cost should be independentof λ but still dependent on the distance |x−x0|, for the same accuracy. However, we proposea method based on kernel decomposition. Our integrands are periodic, and transformationmethod in general would destroy the periodicity and call for a Gauss-Legendre quadrature.We can preserve the periodicity with the kernel decomposition method, and apply the trape-zoidal rule. For the same accuracy, our method will be independent of |x−x0| but dependenton λ.

To illustrate, we take Amn as example and expand A(λr) (see Eq. (12)) at λr = 0,

A(λr) =∞∑

k=0

2(−1)k(k + 1)2

(k + 2)!(λr)k = 1− 4λ

3r +

3λ2

4r2 + · · · . (28)

Then we get

Amn =

∫ 2π

0

cosn φ

rm

(

1− 4λ

3r +

3λ2

4r2 + · · ·

)

dφ = Imn −4λ

3Im−1,n +

3λ2

4Im−2,n + · · · ,

(29)

9

where

Imn =

∫ 2π

0

cosn φ

rmdφ. (30)

Each Imn can be evaluated accurately and efficiently using complete elliptic integrals [41].However, one needs to determine how many terms to retain in the expansion of Eq. (29) suchthat a certain accuracy requirement can be met. To make the computation more efficient,we use a hybrid method and denote the first q + 1 terms in Eq. (28) by Aq:

Aq(λr) =

q∑

k=0

2(−1)k(k + 1)2

(k + 2)!(λr)k, (31)

and the remainder by Rq. Then,

Amn =

∫ 2π

0

cosn φ

rmAdφ =

∫ 2π

0

cosn φ

rmAq dφ+

∫ 2π

0

cosn φ

rmRq dφ. (32)

On the right-hand side of Eq. (32), the second integrand becomes smoother when q grows, andtherefore applying the trapezoidal rule can achieve high accuracy. While the first integrandmay display nearly singular behavior, it can be accurately evaluated using complete ellipticintegrals as explained. In practice, we choose q = m + 1 such that the leading order termin the second integrand is O(r2) to ensure sufficient smoothness. The smoother integrandfor the second integral of Eq. (32) is plotted for q = m + 1 in Fig. 2(b) correspondingto the integrand of Amn as shown in Fig. 2(a). The smoother part is also plotted for anintegrand similar to the one shown in Fig. 2(a), except that x is brought closer to x0:x = (0.29, 0.49),x0 = (0.3, 0.5). It is clear that the smoother part is largely independent ofthe distance between x and x0.

M 8 16 32 64 128errhybrid 1.4864e-06 3.4378e-08 7.2764e-11 2.9941e-15 1.4257e-15errtrap 0.4813 0.0472 4.1804e-04 2.4503e-08 1.2832e-15

Table 1: The relative errors in computing A32 for the integrand shown in Fig. 2(a) with x = (0.2, 0.4) andx0 = (0.3, 0.5) using the hybrid method, errhybrid, and the trapezoidal rule, errtrap.

M 8 16 32 64 128 256 512errhybrid 9.3051e-08 1.7951e-08 2.9889e-09 3.4072e-10 1.7054e-11 1.2330e-13 1.3720e-14errtrap 12.7805 5.8949 2.4680 0.8122 0.1408 0.0048 4.3747e-06

Table 2: The relative errors in computing A32 with λ2 = i, x = (0.29, 0.49) and x0 = (0.3, 0.5) using thehybrid method, errhybrid, and the trapezoidal rule, errtrap.

To demonstrate the efficiency of the hybrid method, we compute the integral A32 for theintegrand as shown in Fig. 2 using the hybrid method based on the kernel decomposition

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(32) and the straightforward implementation of the trapezoidal rule. Tables 1 and 2 showthe relative errors using the two numerical integration methods for the integral A32 with thepair of the points, x = (0.2, 0.4) and x0 = (0.3, 0.5), and another pair of the points that arecloser to each other, x = (0.29, 0.49) and x0 = (0.3, 0.5), respectively.

In the tables, M is the number of subintervals in the trapezoidal rule, which is also usedin the hybrid method for the second integral on the right-hand side of Eq. (32). The costsfor the two methods are comparable for the same value ofM , since the elliptic integrals usedin the hybrid method can be evaluated in negligible time. The tables show that the hybridmethod is much more efficient than the trapezoidal rule, and it requires approximately twoorders of magnitude fewer subintervals to achieve the same accuracy for the target and thesource points that are close to each other. It would be interesting future research trying tocombine the kernel decomposition method and the transformation method.

3.2. Quadrature

The Green’s function tensor (20) is for unsteady Stokes flow or LVE flow. To simplifypresentation of our quadrature method, we start with the simpler Stokes flow by lettingλ = 0, so Eq. (20) becomes

M = σ

(

I10 + x2I30 x(σI30 − σ0I31)x(σI31 − σ0I30) I11 + (σ2 + σ2

0)I31 − σσ0(I30 + I32)

)

. (33)

Each Imn is a combination of the complete elliptic integrals K(η2) and E(η2) of the first andsecond kinds, where

η2 =4σσ0

(x− x0)2 + (σ + σ0)2, (34)

and

K(η2) =

∫ π/2

0

dθ√

1− η2 sin θ, E(η2) =

∫ π/2

0

√

1− η2 sin θ dθ. (35)

The complete elliptic integrals have asymptotic expansions involving logarithmic singularitiesas η tends to 1. For λ 6= 0, Amn and Bmn in the Green’s function tensor Eq. (20) are λ-weighted combinations of Imn’s, as shown in Eq. (29). Therefore, one can similarly show thateach integrand M(α, αl) in the Green’s function tensor Eq. (20) for λ 6= 0 has the followingexpansion about α = αl (η = 1):

M(α, αl) = M(α, αl) +∞∑

k=0

ck(αl)(α− αl)k log |α− αl|, (36)

where M(α, αl) is a smooth function in α. We note that, when αl = 0 or π, M(α, αl) is finitein α without logarithmic singularities. There are various methods to evaluate integrals withlogarithmic singularities, including the hybrid Gauss-trapezoidal quadrature [48] and theextrapolation method [49], among others. Here we use a modified trapezoidal quadrature.

11

For an integrand M(α, αl) defined for α ∈ [a, b] with logarithmic singularities at α = αl, thefollowing quadrature error formula can be shown:

∫ b

a

M(α, αl) dα = h

N∑

k=0k 6=l

′M(αk, αl) + hM(αl, αl) + c0(αl)h logh

2π

+m∑

k=1k odd

γk(M(k)(b, αl)−M (k)(a, αl))h

k+1 +m∑

k=2k even

νkck(αl)hk+1 +O(hm+2),(37)

where m ≥ 0, h = (b− a)/N , αk = a+ kh, and the prime on the summation indicates thatthe two end summands are weighted by half like for the regular trapezoidal rule [50, 42].We have assumed αl is one of the quadrature points. The formula for the constants γk andνk are provided in [50]. Modifying the ’punched-hole’ trapezoidal rule (the first term on theright-hand side of (37)) by adding the leading-order error terms in the expansion of Eq. (37)gives a modified trapezoidal rule dealing with logarithmic singularities.

To implement this quadrature, one needs the expressions for the smooth part M , thederivatives M (k) of the integrand at the end points a, b, and the coefficients ck’s at αl. Onecan obtain a quadrature of arbitrarily high order if all these quantities can be analyticallycomputed, which is the case for steady Stokes flow [42]. For the viscoelastic Green’s func-tion tensor Eq. (20), it is not possible to perform a full asymptotic analysis to obtain allthese quantities, though all the derivatives M (k) at the end points can be computed exactly.Nevertheless, we will show that one can obtain c0 and c1 analytically, and then use thisknowledge to interpolate the smooth part M(αl, αl) to second order accuracy in h. Thus, wecan use the expansion on the right-hand side of Eq. (37) with m = 1 to obtain a third-orderaccurate quadrature. In our case, one does not have a complete knowledge of the logarith-mic singularity in a Green’s function tensor, and thus the order of accuracy of a quadratureviable through Eq. (37) depends on the availability of the coefficients ck.

The derivatives M (k) at the end points can be calculated analytically, and we only needthem for k = 1 for a third-order accurate quadrature. For completeness, we include them inAppendix A. We now explain how to compute c0 and c1 exactly, and use them to interpolateM . Expanding the entries in the axisymmetric Green’s tensor (20) at α = αl reduces toexpanding Amn and Bmn at that point, which itself reduces to expanding Iin for i ≤ m, byvirtue of Eq. (29). We provide the details for A10 as an illustration. From Eq. (29), we have

A10 =

∫ 2π

0

1

r

(

1− 4λ

3r +

3λ2

4r2 + · · ·

)

dφ = I10 −4λ

3I00 +

3λ2

4I−1,0 + · · · . (38)

With simple algebraic manipulations, each of the Ii0’s can be written as a combination ofthe first and second kind complete elliptic integrals K(η2) and E(η2). Expanding these twointegrals near η = 1 (α = αl), we obtain

K(η2) = K(α, αl)− log |α− αl|+∞∑

j=2

cKj (αl)(α− αl)j log |α− αl|,

E(η2) = E(α, αl) +∞∑

j=2

cEj (αl)(α− αl)j log |α− αl|, (39)

12

where K(α, αl) and E(α, αl) are the smooth parts of the functions K(η2) and E(η2) respec-tively, i.e., cK0 = −1, cK1 = 0 for K(η2) and cE0 = 0, cE1 = 0 for E(η2). The higher coefficientscKj , c

Ej with j ≥ 2 are also available analytically, but we do not need them.Now we consider each Ii0 on the right-hand side of the expansion (38). For i even, Ii0

is a smooth function of α. For Ii0 with a negative odd i, its expression in terms of ellipticintegrals will only involve E(η2), because the integrand is clearly bounded. Consequently,only I10 can contribute c0 and c1 towards A10 since it is the only term that contains K(η2).The other Amn’s and Bmn’s can be treated similarly, i.e., only the first m terms of theirexpansions similar to (38) need to be analyzed for computing c0 and c1. Since there is noway to compute c′ks for k ≥ 2, it appears that quadratures of higher orders are not accessiblewith the formulation (37) alone.

For each of the entries in the Green’s tensor (20), using the procedure described above,we have obtained the coefficients c0 and c1 in the quadrature error formula (37),

M11 : c0 = −2, c1 = −xl/σl; M12 : c0 = 0, c1 = 1;

M21 : c0 = 0, c1 = −1; M22 : c0 = −2, c1 = −xl/σl.(40)

Note that the coefficients ck’s in Eq. (40) are simple and independent of λ.Next, we interpolate to obtain the smooth part M(αl, αl) in (37). Rewriting the expansion

(36) by further expanding the smooth function M(α, αl), we have

M(α, αl) = a0+a1(α−αl)+a2(α−αl)2+ · · ·+(c0+c1(α−αl)+c2(α−αl)

2+ · · · ) log |α−αl|.(41)

Thus, the four-point symmetric interpolation scheme has the following expansion

1

6(−M(αl−2h, αl)+4M(αl−h, αl)+4M(αl+h, αl)−M(αl+2h, αl)) = a0−

1

3c0 log 2+c0 log h+O(h

2).

(42)Knowing c0, we can obtain a0 at αl accurate to second order in h. When αl is the quadraturepoint next to 0, the four-point interpolation formula (42) cannot be used. We use thefollowing three-point interpolation scheme

1

3(M(αl − h, αl) + 3M(αl + h, αl)−M(αl + 2h, αl)) (43)

to obtain M(αl, αl). Expansion of the scheme (43) shows that one needs the knowledge ofboth c0 and c1 for an O(h2) interpolation of the smooth part M . Since the errors grow fastnear the pole at α = 0 As we will see in the next section, the numerical error is larger nearthe pole α = 0. Abrupt change from the four-point interpolation scheme to the three-pointone near the pole causes the errors to be non-smooth. In practice, to avoid this, we use thethree-point interpolation (43) for all αl ≤ π/2; for αl > π/2, the interpolation formula isobtained by symmetry.

3.3. Pole correction

For the case of a single particle, the analytic solution to the BIE (27) is given in [46]. Weapply the numerical methods described above to evaluate the velocity v from the boundaryintegral equation (27), when the force f is prescribed as in Eq. (46). In Fig. 3(a), we plot

13

0 π/2 π10

−10

10−8

10−6

10−4

10−2

αl

(a)

0 π/2 π10

−10

10−8

10−6

10−4

10−2

αl

(b)

0 π/2 π10

−10

10−8

10−6

10−4

10−2

αl

(c)

0 π/2 π10

−10

10−8

10−6

10−4

10−2

αl

(d)

Figure 3: (a) The relative error in the computed velocity, when the force is given, is plotted against αl ∈ [0, π]for different spatial resolutions h = π/N with N = 8, 16, 32, 64, 128, M = 64 and λ2 = i(1 + i)/

√2. (b) The

same as (a) except after the pole correction. (c) The same as (a) except for λ2 = i. (d) The same as (c)except for λ2 = i.

the error in the computed velocity v against the parameter α for different spatial resolutionsN = 8, 16, 32, 64, 128,M = 64, and λ2 = i(1 + i)/

√2 (viscoelastic). It can be seen that, for

each fixed value of αl, the error is of third-order accuracy in the spatial resolution h = π/N , asexpected. However, the error increases near the poles, corresponding to αl = 0 and π, and theerror measured in infinity norm only shows an accuracy of order one. In other words, the errordoes not decrease uniformly asO(h3). This numerical behavior, intrinsic to boundary integralmethods applied axisymmetrically, has been discussed in [42]. The problem originates fromthe fact that in the expansion (36) the coefficient ck grows unbounded as α → 0, π. TakingM11 as example, one can find that ck(αl) ∼ α−k

l as αl → 0. The derivatives of the componentsin the Green’s tensor, M (k), in the quadrature error formula (37) show similar behaviors.

The method of getting the pole correction is to approximate the integrands using Taylorexpansions at α = αl = 0 or π. For instance, the integrand M11q1 in Eq. (27) can beapproximated as

M11q1 = M11 +O(α2) ≡ αA10(1 + λ+ λ2) +O(α2). (44)

14

Then the integral is evaluated as

∫ π

0

M11q1 dα =

∫ π

0

(M11q1 − M11) dα +

∫ π

0

M11 dα. (45)

For the first integral on the right-hand side of (45), the integrand becomes less singular,ck(αl) ∼ α−k+2

l , as αl → 0. We use the same quadrature described above, and the error willbe O(α3

l ) as αl → 0. For the second integral on the right-hand side of (45), we precomputeit, and the correction values can be used for all values of λ and the resolution h. Figure 3(b)shows the result after using the pole correction scheme. We can see that the error is uniformlythird-order accurate, except at the two poles where it is fourth-order. The other threeentries in the Green’s function tensor can be treated similarly, by deriving the correspondingapproximate integrands. For comparison, we present the results corresponding to unsteadyStokes flow λ2 = i in Figs. 3(c) and (d). The difference between the two cases, λ2 =i(1 + i)/

√2 and λ2 = i, is small, thus we use the relative l2-error from now on.

4. Results

In this section, we verify our numerical methods against known exact or asymptoticsolutions. When an isolated spherical particle undergoes a single mode oscillation in aviscoelastic flow, the analytic expression for the traction on the particle can be obtained byusing the results for Stokes flow in [46] together with the correspondence principle. We firsttest and verify our numerical method using the analytic solution, and a sweep of frequencieswill be considered. We then move to two particles moving in unsteady Stokes flow. In thiscase, our studies consider two parameters: the frequency and the separation of particles. Wecompare our results to an existing asymptotic solution [51] for small ǫ, the ratio of particleradius to particle distance. We perform quantitative studies to find the ranges of separationwhere the asymptotic solution is valid.

4.1. Accuracy of the scheme

To study the order of convergence of the scheme, we compare with the known exact solu-tion. We consider a single-mode Maxwell (linear viscoelastic) fluid with relaxation modulus

given by G = G0e−t/λ0 or G =

G0λ01 + iωλ0

.

The governing equation (27) in the Fourier domain is solved in the infinite space R3

with a single immersed spherical non-deformable particle, and we assume equal density ρ forthe particle and the fluid. If the particle undergoes a single frequency oscillation, i.e., thevelocity is veiωt, then the traction feiωt on the particle is

f = −3G(ω)

2R

[

(1 + λR)I +λ2R2

3nn

]

· v (46)

where R is the radius of the particle, I is the identity matrix and λ2 = iωR2ρ/G [46]. Welet R = 1, ρ = 1, G0 = 1, λ0 = 1 (so λ2 = iω(1 + iω)), and v = (1, 0). Since the real part G′

(storage modulus) and the imaginary part G′′ (loss modulus) of G stand for the elasticity andfluidity of the material respectively, λ2 = iω(1+ iω) would represent a spectrum of elasticity

15

as ω goes from 0 to ∞. For convenience, we further normalize λ2 by setting λ2 =iω(1 + iω)

|1 + iω|such that ω is the magnitude of λ2. Since v is prescribed, Eq. (46) provides the force f onthe particle. This exact pair (v, f) is employed to check our numerical scheme.

We have also verified that the accuracy of our method, in l2-norm, depends only on themagnitude of λ2, and not on whether λ2 is fully complex (LVE) or purely imaginary (unsteadyStokes). Since v can be freely prescribed, we use Eq. (46) to obtain the exact force f on theparticle. This pair (v, f) is exploited to check our numerical scheme. For a single sphericalparticle undergoing a single-frequency oscillation, the ghost flow v′ inside the particle yieldsf ′ = λ2(x ·v)n, as explained in Sec. 2.2. We use the numerical methods described in previoussections to evaluate q from Eq. (27) and find the force f from f = q+ f ′, then check f againstits exact values. We call this the backward problem. We first test the numerical methodswith a forward problem: using Eq. (27) to numerically recover the particle velocity when thesurface force profile is prescribed by Eq. (46).

4.1.1. Forward problem

ω = 1 4 8 16 32 ω = 10 8 16 32 648 3.0 3.0 3.0 3.0 16 2.9 2.9 2.9 2.916 2.6 3.1 3.1 3.1 32 2.3 3.1 3.1 3.132 0.7 3.0 3.1 3.1 64 0.3 2.9 3.0 3.064 0.0 2.0 3.0 3.1 128 0.0 1.5 3.0 3.0

ω = 100 16 32 64 128 ω = 1000 32 64 128 25632 2.5 2.5 2.5 2.5 64 2.0 2.2 2.2 2.264 1.9 2.9 2.9 2.9 128 1.1 2.7 2.7 2.7128 0.2 2.8 3.0 3.0 256 0.1 2.5 2.9 2.9256 0.0 1.2 2.9 3.0 512 0.0 0.6 2.9 3.0

Table 3: Numerical order of convegence for the forward problem in terms of the resolutions in meridional planeN (vertical) and the azimuthal directionM (horizontal), with λ2 = iω(1+iω)/|1+iω| and ω = 1, 10, 100, 1000.The numerical order is obtained by computing the ratio of the l2-errors of N and N/2 with a fixed value M .

In Table 3, we provide the numerical orders of convergence of our scheme for the rep-resentative values in the magnitude of λ2 or ω = 1, 10, 100, 1000. The numerical errorsare computed by solving the forward problem (finding the velocity given the surface force)numerically and then comparing the numerical results with the analytic solution. The nu-merical order is given by log2 of the ratio of the errors corresponding to N and N/2 for a fixedvalue of M . The table also shows how the order of convergence depends on the numericalparameters N (the number of segments on the semi-circle C in (37)) and M (the numberof quadrature points in evaluating the smooth part of the axisymmetric Green’s functionsin (32)). Table 3 shows that, for larger magnitude of λ2, one has to take larger values ofN (the numbers below each ω in the table) and M (the numbers to the right of each ω inthe table) in order to reach the theorectical order of convergence: order three. For example,when ω takes the value of 1000, the numerical scheme achieves the third-order accuracywhen M ≥ 128 and N ≥ 256.

16

10−2

10−1

100

101

102

10−10

10−8

10−6

10−4

10−2

100

Figure 4: The relative l2-errors for the forward problem are plotted against ω evenly distributed in [10−2, 102]on log scale, using different spatial resolutions π/N with N = 128, 256, 512, 1024(from top to bottom) andM = N/8(−−), N/4(−·), N/2(−). An error threshold line at 10−5 is drawn.

From Table 3, it can be seen that the method reaches the third-order convergence rate ifN is large enough, and this occurs approximately whenM reaches N/2. We now quantify theaccuracy of our method by showing that this is also the value ofM at which further increasingM does not improve the accuracy for that fixed value of N . Figure 4 shows the l2-error of thenumerical method for the forward problem by taking ω = 2m with m = −7,−6, · · · , 7. Thelines correspond to different values of the spatical resolution N = 128, 256, 512, 1024, wherethe solid lines are for M = N/2, the dash-dotted lines are for M = N/4 and the dashedlines are for M = N/8. The magnitudes of λ2 or ω are evenly distributed on log scale,corresponding approximately to the interval [10−2, 102]. The graphs show that the accuracylevels out for ω < 10−2. It can be seen that M ≥ N/4 is sufficient for convergence, at leastfor the range of frequencies considered here. To be conservative, we will use M = N/2 whenapplying our method to motion of two particles.

ω = 0 N = 64 N = 128 ω = 100 N = 64 N = 128

Interpolated M 1.4034e-6 1.8182e-7 Interpolated M 2.5268e-2 3.2380e-3

Exact M 0.9183e-7 1.1701e-8 Exact M 1.9572e-3 2.4485e-4

Table 4: The l2-error in the computed velocity is tabulated for ω = 0, 100, N = 64, 128 and M = 64. Thesmooth part M in Eq. (37) is obtained using two different methods: interpolated or computed analytically.

In our scheme, the smooth part M(αl, αl) in Eq. (37) needs to be interpolated for unsteadyStokes flow or LVE flow. We now study the effect of this interpolation on accuracy. First, wecheck against the steady Stokes flow case (λ2 = 0), where M can be analytically obtained.We solve the forward problem with pole correction for λ2 = 0 using interpolated M , andthen solve the problem again using M analytically computed. The errors of both solutions

17

in l2-norm are tabulated in Table 4. We note that the pole correction values need to bere-computed using the new quadrature with analytic values of M . It can be seen that wegain approximately one order of magnitude in accuracy, when the value of the smooth partM(αl, αl) is analytically computed. We now demonstrate that this ratio of gain in accuracyis largely independent of the size of λ2. For this purpose, we carry out similar computations,but with the higher value of ω = 100. Since the smooth part M for non-zero λ2 can not becomputed analytically, we obtain their ’exact’ values by interpolating them with a sufficientlysmall h such that the interpolation errors are below 10−14. The results in Table 4 show thatwe gain only about one order of magnitude in accuracy by using the ’exact’ values of M . Inthe computations hereafter, we obtain M by interpolation on the fly, because losing one orderof magnitude in accuracy is not significant for our applications. If much higher accuracy isrequired, interpolating M to machine precision in advance can be considered.

4.1.2. Backward problem

We now turn to the more challenging backward problem: use the boundary integralmethod to solve the first-kind Fredholm integral equation (27) for the traction on particlesmoving in unsteady Stokes or viscoelastic flow, if their velocities are known. Since theresulting linear systems from discretizing integral equations are of moderate sizes, we solvethem by the LU decomposition. We find the condition number of the matrix correspondingto the discretized equations of Eq. (27) behaves like O(h−2). Since solving the forwardproblem is of third-order accuracy, we expect that solving the backward problem is of first-order accuracy in the worst case. However, we find that the l2-error of the traction inthe backward problem decays with an order of 2.5, thus we only lose half of an order ofconvergence.

In fact, we can obtain the third-order convergence rate in the total surface force. In ourapplications of studying motion of micro-sized particles in unsteady Stokes or LVE flow, weare only concerned with the total force exerted on the particles by the surrounding fluidwhen their velocities are given. Thus, we present the accuracy of our method using the errorin computing the total force, rather than the l2-error for the traction vector. We integrateEq. (46) over the sphere to obtain the total force exerted on the particle

ftot = −(

6π + 6πλ+2π

3λ2)

v. (47)

Since the traction on the particle is not periodic in the meridional direction, we numericallyintegrate using the Simpson’s rule to obtain the total force ftot on the particle.

We compare the numerical result to the exact solution in Eq. (47). In Fig. 5, we presentthe numerical errors in computing ftot with the same parameter setting (we use M = N/2here) as in Fig. 4. The figure shows a convergence order of three. Though the computedtraction only shows a convergence order of 2.5 in l2-norm, we recover the order of three forthe error in the integral force due to the presence of the sine function in the surface areaintegration factor provding the smoothing effect near the poles. We can refer to Fig. 5 whenselecting the proper values of N and M in simulating motions of two particles in unsteadyStokes flow in the next section. For later use, we draw an error threshold line at 10−5 tosuggest values of N and M for achieving the accuracy for a prescribed frequency magnitude.

18

10−2

10−1

100

101

102

10−10

10−8

10−6

10−4

10−2

100

Figure 5: The errors in the computed total force ftot for the backward problem are plotted with the solidlines (–) against ω evenly distributed in [10−2, 102] on log scale for the spatial resolution π/N with N =128, 256, 512, 1024 (from top to bottom) andM = N/2. An error threshold line at 10−5 is drawn for reference.The results where the pole correction is not implemented are also shown with the dash-dotted lines (−·).

The errors in the total force obtained from the backward problem, as shown in Fig. 5, areonly slightly larger than the l2-errors for the forward problem, as shown in Fig. 4.

To demonstrate the effect of pole correction, we also show the numerical errors obtainedwithout the pole correction (the dash-dotted lines in Fig. 5). The errors without the polecorrections are hundreds of times larger than those with the pole correction for the samevalues of N , M and λ2, and they converge less than the order of three except at highfrequencies.

4.2. Motion of two particles in unsteady Stokes flow

Accurately capturing the hydrodynamic interaction of two particles in purely viscous orviscoelastic flow is of fundamental importance for studying two-particle microrheology andbinding kinetics of particles in those flows. We compare our numerical results with an existingasymptotic solution for motion of two particles in unsteady Stokes flow, which was derivedfor the case of large separation of the particles [51]. For studies of two-particle microrheology,the two particles need to be sufficiently separated (at least one diameter apart), in contrastto studies of binding kinetics of approaching particles. The asymptotic solution of [51] forunsteady Stokes flow has been exploited to study two-particle microrheology in viscoelasticitywith great success [8, 43, 21, 44, 10, 9], by using the correspondence principle. Here, weinvestigate the validity of the asymptotic solution for study of binding kinetics.

For the case of two same-size particles moving in tandem in a steady Stokes flow, Stimsonand Jeffrey [52] used the method of stream-function to derive that the force on each particleis F = 6πµRV β, with the correction coefficient β (from one particle to two particles) given

19

by

β =2

3sinh θ

∞∑

n=1

n(n+ 1)

(2n− 1)(2n+ 3)

1− 4 sinh2(n+ 1/2)θ − (2n+ 1)2 sinh2 θ

2 sinh(2n+ 1)θ + (2n+ 1) sinh 2θ

, (48)

where θ = cosh−1(1/ǫ) with ǫ := R/L being the ratio of the particle radius R to the distancebetween the two particle centers L. If the inertia effect can not be ignored, Ardekani andRangel derived asymptotic solutions for the force on two particles moving in unsteady Stokesflow [51]. The approximate solutions are derived for the case of large separation of theparticles or small values of the ratio ǫ, and are third-order accurate in ǫ. One of their resultsrelevant to our studies is outlined in Appendix B for completeness of presentation.

Inertia effect and viscoelastic effect are similar in that they both cause retardation. Oneadvantage of solving the problems in the Fourier domain is that these two effects can becaptured simultaneously. We use our boundary integral method developed for linear vis-coelasticity to study the accuracy of the asymptotic solution for motion of two particles inunsteady Stokes flow.

2.25 3 4 5 6 810

−4

10−3

10−2

10−1

100

Particle separation

Err

or in

the

asys

mpt

otic

sol

utio

n

Slope=−3

ω=0.01ω=0.1ω=1ω=10ω=100

Figure 6: The relative errors in the asymptotic solution are plotted for λ2 = iω with ω = 0.01, 0.1, 1, 10, 100,and the normalized particle separation L/R = 2.25, 3, 4, 5, 6, 8. A 1% error threshold line is drawn.

We let the two particles undergo motion in tandem along the axis of their centers withvelocity veiωt, and then solve for the traction on them using our numerical method for differ-ent values of the frequency ω and the particle separation ǫ. We take ω = 0.01, 0.1, 1, 10, 100and seek an accuracy of approximately 10−5 in our numerically computed force. Refer-ing to Fig. 5, we select N = 32, 32, 64, 256, 1024 and M = N/2 for the simulations withω = 0.01, 0.1, 1, 10, 100 respectively. While the accuracy may also depend on ǫ, we havefound that the condition number of the discretization matrix for the two particle problem isindependent of ǫ, at least when its value is smaller than 4/9 (the two particles are at least aquarter radius apart). We note that, just like the case for one particle, the condition numberalso scales like O(h−2). Because of the relative independence of the condition number on ǫ,we use the same N in our simulations for all values of ǫ considered, when ω is fixed. Becauseof the high accuracy of our boundary integral method, we take our numerical results as theexact solution and compute the relative errors in the asymptotic solutions. We present theresults in Fig. 6 for the normalized particle separation L/R = 2.25, 3, 4, 5, 6, 8. We can seethat, when ǫ = 1/4, the error in the asymptotic solution is approximately 1%. Moreover,

20

the results show that the errors are weakly dependent on ω. The asymptotic solution has anerror larger than 6% when ǫ is larger than 4/9. This indicates that when the two particlesare close to each other, the asymptotic solution should be used with caution. We also drawa line of slope of −3 in the figure since the asymptotic solution is of third-order accuratein ǫ. Interestingly, the errors in the asymptotic solution display an order of approximatelythree even for large values of ǫ.

0 π/2 π0

0.5

1

1.5

2

direction of the

other particle

ω=0.01ω=0.1ω=1ω=10ω=100

Figure 7: The profile of the traction on the particle surface is plotted for the same ω values as in Fig. 6 andthe particle separation L/R = 3.

As another demonstration of our numerical scheme, we also plot the profile of the tractionon one particle surface for all the frequencies studied. We choose the small particle separa-tion, ǫ = 1/3, so that we could examine the effect of particle interaction. For convenience ofpresentation, we normalize each profile by its largest value and present the results in Fig. 7.It can be seen that the force becomes more symmetric with respect to the plane separatingthe fore and rear halves of the particle, as the frequency ω increases. At high frequencies,the force due to the added mass dominates, which is symmetric since it does not distinguisha front side from a rear side in motion. At lower frequencies, our results show that the polefacing the other particle feels larger drag.

5. Conclusion

In this work, we have developed an accurate and efficient boundary integral method tocompute the forces on interacting spherical particles in an unsteady viscoelastic flow, if themotions of the particles are known. Our formulation is simple as it only involves single-layerpotentials. Although the formulation results in a Fredholm integral equation of the firstkind, the right-hand side of the equation is given and without any perturbation (except thecomputer round-off errors). The formulation could be applied to the cases in which multipleparticles are present, and reduces the computational cost significantly comparing with thestandard formulation. The computational complexity of our boundary integral method scaleswith O(M2N2) for the formulations with or without the double-layer integrals, whereM andN are the number of quadrature points in evaluating the Green’s function and the numberof the marker points along the particle contour respectively. By eliminating the double-layerintegrals, assuming that we are solving using Gaussian elimination, the computational costis reduced by a factor of at least four, because the number of terms in the Green’s tensor

21

(Tijk is four times larger than that for the Green’s tensor (Gij) and their expressions aremore complicated.

Without any fast algorithm applied to the boundary integral method, almost all the com-putational cost resides in evaluating the Green’s functions. To this end, we have presented ahybrid integration scheme that evaluates the axisymmetric Green’s functions with seven or-ders of magnitude more accuracy than the trapezoidal rule, even though the trapezoidal ruleis spectrally accurate for evaluating integrals with periodic and smooth integrands. For un-steady Stokes and viscoelastic flow, we also have provided the correction schemes to treat theintrisic singular behavior of the boundary integral methods near the poles of axisymmetricsurfaces.

We have verified the accuracy of the proposed numerical schemes against an exact solutionfor an isolated particle in a linear viscoelastic flow. We have compared our results with anexisting asymptotic solution for two spherical particles translating along their centers in anunsteady Stokes flow.

Acknowledgments

This work was supported by the NSF through grant DMS-0923111, DMS-0914923, DMS-1217277, the ARO through grant W911NF-08-2-0058, W911NF-09-2-0071, and an IllinoisInstitute of Technology post-doctoral traveling allowance. We thank Prof. John Lowengrubof University of California, Irvine, for helpful comments.

Appendix A. Appendix A. Derivatives of the Green’s functions at the end points

To implement the third-order accurate scheme on Eq. (37), one needs the derivativesM (k) for k = 1 at the end points. They can be calculated analytically as follows. We firstconsider the case when both x and x0 are on the same particle. Let

d0 =√

(1− xl)2 + σ2l , dπ =

√

(−1− xl)2 + σ2l ,

A0 = A(λd0), B0 = B(λd0), Aπ = A(λdπ), Bπ = B(λdπ), (A.1)

where (xl, σl) is the polar coordinates of the point corresponding to α = αl. Then, we havefor any αl

M ′12(0, αl) =M ′

12(π, αl) =M ′22(0, αl) =M ′

22(π, αl) = 0; (A.2)

for αl = 0, π (x and x0 may coincide)

M ′11(0, 0) = −8πλ/3, M ′

11(π, 0) = −π(Aπ + Bπ), M ′11(0, π) = π(A0 +B0), M ′

11(π, π) = 8πλ/3,

M ′21(0, 0) = 0, M ′

21(π, 0) = 0, M ′21(0, π) = 0, M ′

21(π, π) = 0;(A.3)

and for αl 6= 0, π

M ′11(0, αl) = 2π

d20A0 + (1− xl)2B0

d30, M ′

11(π, αl) = 2π−d2πAπ − (−1− xl)

2Bπ

d3π,

M ′21(0, αl) = 2π

−σl(1− xl)B0

d30, M ′

21(π, αl) = 2πσl(−1− xl)Bπ

d3π. (A.4)

22

When x is on the right particle while x0 stays on the left particle, the formula (A.2) and(A.4) are valid with xl replaced with L + xl, where L is the distance between the particlecenters. In this case, the special cases (A.3) are replaced by (A.4), because x and x0 areresiding on different particles.

Appendix B. Appendix B. Asymptotic solution for two particles in unsteady

Stokes flow

For two spherical particles of the same size moving in tandem (along their line of centers)in unsteady Stokes flow, Ardekani and Rangel [51] derived an asymptotic solution for theforce on each particle if their velocities are known. For simplicity, we choose physical scalessuch that the relevant physical quantities are of unit size. Furthermore, we assume that theparticles and the fluid have the same density. We emphasize that these assumptions do notsacrifice generality. With these simplifications, the force on each particle is given by

F = −6π

V

1 + 3ǫ/2− ǫ3+

1

9

dV/dt

1 + 3ǫ3+

∫ t

−∞

dV

dτh(t− τ) dτ

, (B.1)

where ǫ is the ratio between the particle radius and the particle separation, and the inertialmemory kernel h(t) has the following Laplace transform

h(s) =1 +

√s+

s

9

s+ 3ǫ3[

1 +√s+

s

3− e−

√s/ǫ+

√s

(

1 +

√s

ǫ

)] . (B.2)

The asymptotic solution is derived for the limiting case of ǫ→ 0, and is third-order accuratein ǫ. Since we compare in the Fourier domain, we take the Fourier transform in time ofEq. (B.1) to obtain

F (ω) = −6π

1

1 + 3ǫ/2− ǫ3+

1

9

iω

1 + 3ǫ3+ iωh(ω)

V (ω), (B.3)

where the Fourier transform h(ω) is provided by

h(ω) =1

2πi

∫ ∞

0

1

s+ iω[h(se−iπ)− h(seiπ)] ds. (B.4)

This integral in (B.4) is evaluated numerically.

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