A transform method for the biharmonic equation in multiply ...

IMA Journal of Applied Mathematics (2018) Page 1 of 33doi:10.1093/imamat/xxx000

A transform method for the biharmonic equation in multiply connectedcircular domains

ELENA LUCA∗

Department of Mechanical and Aerospace Engineering, University of California, San Diego,9500 Gilman Drive, La Jolla, CA 92093, USA∗Corresponding author: [email protected]

AND

DARREN G. CROWDYDepartment of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ,

[email protected]

[Received on 29 June 2018]

A new transform approach for solving mixed boundary value problems for the biharmonic equation insimply and multiply connected circular domains is presented. This work is a sequel to Crowdy [2015,IMA J. Appl. Math. 80, 1902–1931] where new transform techniques were developed for boundary valueproblems for Laplace’s equation in circular domains. A circular domain is defined to be a domain, whichcan be simply or multiply connected, having boundaries that are a union of circular arc segments. Themethod provides a flexible approach to finding quasi-analytical solutions to a wide range of problems influid dynamics and plane elasticity. Three example problems involving slow viscous flows are solved indetail to illustrate how to apply the method; these concern flow towards a semicircular ridge, a translatingand rotating cylinder near a wall as well as in a channel geometry.

Keywords: biharmonic equation; transform method; mixed boundary value problem; circular domain.

1. Introduction

In a recent paper one of the authors (Crowdy , 2015c) presented a transform method for solving a vari-ety of mixed boundary value problems for Laplace’s equation in the class of two-dimensional circulardomains. A circular domain is defined to be a planar domain, which can be simply or multiply con-nected, having boundaries that are a union of circular arc segments; that is, the boundary is made up ofa finite set of (possibly disconnected) segments with piecewise constant curvature. A polygon is a veryspecial case in which all boundary segments have zero curvature. Figure 1 illustrates the diversity ofthe class of circular domains. A transform method applicable to boundary value problems for Laplace’sequation in simply connected polygonal domains was developed by Fokas & Kapaev (2003); it wasextended to boundary value problems for biharmonic fields in the same class of domains by Crowdy& Fokas (2004), with further developments made more recently (Dimakos & Fokas , 2015). All thesetransform techniques fit broadly into the so-called unified transform method pioneered by Fokas andcollaborators (Fokas , 2008) and often called the Fokas method. The present paper is a generalizationof the work of Crowdy & Fokas (2004), which pertained to the biharmonic equation only in polygonaldomains, to the biharmonic equation in the much more general class of circular domains.

c© The author 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 of 33 E. LUCA & D. CROWDY

FIG. 1: Examples of circular domains. A simply connected polygon (left), a simply connected circulardomain (middle) and the doubly connected “disc-in-channel” geometry (right).

It is well-known that, in planar multiply connected domains, the problem of finding a harmonicfield φ(x,y) satisfying some boundary value problem for Laplace’s equation in that domain can betransformed to that of finding a complex potential w(z) which is an analytic (but not necessarily single-valued) function in the domain with real part equal to φ . The present paper is a natural sequel toCrowdy (2015c) which was based on application of “Fourier-Mellin transform pairs” for such complexpotentials w(z) defined over the class of circular domains. Those Fourier-Mellin transform pairs forcircular domains were first derived by Crowdy (2015a). By making use of the so-called global relationsnaturally associated with such transform pairs it is possible to find the complex potential w(z) and, hence,to solve a given boundary value problem for a harmonic function φ .

If ψ(x,y) is instead a function that satisfies the biharmonic equation,

∇4ψ = 0, (1.1)

in the same class of domains then ψ can be written as

ψ = Im[z f (z)+g(z)], (1.2)

where f (z) and g(z) constitute a pair of complex potentials. These two functions are sometimes calledGoursat functions. In the present paper we follow the ideas of Crowdy (2015c) and use the Fourier-Mellin transform pairs of Crowdy (2015a) to represent the two Goursat functions f (z) and g(z) and, byanalysis of the relevant global relations and boundary conditions, to solve a variety of mixed boundaryvalue problems for the biharmonic equation. Naturally, the formulation is more complicated for bihar-monic fields because it is necessary to simultaneously solve for two analytic functions instead of justone and the boundary values of those two functions are coupled by the boundary data.

The biharmonic equation is ubiquitous in the study of slow viscous (Stokes) flows and in planeelasticity (Goodier , 1934). In Stokes flows, for example, there are a rich variety of numerical meth-ods, especially boundary integral methods (Pozrikidis , 1992), available for the solution of boundaryvalue problems for the biharmonic equation; many analytical techniques also exist, although they oftenapply only to special geometries (Happel & Brenner , 1965). The transform method presented here hasthe virtue that it can be applied algorithmically to a very large class of domains (the class of circulardomains, which includes the subclass of polygonal domains) and, moreover, it can be viewed theoreti-cally as a modification of boundary integral methods in which unknown boundary data are determined,

A NEW TRANSFORM METHOD FOR THE BIHARMONIC EQUATION 3 of 33

not by solving a set of singular integral equations defined in physical space, but by making use of globalrelations in a “spectral space” to determine the coefficients in, for example, Fourier or Chebyshev expan-sions of unknown boundary data.

This pedagogical paper presents, using a series of examples, the key elements of the new methodol-ogy. We have chosen to showcase it by solving three example problems for slow viscous flows alreadysolved by other methods. This affords the reader the opportunity to compare and contrast the approachwith extant methods, while also affording us a check on the solutions obtained by the new method. Bystudying the suite of examples it should be clear to the reader how to generalize the method to any newbiharmonic problems of interest defined in the general class of circular domains.

2. Fourier-Mellin representations of analytic functions in circular domains

Since the general idea of using the Fourier-Mellin transform pairs for analytic functions in multiplyconnected circular domains to solve boundary value problems is borrowed from Crowdy (2015c), wherea full derivation of the transform pairs is given, we will not repeat the derivation here. Instead we simplystate the transform pairs for some example domains and refer the reader to Crowdy (2015a,c) for detailsof the derivations.

Consider first a bounded polygon P with N sides. If side S j has inclination χ j then any function w(z)that is analytic in P has the representation

w(z) =1

2π

N

∑j=1

∫L

ρ j j(k)e−iχ j eie−iχ j kzdk, (2.1)

where, for integers m,n between 1 and N, we define the spectral matrix (Crowdy , 2015a) to be

ρmn(k)≡∫

Sn

w(z′)e−ie−iχm kz′dz′, (2.2)

and where L = [0,∞) is the fundamental contour for straight line edges. The global relations are that,for any k ∈ C, and for any m = 1, ...,N,

N

∑n=1

ρmn(k) =N

∑n=1

∫Sn

w(z′)e−ie−iχm kz′dz′ =∫

∂Pw(z′)e−ie−iχm kz′dz′ = 0. (2.3)

This result was first derived (in a slightly modified form, and without reference to the spectral matrix)by Fokas & Kapaev (2003) using the spectral analysis of a Lax pair and Riemann-Hilbert methods. Ithas since been rederived in a number of different ways (see, for example, Fokas & Spence (2012)).An elementary geometrical derivation based on use of Cauchy’s integral formula was given in Crowdy(2015a); this derivation was significant in that it showed the route to generalization to find the analogoustransform pairs relevant to analytic functions defined in the much more general class of circular domains,including multiply connected cases.

Consider now a multiply connected circular region D bounded by S+ 1 circles C j| j = 0, ...,S;Figure 2 shows a schematic for S = 2. Circle C0 is defined to be |z|= 1 and C j, j = 1, ...,S is centred atδ j and has radius q j. Domain D can be thought as the intersection of the interior of the unit circle C0 andthe exterior of circles C j| j = 1, ...,S. A representation for a function w(z) analytic and single-valued


1

C0

C1

C2

D

δ1

δ2

q1

q2

FIG. 2: A multiply connected circular domain D (for S = 2).

in D is

w(z) =1

2πi

∫L1

ρ00(k)1− e2πik zkdk+

∫L2

ρ00(k)zkdk+∫

L3

ρ00(k)e2πik

1− e2πik zkdk

− 12πi

S

∑j=1

∫L1

ρ j j(k)1− e2πik

[q j

z−δ j

]k+1

dk+∫

L2

ρ j j(k)[

q j

z−δ j

]k+1

dk+∫

L3

ρ j j(k)e2πik

1− e2πik

[q j

z−δ j

]k+1

dk,

(2.4)

where now the spectral matrix (Crowdy , 2015a) is defined, for n = 0,1, , ...,S, by

ρ0n(k)≡∮

Cn

w(z′)[z′]−k−1 dz′,

ρmn(k)≡−1

qm

∮Cn

w(z′)[

z′−δm

qm

]k

dz′, m = 1, ...,S,(2.5)

and where the set L j| j = 1,2,3 constitutes the fundamental contour for circular edges (Figure 3). Thecontour L1 is the union of the negative imaginary axis (−i∞,−ir], where 0 < r < 1, and the arc of thequarter circle |k| = r in the third quadrant traversed in a clockwise sense; the contour L2 is the realinterval [−r,∞); the contour L3 is the arc of the quarter circle |k|= r in the second quadrant traversed ina clockwise sense together with the portion of the positive imaginary axis [ir, i∞). The global relationsare

S

∑n=0

ρmn(k) = 0, k ∈ −N, m = 1, ...,S. (2.6)


0 1 2 3

L2

L3

L1

−r−1

−ir

ir

FIG. 3: The fundamental contour for circular edges with 0 < r < 1.

The geometrical derivation of Crowdy (2015a) is readily generalizable to hybrid domains having amixture of straight-line and circular edges, such as the “disc-in-channel” geometry D shown far rightin Figure 1. This geometry has been considered, for various different physical problems, by Richmond(1923), Poritsky (1960), Martin & Dalrymple (1988), and Crowdy (2015b,c). The transform repre-sentation of a function w(z) that is analytic and single-valued in such a domain (for a disc of unit radiuscentred at the origin) can be shown (Crowdy , 2015c) to be

w(z) =1

2π

∫∞

0ρ11(k)eikzdk+

12π

∫ −∞

0ρ33(k)eikzdk︸︷︷︸

Fourier−type transform

− 12πi

∫L1

ρ22(k)1− e2πik

1zk+1 dk+

∫L2

ρ22(k)1

zk+1 dk+∫

L3

ρ22(k)e2πik

1− e2πik1

zk+1 dk

︸︷︷︸Mellin−type transform

,

(2.7)

where the simultaneous appearance of both “Fourier-type” and “Mellin-type” contributions naturallyreflects the hybrid geometry of the domain and motivates the designation “Fourier-Mellin transforms”(Crowdy , 2015a). The elements of the spectral matrix are defined as follows:

ρ11(k) =∫ +∞−il

−∞−ilw(z)e−ikzdz = ρ31(k) ρ22(k) =−

∮|z|=1

w(z)zkdz, (2.8)

and

ρ33(k) =∫ −∞+il

∞+ilw(z)e−ikzdz = ρ13(k), (2.9)

with

ρ21(k) =∫ +∞−il

−∞−ilw(z)zkdz, ρ23(k) =

∫ −∞+il

∞+ilw(z)zkdz, (2.10)

and

ρ12(k) = ρ32(k) =−∮|z|=1

w(z)e−ikzdz. (2.11)


The functions appearing in the spectral matrix satisfy the global relations

ρ11(k)+ρ12(k)+ρ13(k) = 0, k ∈ R,ρ31(k)+ρ32(k)+ρ33(k) = 0, k ∈ R,

(2.12)

which are equivalent, and

ρ21(k)+ρ22(k)+ρ23(k) = 0, k ∈ −N. (2.13)

As discussed in Crowdy (2015c), the doubly connected nature of the domain means that both (2.12)and (2.13) must be analyzed to find the unknown spectral functions.

3. Stokes flows and the biharmonic equation

The study of two-dimensional incompressible slow viscous Stokes flows results in the need to solve abiharmonic equation for an associated stream function. The two-dimensional Stokes equations are givenby

∇p = µ∇2u, ∇ ·u = 0, (3.1)

where u = (u,v) is the velocity field, p is the fluid pressure and µ is the viscosity. The incompressibilitycondition permits the introduction of a stream function ψ such that

u =∂ψ

∂y, v =−∂ψ

∂x. (3.2)

By taking the curl of the first equation in (3.1) it can be shown that the stream function satisfies thebiharmonic equation

∇4ψ = 0, (3.3)

where ∇2 is the two-dimensional Laplacian operator. The general solution of (3.3) (Langlois , 1967) isgiven by

ψ = Im[z f (z)+g(z)], (3.4)

where f (z) and g(z) (the Goursat functions) are analytic functions of the complex variable z = x+ iy,but are allowed to have isolated singularities to model flows of interest. To solve a Stokes flow problemin two dimensions, it is sufficient to determine these functions and this is done by making use of theboundary conditions. It should be noted that expression (3.4) also appears in plane elasticity with ψ inthat case being the Airy stress function (Muskhelishvili (1977); Goodier (1934)).

It can be shown (Langlois , 1967) that all the physical quantities of interest are expressed in termsof functions f (z) and g(z) as follows:

pµ− iω = 4 f ′(z), u− iv =− f (z)+ z f ′(z)+g′(z), e11 + ie12 = z f ′′(z)+g′′(z), (3.5)

where ω is the fluid vorticity and ei j is the fluid rate-of-strain tensor. The complex form of the fluidstress on a boundary component of a fluid region can be expressed as

−pni +2µei jn j = 2µidHds

, where H(z,z)≡ f (z)+ z f ′(z)+g′(z), (3.6)

where ni is outward normal to the boundary and s is arclength along the boundary.


The integral of the fluid stress around the boundary of a body gives the Stokes drag force F actingon it. The complex form of the Stokes drag force on a body A with boundary ∂A is given by

F =∮

∂A(−pni +2µei jn j)ds = 2µi

∮∂A

dHds

ds = 2µi[H]∂A, (3.7)

where the square brackets with subscript denote the change in H on traversing ∂A. Thus logarithmicsingularities of the Goursat functions inside the body are associated with non-zero net external forces onit. The torque on a body centred at z0 is defined as the integral (x− x0)∧F around its boundary, whereF is the hydrodynamic force on the body at position x−x0. Since the cross product of two vectors a andb is Im[ab] (where a and b denotes the complex forms of the corresponding vectors), the torque can bewritten in complex form as

T = Im[

2µi∮

∂A(z− z0)

dHds

ds]= Im

[2µi

∮∂A

(z− z0)dH]. (3.8)

In the remainder of the paper we show how to apply the new Fourier-Mellin transform pairs describedin §2 to a variety of Stokes flow problems.

4. Stagnation point flow towards a semicircular ridge

Consider a stagnation point flow towards a plane with a cylindrical ridge as shown in Figure 4. Davis &O’Neill (1977b) analyzed this problem for a general angle of intersection but, for ease of presentation,we focus on the case where the cylinder of unit radius intersects the plane boundary at a corner of angleπ/2. Generalization of our approach to other angles is straightforward. Davis & O’Neill (1977b) solvedthe problem by employing bipolar coordinates to map the fluid domain to a channel geometry where it ispossible to use standard Fourier transform techniques. To exemplify the main steps in the new transformapproach for circular domains we rederive this solution.

4.1 Problem formulation

x

y

−1 10

FIG. 4: Schematic of the configuration. A stagnation point flow towards a plane with a cylindrical ridgewith angle of intersection between the two boundaries equal to π/2.

In the far-field, the stagnation point flow has associated velocity profile

(u,v) = (2axy,−ay2), (4.1)


where a ∈ R is a constant determining the strength of the flow. In the absence of the semicircularboundary, the flow is a stagnation point flow described by (4.1); however, the presence of the no-slipsemicircular boundary alters the resulting fluid flow.

4.2 Goursat functions and transform representation

By the linearity of the problem it is natural to write

f (z) = fs(z)+ fR(z), g′(z) = g′s(z)+g′R(z), (4.2)

where

fs(z) =iaz2

4, g′s(z) =−

3iaz2

4(4.3)

are associated with the far-field flow (4.1) and fR(z), g′R(z) are analytic functions in the flow regionwhich vanish in the far-field.

The fluid domain can be thought as the intersection of the upper half-plane and the exterior of theunit disc centred at the origin. Therefore, we can write the following integral representation

fR(z) =1

2π

∫L

ρ11(k)eikzdk− 12πi

∫L1

ρ22(k)1− e2πik

1zk+1 dk+

∫L2

ρ22(k)1

zk+1 dk+∫

L3

ρ22(k)e2πik

1− e2πik1

zk+1 dk,

(4.4)

where L = [0,∞) is the fundamental contour for straight line edges and the set L j| j = 1,2,3 consti-tutes the fundamental contour for circular edges. The two spectral functions are given by

ρ11(k) =∫

LfR(z)e−ikzdz, ρ22(k) =−

∫C

fR(z)zkdz, (4.5)

with L = [1,∞)∪ (−∞,−1] and C = z : |z| = 1, Im[z] > 0 (contour C is traversed counterclockwise).The other elements of the spectral matrix are

ρ12(k) =−∫

CfR(z)e−ikzdz, ρ21(k) =

∫L

fR(z)zkdz. (4.6)

The global relations areρ11(k)+ρ12(k) = 0, k < 0 (4.7)

andρ21(k)+ρ22(k) = 0, k ∈ −N. (4.8)

The global relations (4.7)-(4.8) are equivalent statements of analyticity of fR(z) in the fluid domain.In the same way, the analytic function g′R(z) can be represented by

g′R(z) =1

2π

∫L

ρ11(k)eikzdk− 12πi

∫L1

ρ22(k)1− e2πik

1zk+1 dk+

∫L2

ρ22(k)1

zk+1 dk+∫

L3

ρ22(k)e2πik

1− e2πik1

zk+1 dk,

(4.9)

with elements of the spectral matrix given by

ρ11(k) =∫

Lg′R(z)e

−ikzdz, ρ12(k) =−∫

Cg′R(z)e

−ikzdz,

ρ21(k) =∫

Lg′R(z)z

kdz, ρ22(k) =−∫

Cg′R(z)z

kdz.(4.10)


The global relations areρ11(k)+ ρ12(k) = 0, k < 0 (4.11)

andρ21(k)+ ρ22(k) = 0, k ∈ −N. (4.12)

4.3 Boundary conditions

The no-slip boundary condition on boundary L = [1,∞)∪ (−∞,−1], where z = z, can be written as

− f (z)+ z f ′(z)+g′(z) = 0. (4.13)

Substitution of (4.2) into (4.13) gives

− fR(z)+ z f ′R(z)+g′R(z) = fs(z)− z f ′s(z)−g′s(z) = 0. (4.14)

Similarly, on the semicircular boundary C = z : |z|= 1, Im[z]> 0, where z = 1/z, we have

− f (z)+(1/z) f ′(z)+g′(z) = 0. (4.15)


− fR(z)+(1/z) f ′R(z)+g′R(z) = fs(z)− (1/z) f ′s(z)−g′s(z). (4.16)

4.4 Spectral analysis

We now multiply (4.14) by zk and integrate along L:

−∫

LfR(z)zkdz+

∫L

z f ′R(z)zkdz+

∫L

g′R(z)zkdz = R21(k), (4.17)

whereR21(k)≡

∫L

[fs(z)− z f ′s(z)−g′s(z)

]zkdz = 0. (4.18)

This can be written in terms of the spectral functions as

−ρ21(k)− (k+1)ρ21(k)+ ρ21(k)+(−1)k+1 fR(−1)− fR(1) = 0. (4.19)

Next, we multiply (4.16) by zk and integrate along C (orientation is counterclockwise):∫C

fR(z)zkdz−∫

C(1/z) f ′R(z)z

kdz−∫

Cg′R(z)z

kdz = R22(k), (4.20)

where

R22(k)≡−∫

C

[fs(z)− (1/z) f ′s(z)−g′s(z)

]zkdz =

ia4

∫C

zk−2dz+ia2

∫C

zkdz− 3ia4

∫C

zk+2dz, (4.21)

using (4.3); we find

R22(k) =−2iak(1+ eiπk)

(k+3)(k2−1), k 6=−3,±1, (4.22)

andR22(−3) =

3πa4

, R22(−1) =−πa2, R22(1) =−

πa4. (4.23)

Expression (4.20) can be written in terms of the spectral functions as

ρ22(−k−2)− (k−1)ρ22(k−2)+ ρ22(k)+ fR(1)− (−1)k−1 fR(−1) = R22(k). (4.24)


4.5 Solution scheme

Addition of (4.19) and (4.24) and use of the global relations (4.8) and (4.12) to eliminate the spectralfunctions ρ21(k), ρ21(k), ρ22(k) gives

(k+1)ρ22(k)+ρ22(k)− (k−1)ρ22(k−2)+ρ22(−k−2) = R22(k), for k ∈ −N. (4.25)

As we will see, this infinite set of conditions is sufficient to determine the unknown spectral functionρ22(k).

4.5.1 Function representation Conditions (4.25) contain only the spectral function ρ22(k) which isassociated with the semicircular boundary C. The next step is to represent fR(z) on this boundary usinga Chebyshev basis expansion. The circular boundary C can be parametrized by

z(s) = eiπ(s+1)/2, for s ∈ [−1,1], (4.26)

with z(−1) = 1 and z(1) =−1. We write

fR(z(s)) =∞

∑m=0

amTm(s), (4.27)

for some set of coefficients am to be found and where Tm(s) = cos(m cos−1(s)), s ∈ [−1,1] are theChebyshev polynomials of the first kind. Substitution of (4.27) into the expression for ρ22(k) gives

ρ22(k) =∞

∑m=0

amZ(k,m), Z(k,m) =− iπ2

∫ 1

−1Tm(s)eiπ(k+1)(s+1)/2ds, (4.28)

where Z(k,m) can be computed explicitly using the formulae given by Fokas & Smitheman (2012). Thesum in (4.28) is truncated to include only terms m = 0, ...,M. Substitution of (4.28) into (4.25) gives

M

∑m=0

am

[(k+1)Z(k,m)− (k−1)Z(k−2,m)

]+

M

∑m=0

am

[Z(k,m)+Z(−k−2,m)

]=R22(k), for k∈−N.

(4.29)The unknown coefficients am|m = 0, ...,M, and their complex conjugates, are computed by solvingan overdetermined linear system for suitable number of k ∈ −N. Once the coefficients am are found,ρ22(k) can be computed using (4.28).

4.5.2 Finding the other spectral functions Once the coefficients am|m = 0, ...,M are found, ρ22(k)can be computed using (4.28). Expansion (4.27) can then be substituted into (4.16) to give an expressionfor g′R(z) on the semicircular boundary. Substitution of this expression into ρ22(k) will give us this spec-tral function. But the transform representations for fR(z) and g′R(z) given by (4.4) and (4.9) respectivelyalso contain the spectral functions ρ11(k) and ρ11(k) (integrated on k > 0). To find these, we proceed asfollows.

We multiply (4.14) by e−ikz and integrate along L:

−∫

LfR(z)e−ikzdz+

∫L

z f ′R(z)e−ikzdz+

∫L

g′R(z)e−ikzdz = 0. (4.30)


This expression can be written in terms of the spectral functions as

−ρ11(−k)− ∂ [kρ11(k)]∂k

+ ρ11(k)− fR(−1)eik− fR(1)e−ik = 0. (4.31)

Solving for ρ11(−k) and using the global relations (4.7) and (4.11), this becomes

ρ11(−k) =∂ [kρ12(k)]

∂k− ρ12(k)− fR(−1)eik− fR(1)e−ik, for k < 0. (4.32)

If we take Schwarz conjugate and let k 7→ −k, we find

ρ11(k) =∂ [kρ12(−k)]

∂k− ρ12(−k)− fR(−1)eik− fR(1)e−ik, for k > 0. (4.33)

This gives an expression for the unknown spectral function ρ11(k) (for k > 0) in terms of quantitiesassociated with the semicircular boundary; all these can be computed using the expressions for fR(z)and g′R(z) on this boundary found above.

Finally, to find ρ11(k), we rearrange (4.31):

ρ11(k) = ρ11(−k)+∂ [kρ11(k)]

∂k+ fR(−1)eik + fR(1)e−ik, (4.34)

which, on use of the global relations, can be written as

ρ11(k) =−ρ12(−k)+∂ [kρ11(k)]

∂k+ fR(−1)eik + fR(1)e−ik, for k > 0. (4.35)

Again, this is an expression for the unknown spectral function ρ11(k) (for k > 0) in terms of quantitiesassociated with the semicircular boundary (having used (4.33)).

4.6 Results

We have now derived all spectral functions needed for the computation of the functions fR(z) and g′R(z).To validate the approach, Figure 5 shows the relative error |ωDO−ωT M| along the semicircular ridgez = eiθ , θ = [0,π] as a function of θ , where ωDO is used to denote the vorticity values computedusing the expressions given by Davis & O’Neill (1977b), while ωT M using our transform method.The results shown are for different truncation parameters M = 16,32,64 in (4.27). Davis & O’Neill(1977b) derived a closed-form expression for the stream function as an infinite sum which we have usedto compute the associated vorticity field. Although few modes (in their expression) are sufficient toobtain good accuracy near the corner points, many more modes are needed to maintain accuracy forpoints near z = eiπ/2 = i (the results shown are for 100 modes in their expansion). On the other hand,our solution produces accurate results away from the corner points for small truncation parameters M,but more terms are required to capture the singularity behaviour near the corners (Moffatt , 1964). Wehave found that our transform approach captures the corner singularities with accuracy ranging fromO(10−2) for M = 16 to O(10−6) for M = 64 (and improves as M is increased), when compared to thesolution of Davis & O’Neill (1977b) which exactly captures the singularity structure at those points. Wenote that Fornberg & Flyer (2011) who presented a numerical method for solving Laplace’s equation inpolygonal domains using the unified transform method showed that if corner singularities are present,then including leading-order singular terms of known type improves the accuracy. Although this was


not pursued here, similar ideas can be used in the present study to improve the resolution near the cornerpoints.

Figure 6 shows the pressure and vorticity fields along the semicircular ridge z = eiθ , θ = [0,π] asa function of θ computed using our transform method. The results shown are for truncation parameterM = 16 (in (4.27)).

0 /2

-15

-10

-5

0

5

|D

O-

TM

|

10-3

M=16

M=32

M=64

FIG. 5: Computation of |ωDO−ωT M| along the semicircular ridge z = eiθ , θ = [0,π] as a function ofθ , where ωDO is used to denote the vorticity values computed using the expressions given by Davis &O’Neill (1977b), while ωT M using our transform method for truncation parameters M = 16,32,64. Thestrength of the flow is a = 1/2. Our transform approach captures the corner singularities with accuracyranging from O(10−2) for M = 16 to O(10−6) for M = 64 (and improves as M is increased), whencompared to the solution of Davis & O’Neill (1977b) which exactly captures the singularity structureat those points.

0 /2

-3

-2

-1

0

1

2

p

0 /2-4

-2

0

2

4

FIG. 6: Pressure (left) and vorticity (right) values along the semicircular ridge z = eiθ , θ = [0,π] as afunction of θ , computed using our transform method. The results shown are for truncation parameterM = 16 (in (4.27)). The strength of the flow is a = 1/2.


5. A translating and rotating cylinder near a wall

The problem of a translating and rotating cylinder near a wall is a classical problem in Stokes flowsand it is one of few problems which admits an exact solution, even though the fluid domain is doublyconnected. This problem was solved by Jeffrey & Onishi (1981) who employed bipolar coordinates tosolve for the resulting flow and to compute forces and torque acting on the cylinder. Recently, Crowdy(2011) rederived that solution using complex variable techniques and combined it with the reciprocaltheorem to compute the dynamics of a circular “treadmilling” swimmer above a wall. Although ourfocus here will be the Jeffrey & Onishi (1981) problem, a Stokes flow in the same geometry wasoriginally analyzed by Davis & O’Neill (1977a) who, using bipolar coordinates, solved the problem ofa shear flow past the cylinder above a wall.

We now apply the transform approach to the problem of a translating and rotating cylinder near awall without any background flow (although it is easy to add it in) with the aim of computing the forcesand torque acting on the cylinder for given translational/angular velocities. Since Stokes equations arelinear, forces and torque are linearly related to translational and angular velocities and this relation canbe expressed through a tensor which is known as the mobility or resistance matrix (Happel & Brenner ,1965).


Consider a circular cylinder of unit radius centred at z0 = iy0, with y0 > 1, above a wall along the realaxis in a z-plane (Figure 7). The cylinder is translating with complex speed U =Ux + iUy (Ux,Uy ∈ R)and rotating with angular velocity Ω and experiencing a non-zero net force F = Fx + iFy (Fx,Fy ∈ R)and torque T .

y

x

z0

UΩ

FIG. 7: Schematic of the configuration: a translating and rotating cylinder of unit radius centred atz0 = iy0, with y0 > 1, above a wall along the real axis.

The relation between U,Ω and F,T acting on the cylinder can be expressed through the mobilitymatrix A: Ux

UyΩ

= A

FxFyT

, where A =

A11 0 00 A22 00 0 A33

, (5.1)


with A j j ∈ R, j = 1,2,3. In general, A is a 3× 3 full-element tensor, but for this particular problem ithas a diagonal form (Jeffrey & Onishi , 1981). Equivalently, (5.1) can be written as Fx

FyT

= B

UxUyΩ

, (5.2)

where B = A−1 is called the resistance matrix (Happel & Brenner , 1965).


As before, the Goursat functions are decomposed as


where fs(z), g′s(z) are now defined by

fs(z) = λ log(

z− z0

z+ z0

), g′s(z) =−λ log

(z− z0

z+ z0

), (5.4)

where λ ∈C is an unknown constant which will be found as part of the solution and fR(z), g′R(z) are thecorrection functions to be found using the transform method. The above form of fs(z) and g′s(z) has beenchosen for the following reasons. First, the logarithmic singularities at z0 are included to ensure thatthere is a non-zero contribution in function H(z), defined by (3.6), on traversing the cylinder |z−z0|= 1;this produces a net force on the cylinder. The coefficients of the logarithmic terms were forced tohave the above form by requiring that velocity is single-valued. The appropriate image singularitiesat z0 = −z0 have been included to facilitate the computation of integral expressions appearing in thesubsequent analysis; with this choice, it can be shown that the velocity decays at infinity.

Geometrically, the fluid domain can be thought of as the intersection of the upper half-plane and theexterior of the unit disc centred at z0. Therefore, following Crowdy (2015c), we can write the followingintegral representation

fR(z) =1

2π

∫L

ρ11(k)eikzdk

− 12πi

∫L1

ρ22(k)1− e2πik

1(z− z0)k+1 dk+

∫L2

ρ22(k)1

(z− z0)k+1 dk+∫

L3

ρ22(k)e2πik

1− e2πik1

(z− z0)k+1 dk,

(5.5)

where L = [0,∞) and the set L j| j = 1,2,3 constitutes the fundamental contour for circular edges.The two spectral functions are given by

ρ11(k) =∫

∞

−∞

fR(z)e−ikzdz, ρ22(k) =−∮|z−z0|=1

fR(z)(z− z0)kdz. (5.6)

The other elements of the spectral matrix are given by

ρ12(k) =−∮|z−z0|=1

fR(z)e−ikzdz, ρ21(k) =∫

∞

−∞

fR(z)(z− z0)kdz. (5.7)


The global relations areρ11(k)+ρ12(k) = 0, k < 0 (5.8)

andρ21(k)+ρ22(k) = 0, k ∈ −N. (5.9)

Similarly, we can write an integral representation for g′R(z) and, as in the previous section, we distinguishthe spectral functions associated with this function by adding a hat.


The no-slip boundary condition on z = z can be written as

− f (z)+ z f ′(z)+g′(z) = 0. (5.10)


− fR(z)+ z f ′R(z)+g′R(z) = fs(z)− z f ′s(z)−g′s(z). (5.11)

The boundary condition on |z− z0|= 1 can be written as

− f (z)+(

z0 +1

z− z0

)f ′(z)+g′(z) =U− iΩ

1z− z0

. (5.12)

Substitution of (5.3) and solving for g′R(z) gives

g′R(z) = fR(z)−(

z0 +1

z− z0

)f ′R(z)+ fs(z)−

(z0 +

1z− z0

)f ′s(z)−g′s(z)+U− iΩ

1z− z0

. (5.13)


We multiply (5.11) by e−ikz and integrate along the lower boundary:

−∫

∞

−∞

fR(z)e−ikzdz+∫

∞

−∞


∫∞

−∞

g′R(z)e−ikzdz = R(k), (5.14)

whereR(k)≡

∫∞

−∞


]e−ikzdz. (5.15)

Using residue calculus, we can compute R(k):

R(k) = λ

2πiz0 eikz0 , k > 0,0, k = 0,−2πiz0 e−ikz0 , k < 0.

(5.16)

Expression (5.14) can be written in terms of the spectral functions as

−ρ11(−k)− ∂ [kρ11(k)]∂k

+ ρ11(k) = R(k), (5.17)


which can equivalently be written as

ρ11(−k) =−∂ [kρ11(k)]∂k

+ ρ11(k)−R(k), or ρ11(k) =−∂ [kρ11(−k)]

∂k+ ρ11(−k)−R(−k). (5.18)

On use of the global relation (5.8), we can write

ρ11(k) =∂ [kρ12(−k)]

∂k− ρ12(−k)−R(−k), for k > 0. (5.19)

Next, we observe that

ρ11(k) =∫

∞

−∞

fR(x)e−ikxdx =∂ [kρ12(−k)]

∂k− ρ12(−k)−R(−k), for k > 0, (5.20)

which means that taking the inverse Fourier transform for the upper half-plane (Crowdy , 2015c), wecan write

fR(x) =1

2π

∫∞

0

[∂ [kρ12(−k)]

∂k− ρ12(−k)−R(−k)

]eikxdk, for x ∈ R. (5.21)

This gives a relation between the unknown function fR(x) on the real axis in terms of spectral functionsassociated with the cylindrical boundary. Note that, using (5.21) in (5.11), the correction function g′R(z)on the lower boundary can be also expressed in terms of spectral functions associated with the cylinder.

The global relation given by (5.9) can be expressed as∫∞

−∞

fR(z)(z− z0)−ndz−

∮|z−z0|=1

fR(z)(z− z0)−ndz = 0, n ∈ N. (5.22)

Using (5.21) for fR(z) on the real axis, this becomes∮|z−z0|=1

fR(z)(z− z0)−ndz =

∫∞

0

[∂ [kρ12(−k)]

∂k− ρ12(−k)−R(−k)

]I(k,n)dk, (5.23)

for n ∈ N, where we have defined

I(k,n)≡ 12π

∫∞

−∞

eikz

(z− z0)n dz. (5.24)

It can be shown that, for n = 1,

I(k,1)≡ 12π

∫∞

−∞

eikz

(z− z0)n dz =

ieikz0 , k > 0,i/2, k = 0,0, k < 0,

(5.25)

and, for n > 2,

I(k,n)≡ 12π

∫∞

−∞

eikz

(z− z0)n dz =

inkn−1eikz0

(n−1)!, k > 0,

0, k < 0.

(5.26)


The second global relation for g′R(z):

ρ21(k)+ ρ22(k) = 0, k ∈ −N, (5.27)

can be equivalently expressed as∮|z−z0|=1

g′R(z)(z− z0)−ndz =

∫∞

−∞

g′R(z)(z− z0)−ndz, n ∈ N. (5.28)

But for g′R(z) on the lower boundary, we can use the boundary condition (5.11) and the (inverse Fouriertransform) representation for fR(z) given by (5.21) to express g′R(z) in terms of quantities integrated onthe cylindrical boundary. It can be shown that∮

|z−z0|=1g′R(z)(z− z0)

−ndz = (n−1)ρ22(−n)+nz0 ρ22(−n−1)+A(n), n ∈ N, (5.29)

withA(n)≡

∫∞

−∞

[ fs(z)− z f ′s(z)−g′s(z)](z− z0)−ndz, (5.30)

where we have used the global relation (5.9).

5.5 Solution scheme

From the spectral analysis, we have found two sets of conditions given by (5.23) and (5.29) both validfor n ∈N. We now show how to use these conditions together with a Laurent series expansion for fR(z)on the cylindrical boundary to formulate a linear system for the unknown coefficients and parameter λ .The solution of this linear system can be used to compute the resistance matrix (which can be invertedto give the mobility matrix using expressions (5.1)-(5.2)). In addition, if the resulting flow in the domaininterior is required, this can be found by computing all associated spectral functions by back substitutioninto various relations.

5.5.1 Function representation We use a Laurent series expansion to represent fR(z) on the cylinder|z− z0|= 1:

fR(z) =∞

∑m=−∞

am(z− z0)m, (5.31)

where the coefficients am are to be found. Using (5.31), we find

ρ12(k) =−∮|z−z0|=1

fR(z)e−ikzdz =∞

∑m=−∞

am[T (k,m)], (5.32)

where

T (k,m)≡−∮|z−z0|=1

(z− z0)me−ikzdz =

0, m > 0,

−2πi(−ik)−m−1e−ikz0

(−m−1)!, m < 0.

(5.33)

On use of the boundary condition (5.13), we can write

ρ12(k) =−∮|z−z0|=1

g′R(z)e−ikzdz =−

∮|z−z0|=1

[fR(z)−

(z0 +

1z− z0

)f ′R(z)

]e−ikzdz+S(k), (5.34)


where

S(k)≡−∮|z−z0|=1

[fs(z)−

(z0 +

1z− z0


1z− z0

]e−ikzdz. (5.35)

On substitution of (5.31) into (5.34), we find:

ρ12(k) =∞

∑m=−∞

am[Y (k,m)]+∞

∑m=−∞

am[V (k,m)]+S(k), (5.36)

with

Y (k,m) = mz0 T (k,m−1)−mT (k,m−2), (5.37)V (k,m) = T (k,−m). (5.38)

5.5.2 Formulation of the linear system Using (5.31), (5.32) and (5.36) truncated to include onlyterms m=−M, ...,M, conditions (5.23) and (5.29) can be expressed in terms of the unknown coefficientsam, parameter λ and known quantities. Conditions (5.23) can be expressed as

an−1 =M

∑m=−M

amPnm +M

∑m=−M

amQnm +Rn, n ∈ N, (5.39)

where

Pnm =− 12πi

∫∞

0V (−k,m)I(k,n)dk,

Qnm =1

2πi

∫∞

0

[T (−k,m)+ k

∂T (−k,m)

∂k−Y (−k,m)

]I(k,n)dk,

Rn =−1

2πi

∫∞

0

[R(−k)+S(−k)

]I(k,n)dk.

(5.40)

Note that terms Rn|n ∈ N contain the unknown complex parameter λ . Conditions (5.29) can beexpressed as:

(n−1)an−1 +2nz0 an− (n+1)an+1 +a−n+1 =1

2πi[A(n)−B(n)] , n ∈ N, (5.41)

where

B(n)≡∮|z−z0|=1

[fs(z)−

(z0 +

1z− z0


1z− z0

](z− z0)

−ndz. (5.42)

A truncated linear system consisting of (5.39) for n = 1, ...,M+1 and (5.41) for n = 1, ...,M+1 issolved for coefficients am|m = −M, ...,M and λ ∈ C and their complex conjugates. It is found thatthe coefficient matrix is well-conditioned and coefficients am decay rapidly (Figure 8).


5.6 Computation of the resistance matrix

Once am|m =−M, ...,M and λ are computed, all elements of the resistance matrix can be calculated,i.e. forces and torque acting on the cylinder for given translational/angular velocities (the mobility matrixcan be computed by inverting the resistance matrix; expressions (5.1)-(5.2)). To determine these, weconsider the following cylinder motions: (a) motion parallel to the wall, i.e. U =U , Ω = 0, (b) motionaway from the wall, i.e. U =−U , Ω = 0 and (c) pure rotation, i.e. U = 0, Ω 6= 0. The resistance matrixfor this problem has a diagonal form (5.2) which means that for case (a) we expect a horizontal forceacting on the cylinder and no torque, for (b) a vertical force and no torque and finally for (c) a torqueand no force.

The force F on the cylinder is given by

F = [2µiH(z)]|z−z0|=1 = [2µi(2λ log(z− z0)]|z−z0|=1 =−8πµλ , (5.43)

where square brackets denote the change in the quantity they contain on a single counterclockwisetraversal of |z− z0| = 1 and expression H(z) is defined by (3.6). Note that, if the cylinder is in purerotation, the solution of the linear system gives λ equal to zero which implies that, as expected, there isno force exerted on the cylinder.

The torque T on the cylinder is given by

T = 2µ Re[

2πiλ z0 +∮|z−z0|=1

(z− z0)g′′(z)dz], (5.44)

where Re[.] denotes the real part of the expression in square brackets. Derivation of this formula is givenin the Appendix. Using (5.13) and (5.31) in (5.44), we find that

T = 4πµ

[Re[iλ z0]−Ω −2Im[a1]

], (5.45)

in terms of parameter λ , angular velocity Ω and coefficient a1. If cylinder is moving parallel or per-pendicular to the wall, the solution of the linear system gives T = 0. If the cylinder is in pure rotation,parameter λ is found to be equal to zero and therefore (5.45) simplifies to

T =−4πµ

[Ω +2Im[a1]

]. (5.46)

5.7 Results

The graphs in Figure 8 show the absolute values of the coefficients am for different cylinder positionsand motions. The results show the rapid decay of coefficients which suggests that few terms are, ingeneral, required for high accuracy. However, if the cylinder approaches the no-slip wall, i.e. y0 → 1,the decay of coefficients is decelerated and more terms are required.

Figure 9 shows comparison of values computed using our transform approach and exact solution byJeffrey & Onishi (1981). Given the rapid decay of coefficients as illustrated in Figure 8, the resultsshown in Figure 9 are for truncation parameter M = 16. For the values shown, our results are foundto be in agreement with maximum relative errors 10−4% compared to the exact solution of Jeffrey &Onishi (1981).


-20 0 20m

10-15

10-10

10-5

100

|m

|

(a)

y0=1.1

y0=1.5

y0=2

-20 0 20m

10-15

10-10

10-5

100

|m

|

(b)

y0=1.1

y0=1.5

y0=2

-20 0 20m

10-15

10-10

10-5

100

|m

|

(c)

y0=1.1

y0=1.5

y0=2

FIG. 8: Absolute values of the coefficients am for cylinder positions z0 = iy0, y0 = 1.1,1.5,2 whencylinder is: (a) moving parallel to the wall with U = 1, (b) moving away from the wall with U = i, (c)purely rotating with Ω = 1. The graphs show the rapid decay of coefficients which implies that fewterms are, in general, required for high accuracy. However, if the cylinder approaches the no-slip wall,i.e. y0→ 1, the decay of coefficients is decelerated and more terms are required.

1 1.5 2 2.5y

0

-40

-35

-30

-25

-20

-15

-10

-5

Fx

(a)

1 1.5 2 2.5y

0

-250

-200

-150

-100

-50

0

Fy

(b)

1 1.5 2 2.5y

0

-35

-30

-25

-20

-15

-10

T

(c)

FIG. 9: Computation of the resistance matrix for unit viscosity; comparison of our transform approach(circles) and exact solution by Jeffrey & Onishi (1981) (solid lines): (a) Force F = Fx when cylinder ismoving parallel to the wall with U = 1 as a function of the distance of its centre from the wall y0, (b)Force Fy (F = iFy) when the cylinder is moving away from the wall with U = i as a function of y0 and(c) Torque T when the cylinder is in pure rotation with Ω = 1 as a function of y0. The results shownare for truncation parameter M = 16. Our results are found to be in agreement with maximum relativeerrors 10−4% compared to the exact solution of Jeffrey & Onishi (1981).


6. A translating and rotating cylinder in a channel

Jeong & Yoon (2014) and Jeong & Jang (2014) have recently analyzed Stokes flow problems for acylinder in a channel. Jeong & Yoon (2014) study the problem of a cylinder translating along thechannel centreline subject to a background pressure-driven Poiseuille flow using a Papkovich-Fadleeigenfunction expansion and a least-squares method. Using similar techniques, Jeong & Jang (2014)generalized this to a translating and rotating cylinder not necessarily placed along the centreline. Wenow show how to analyze such problems using the transform method described above. Here we omit thebackground Poiseuille flow considered by Jeong & Yoon (2014) and Jeong & Jang (2014) (although itis easy to add it in) and instead aim to compute the resistance matrix – a challenge that has already beenaddressed, in this geometry using numerical methods based on boundary fitted coordinate systems, byDvinsky & Popel (1987).


The new transform approach to this biharmonic problem closely follows a formulation for harmonicfields in the same “disc-in-channel” geometry given in Crowdy (2015c). Consider a circular cylinderof unit radius centred at z0 = iy0, 1 < y0 < h− 1 in a channel −∞ < x < ∞, 0 < y < h (Figure 10).Similarly to the problem considered in the previous section, the cylinder is translating with complexspeed U =Ux + iUy (Ux,Uy ∈ R) and rotating with angular velocity Ω and experiencing a non-zero netforce F = Fx + iFy (Fx,Fy ∈ R) and torque T .

y

x

h

0

UΩ

z0

FIG. 10: Schematic of the configuration: a translating and rotating cylinder of unit radius centred atz0 = iy0, with 1 < y0 < h−1, in a channel geometry −∞ < x < ∞, 0 < y < h.


The Goursat functions are respresented by



where fs(z), g′s(z) are defined by

fs(z) = λ log[tanh

(π

2h(z− z0)

)], g′s(z) =−λ log

[tanh

(π

2h(z− z0)

)], (6.2)

where λ ∈ C is an unknown constant which will be found as part of the solution and fR(z), g′R(z) arethe correction functions to be found. Note that fs(z), g′s(z) are 2hi-periodic and this will be useful later.

In this problem, the fluid domain can be thought as the intersection of two half-planes (upper andshifted lower) and the exterior of the unit disc centred at z0. Therefore, we can write the followingintegral representation

fR(z) =1

2π

∫L1

ρ11(k)eikzdk+1

2π

∫L2

ρ33(k)eikzdk

− 12πi

∫L1

ρ22(k)1− e2πik

1(z− z0)k+1 dk+

∫L2

ρ22(k)(z− z0)k+1 dk+

∫L3

ρ22(k)e2πik

1− e2πik1

(z− z0)k+1 dk,

(6.3)

where L1 = [0,∞) and L2 = [0,−∞) and the set L j| j = 1,2,3 constitutes the fundamental contourfor circular edges. The three spectral functions are given by

ρ11(k) =∫

∞

−∞

fR(z)e−ikzdz, ρ22(k) =−∮|z−z0|=1

fR(z)(z− z0)kdz (6.4)

and

ρ33(k) =∫ −∞+ih

∞+ihfR(z)e−ikzdz. (6.5)

The remaining elements of the matrix of spectral functions are given by

ρ21(k) =∫

∞

−∞

fR(z)(z− z0)kdz, ρ23(k) =

∫ −∞+ih

∞+ihfR(z)(z− z0)

kdz, (6.6)

ρ12(k) = ρ32(k) =−∮|z−z0|=1

fR(z)e−ikzdz (6.7)

and ρ31(k) = ρ11(k) and ρ13(k) = ρ33(k).The global relations are

ρ11(k)+ρ12(k)+ρ13(k) = 0, k ∈ R,ρ31(k)+ρ32(k)+ρ33(k) = 0, k ∈ R,

(6.8)

which are equivalent, and

ρ21(k)+ρ22(k)+ρ23(k) = 0, k ∈ −N. (6.9)

Similar expressions to (6.3)-(6.9) can be written for g′R(z) and its associated spectral functions ρmn(k),m,n = 1,2,3.



The no-slip boundary condition on z = z can be written as

− f (z)+ z f ′(z)+g′(z) = 0. (6.10)


− fR(z)+ z f ′R(z)+g′R(z) = fs(z)− z f ′s(z)−g′s(z). (6.11)

Similarly, the no-slip boundary condition on z = z−2ih can be written as

− fR(z)+(z−2ih) f ′R(z)+g′R(z) = fs(z)− (z−2ih) f ′s(z)−g′s(z). (6.12)

Finally, the boundary condition on |z− z0|= 1 can be written as

− f (z)+(

z0 +1

z− z0

)f ′(z)+g′(z) =U− iΩ

1z− z0

. (6.13)

Substitution of (6.1) and solving for g′R(z) gives

g′R(z) = fR(z)−(

z0 +1

z− z0

)f ′R(z)+ fs(z)−

(z0 +

1z− z0


1z− z0

. (6.14)


We multiply (6.11) by e−ikz and integrate along the lower boundary:

−∫

∞

−∞

fR(z)e−ikzdz+∫

∞

−∞


∫∞

−∞

g′R(z)e−ikzdz = R1(k), (6.15)

whereR1(k)≡

∫∞

−∞


]e−ikzdz. (6.16)

This can be written in terms of the spectral functions as

−ρ11(−k)− ∂ [kρ11(k)]∂k

+ ρ11(k) = R1(k). (6.17)

Similarly if we multiply (6.12) by e−ikz and integrate along the upper boundary we find

−e2khρ13(−k)− ∂ [kρ13(k)]

∂k+2khρ13(k)+ ρ13(k) = R3(k), (6.18)

where

R3(k)≡∫ −∞+ih

∞+ih

[fs(z)− (z−2ih) f ′s(z)−g′s(z)

]e−ikzdz. (6.19)

Addition of (6.17) and (6.18) and use of the first global relation (6.8) gives (after some algebra):

ρ11(k) =2khW (k)− (e2kh−1)W (−k)

∆(k), for k ∈ R, (6.20)


where

W (k) =−e2khρ12(−k)− ∂ [kρ12(k)]

∂k+2khρ12(k)+ ρ12(k)+R1(k)+R3(k), (6.21)

and∆(k)≡ 4(sinh2(hk)−h2k2). (6.22)

But

ρ11(k) =∫

∞

−∞

fR(z)e−ikzdz (6.23)

and therefore taking inverse Fourier transform, we find

fR(z) =1

2π

∫∞

−∞

ρ11(k)eikzdk =1

2π

∫∞

−∞

[2khW (k)− (e2kh−1)W (−k)

∆(k)

]eikzdk, (6.24)

for z = x ∈ R. This expression gives a relation between the unknown function fR(z) on the lowerchannel wall in terms of quantities on the cylinder. Note that, using (6.24) in (6.11), the correctionfunction g′R(z) on the lower channel wall can be also expressed in terms of spectral functions associatedwith the cylinder. But, near k = 0,

∆(k)∼ O(k4) (6.25)

which means that, if we define

X(k)≡ 2khW (k)− (e2kh−1)W (−k), (6.26)

then we must requireX(0) = X ′(0) = X ′′(0) = X ′′′(0) = 0 (6.27)

in order to remove the singularity at k = 0 of the integrand in (6.24).A similar expression to (6.24) can be written for fR(z) on the upper channel wall. In fact, using the

first global relation in (6.8), we can write

ρ13(k) =∫ −∞+ih

∞+ihfR(z)e−ikzdz =−ρ11(k)−ρ12(k), for k ∈ R. (6.28)

Again, taking inverse Fourier transform, we find

fR(z) =1

2π

∫∞

−∞

[ρ11(k)+ρ12(k)]eikzdk, (6.29)

for z = x+ ih, x ∈R. Using (6.20), this expression gives a relation between the unknown function fR(z)on the upper channel wall in terms of quantities on the cylinder. Similarly, using (6.29) in (6.12), thecorrection function g′R(z) on the upper channel wall can be expressed in terms of spectral functionsassociated with the cylinder.

The second global relation given by (6.9) can be written as∫∞

−∞

fR(z)(z− z0)−ndz+

∫ −∞+ih

∞+ihfR(z)(z− z0)

−ndz−∮|z−z0|=1

fR(z)(z− z0)−ndz = 0, n ∈ N. (6.30)


Substitution of (6.24) and (6.29) on their respective integrals gives∮|z−z0|=1


∫∞

−∞

[1

2π

∫∞

−∞

ρ11(k)eikzdk](z− z0)

−ndz

+∫ −∞+ih

∞+ih

[1

2π

∫∞

−∞

[ρ11(k)+ρ12(k)]eikzdk](z− z0)

−ndz,(6.31)

which, after changing the order of integration, can be written as∮|z−z0|=1


∫∞

−∞

ρ11(k)I(k,n)dk+∫

∞

−∞

[ρ11(k)+ρ12(k)]J(k,n)dk, (6.32)

for n ∈ N, where we have defined

I(k,n)≡ 12π

∫∞

−∞

eikz

(z− z0)n dz, J(k,n)≡ 12π

∫ −∞+ih

∞+ih

eikz

(z− z0)n dz. (6.33)

Using residue calculus, it can be shown that, for n = 1,

I(k,1) =1

2π

∫∞

−∞

eikz

(z− z0)dz =

ieikz0 , k > 0,i/2, k = 0,0, k < 0,

(6.34)

and, for n > 2,

I(k,n) =1

2π

∫∞

−∞

eikz

(z− z0)n dz =

inkn−1eikz0

(n−1)!, k > 0,

0, k < 0.

(6.35)

Similarly, for n = 1, we have

J(k,1) =1

2π

∫ −∞+ih

∞+ih

eikz

(z− z0)dz =

0, k > 0,i/2, k = 0,ieikz0 , k < 0,

(6.36)

and, for n > 2,

J(k,n) =1

2π

∫ −∞+ih

∞+ih

eikz

(z− z0)n dz =

0, k > 0,

inkn−1eikz0

(n−1)!, k < 0.

(6.37)

The second global relation for g′R(z), which is similar to (6.9), can be expressed as

∮|z−z0|=1

g′R(z)(z− z0)−ndz =

∫∞

−∞

g′R(z)(z− z0)−ndz+

∫ −∞+ih

∞+ihg′R(z)(z− z0)

−ndz, n ∈ N. (6.38)

But for g′R(z) on the channel boundaries, we can use the boundary conditions (6.11) and (6.12) and


the (inverse Fourier transform) representations for fR(z) found previously to express g′R(z) in terms ofquantities integrated on the cylindrical boundary. It can be shown that∮

|z−z0|=1g′R(z)(z− z0)

−ndz =∫

∞

−∞

ρ11(k)I(−k,n)dk+∫

∞

−∞

[ρ11(k)+ρ12(k)

]e−2khJ(−k,n)dk

+2inh∫

∞

−∞

[ρ11(k)+ρ12(k)

]J(k,n+1)dk

+(n−1)ρ22(−n)+nz0 ρ22(−n−1)+C(n)+D(n), n ∈ N,

(6.39)

with

C(n)≡∫

∞

−∞


](z− z0)

−ndz,

D(n)≡∫ −∞+ih

∞+ih

[fs(z)− (z−2ih) f ′s(z)−g′s(z)

](z− z0)

−ndz,(6.40)

where we have used the global relation (6.9).

6.5 Solution scheme

From the spectral analysis of the previous section, we have found conditions (6.32) and (6.39) which areboth valid for n∈N. In this section, we show how using these conditions and a Laurent series expansionfor fR(z) on the cylindrical boundary, one can formulate a linear system for the unknown coefficientsand parameter λ . The solution of this linear system gives the unknown boundary data on the cylinderand this is sufficient to compute the resistance matrix. Again, if the resulting flow in the domain interioris required, this can be found by computing all associated spectral functions (by back substitution intovarious relations).

6.5.1 Function representation We use a Laurent series expansion to represent fR(z) on |z− z0|= 1:

fR(z) =∞

∑m=−∞

am(z− z0)m, (6.41)

for some unknown coefficients am|m = −M, ...,M to be found. Using (6.41), the spectral functionsρ12(k) and ρ12(k) can be expressed as

ρ12(k) =∞

∑m=−∞

am[T (k,m)],

ρ12(k) =∞

∑m=−∞

am[Y (k,m)]+∞

∑m=−∞

am[V (k,m)]+L(k),(6.42)

where expressions T (k,m), Y (k,m) and V (k,m) are defined by (5.33), (5.37) and (5.38) and L(k) isdefined by

L(k)≡−∮|z−z0|=1

[fs(z)−

(z0 +

1z− z0


1z− z0

]e−ikzdz, (6.43)

for fs(z) and g′s(z) given by (6.2).


6.5.2 Formulation of the linear system Using (6.41)-(6.42) truncated to include only terms m =−M, ...,M, conditions (6.32) and (6.39) can be expressed in terms of the unknown coefficients am,parameter λ and known quantities. Conditions (6.32) can be expressed as

an−1 =M

∑m=−M

amAnm +M

∑m=−M

amBnm +Cn, n ∈ N, (6.44)

where

Anm =1

2πi

∫∞

−∞

P(k,m)I(k,n)dk+∫

∞

−∞

[P(k,m)+T (k,m)]J(k,n)dk,

Bnm =1

2πi

∫∞

−∞

Q(k,m)[I(k,n)+ J(k,n)]dk,

Cn =1

2πi

∫∞

−∞

N(k)[I(k,n)+ J(k,n)]dk,

(6.45)

with

P(k,m) =2khw1(k,m)− (e2kh−1)w2(−k,m)

∆(k),

Q(k,m) =2khw2(k,m)− (e2kh−1)w1(−k,m)

∆(k),

N(k) =2khw3(k)− (e2kh−1)w3(−k)

∆(k)

(6.46)

and

w1(k,m) =−∂ [kT (k,m)]

∂k+2khT (k,m)+Y (k,m),

w2(k,m) =−e2kh T (−k,m)+V (k,m),

w3(k) = L(k)+R1(k)+R3(k).

(6.47)

Note that terms Cn|n ∈ N contain the unknown complex parameter λ . Conditions (6.39) can beexpressed as

(n−1)an−1 +2nz0 an− (n+1)an+1 +a−n+1 =M

∑m=−M

amDnm +M

∑m=−M

amEnm +Fn, (6.48)

for n ∈ N, and where

Dnm =1

2πi

∫∞

−∞

Q(−k,m)[I(k,n)+ e2khJ(k,n)

]dk+2nhi

∫∞

−∞

[P(k,m)+T (k,m)]J(k,n+1)dk,

Enm =1

2πi

∫∞

−∞

P(−k,m)I(k,n)dk+∫

∞

−∞

[P(−k,m)+T (−k,m)

]e2khJ(k,n)dk+2nhi

∫∞

−∞

Q(k,m)J(k,n+1)dk,

Fn =1

2πi

∫∞

−∞

N(−k)[I(k,n)+ e2khJ(k,n)

]dk+2nhi

∫∞

−∞

N(k)J(k,n+1)dk+C(n)+D(n)−E(n),

(6.49)


with

E(n)≡∮|z−z0|=1

[fs(z)−

(z0 +

1z− z0


1z− z0

](z− z0)

−ndz. (6.50)

Finally, we must require the following conditions at k = 0:

X(0) = X ′(0) = X ′′(0) = X ′′′(0) = 0, (6.51)

with X(k) defined by (6.26). These conditions require computation of the first few terms in the Taylorexpansion of X(k). While computation of these coefficients is possible, it is cumbersome and the fol-lowing equivalent conditions, which can be computed numerically, can instead be added to the linearsystem: ∮

|k|=ε

X(k)k j dk = 0, for j = 1,2,3,4, (6.52)

where ε is a small constant such that |k|= ε does not enclose any roots of ∆(k) = 0 other than k = 0.A truncated linear system consisting of (6.44) for n= 1, ...,M+1, (6.48) for n= 1, ...,M+1 and con-

ditions (6.52) is solved for coefficients am|m = −M, ...,M and λ ∈ C and their complex conjugates.It is found that the coefficient matrix is well-conditioned and coefficients am decay rapidly.

6.6 Computation of the resistance matrix

Once am|m =−M, ...,M and λ are computed, we can calculate all elements of the resistance matrix,i.e. forces and torque acting on the cylinder for given translational/angular velocities. The formulae forcomputing the forces and torque on the cylinder are given by (5.43) and (5.44):

F =−8πµλ ,

T = 2µ Re[

2πiz0λ +∮|z−z0|=1

(z− z0)g′′(z)dz].

(6.53)

To compute the resistance matrix, we consider, as previously, the following cylinder motions: (a) motionparallel to the channel walls, i.e. U = U , Ω = 0, (b) motion perpendicular to the channel walls, i.e.U = −U , Ω = 0 and (c) pure rotation, i.e. U = 0, Ω 6= 0. The resistance matrix for this problem issymmetric and has the following form: Fx

FyT

= B

UxUyΩ

, where B =

B11 0 B130 B22 0

B31 0 B33

, (6.54)

where Bi j ∈R, i, j = 1,2,3. (The mobility matrix A can be computed by inverting the resistance matrixB, i.e. A = B−1.)

6.7 Results

Figures 11-13 show computation of the resistance matrix for cylinder motions (a)-(c) presented abovefor channel height h= 4. Our results show qualitative agreement to the results obtained by Jeong & Jang(2014) and Dvinsky & Popel (1987). In addition, if the cylinder is placed near the lower channel wall


and the upper boundary is located at a large distance h away from the lower boundary, then the resistancematrix converges to the results by Jeffrey & Onishi (1981) for a translating and rotating cylinder abovea wall.

1 1.5 2

y0

60

70

80

90

100

110

120

130

140

-Fx

(a)

1 1.5 2

y0

0

0.5

1

1.5

2

2.5

3

3.5

4

-T/(

h/2

)

(b)

FIG. 11: Computation of the resistance matrix (for µ = 1) when cylinder of unit radius is movingparallel to the channel walls (U = 1) for channel height h = 4: Non-dimensionalized (a) Force F = Fxas a function of the distance of its centre from the lower wall y0 and (b) Torque T/(h/2) as a function ofy0. Clearly there is a symmetry with respect to the centreline y = h/2 and for this reason, we restrict ourattention to 1 < y0 < h/2. The results shown are for truncation parameter M = 16. The minimum valueof the force in (a) implies that there is a cylinder position for which the drag force becomes minimum;this agrees with the results of Dvinsky & Popel (1987) and Jeong & Jang (2014).


1 1.5 2

y0

102

103

104

-Fy

FIG. 12: Computation of the resistance matrix (for µ = 1) when cylinder of unit radius is movingperpendicular to the channel walls (U = i) for channel height h = 4: Non-dimensionalized Force Fy(F = iFy) as a function of y0. The results shown are for truncation parameter M = 16.

1 1.5 2

y0

0

0.5

1

1.5

2

2.5

3

3.5

4

-Fx/(

h/2

)

(a)

1 1.5 2

y0

0

5

10

15

20

25

30

35

-T/(

h/2

)2

(b)

FIG. 13: Computation of the resistance matrix (for µ = 1) when cylinder of unit radius is in purerotation (Ω = 1) for channel height h = 4: Non-dimensionalized (a) Force F = Fx/(h/2) and (b) TorqueT/(h/2)2 as functions of y0. The results shown are for truncation parameter M = 16.


7. Discussion

A new transform approach to boundary value problems for biharmonic fields in circular domains hasbeen described, and its application to three example problems involving slow viscous Stokes flowspresented. Its flexibility renders it a powerful addition to the armoury of techniques for solving boundaryvalue problems for the biharmonic equation in this very general class of domains.

We conclude by emphasizing the novel features of the approach, and how it differs from existingmethods. The channel geometry example just considered here offers a clear view of this. In otherapproaches, it is common to make use of a special sets of eigenfunctions of the biharmonic operator –such as the Papkovich-Fadle eigenfunctions relevant to a channel geometry – to represent the unknownfield with the boundary conditions then used to fix the coefficients in such an eigenfunction expan-sion. In contrast, the transform method here allows us to represent the unknown boundary data in anyexpansion basis; here we used either Fourier series (Laurent expansions) or the Chebyshev polynomialswhich, it should be emphasized, have no significance as eigenfunctions of the biharmonic operator. Thecoefficients in these expansions are then found using the global relations associated with the Fourier-Mellin representations of the Goursat functions derived in Crowdy (2015a). Once this boundary datais determined, the result is an integral representation in a complex k plane of the required biharmonicfield.

An advantage of this approach is that it can be applied to a very large class of circular domain geome-tries in an algorithmic way; all that changes between examples is the analysis of the global relations.Moreover, we have found that in many cases, to obtain high accuracy, only a few coefficients in a Fourier(Laurent) or Chebyshev expansion of a single unknown boundary function need to be determined, withall other spectral functions following by back substitution. Thus the approach might therefore be calledquasi-analytical in that only a small number of unknown coefficients are needed to represent the globalsolution even though those coefficients must be found numerically by solving a low-order linear system.

We have focussed here on problems of slow viscous Stokes flows. However, all the same mathe-matical technology can be applied to boundary value problems in plane elasticity (Goodier , 1934). Forexample, various elasticity problems in similar geometries to those considered here have been analyzed,using a variety of mathematical techniques, by Ling (1947a) (the “notched plate”) and Ling (1947b)(the “notched strip”) while problems involving a circular hole in a plane elastic strip have been consid-ered by Howland (1930) and Howland & Knight (1933). The approach advocated here also offers analternative means of solution to all those problems, and many others like it.

Acknowledgments

DGC is supported by an EPSRC Established Career Fellowship (EP/K019430/10) and a Royal SocietyWolfson Research Merit Award. Both authors acknowledge financial support from a Research Grantfrom the Leverhulme Trust which provided a graduate studentship for E. Luca.

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Appendix. Calculation of torque on a cylinder

We present a derivation of expression (5.44). The torque on a cylinder of unit radius centred at z0 isgiven by

T = Im[

2µi∮|z−z0|=1

(z− z0)dH], (A.1)

where H is given by (3.6). On integration by parts, this can be written as

T = µ[H(z− z0)+H(z− z0)

]|z−z0|=1− Im

[2µi

∮|z−z0|=1

Hdz], (A.2)

where the square brackets with subscript denote the change in the quantity they contain on traversing|z− z0| = 1 counterclockwise. Using (3.6), we find, after some algebra, that expression (A.2) can besimplified to

T = 2µ Re[∮|z−z0|=1

(z− z0)g′′(z)dz]−µ

[∮|z−z0|=1

z0 d f + z0 d f]. (A.3)

For the translating and rotating cylinder presented in Sections 5 and 6, function f (z) has local expansionat z0 of the form

f (z) = λ log(z− z0)+ single-valued components, (A.4)

which means that expression (A.3) can be further simplified to

T = 2µ Re[

2πiλ z0 +∮|z−z0|=1

(z− z0)g′′(z)dz]. (A.5)

A transform method for the biharmonic equation in multiply ...

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