exact solution of the displacement boundary-value problem of elasticity...

Exact Solution of the Displacement Boundary-ValueProblem of Elasticity for a Torus

P. KrokhmalDepartment of Industrial and Systems EngineeringUniversity of FloridaP. O. Box 116595, Gainesville, FL 32611-6595, USA

December 11, 2002

Abstract. The paper presents an exact analytical solution to the displacement boundary-valueproblem of elasticity for a torus. The introduced form of the general solution of elasto-statics equations allows for exact solving of a broad class of boundary-value problems incoordinate systems with incomplete separation of variables in the harmonic equation. Theoriginal boundary-value problem for a torus is reduced to infinite systems of linear algebraicequations with tridiagonal matrices. An analytical technique for solving systems of diagonalform is developed. Uniqueness of the solutions of vector boundary-value problems involvingthe generalized Cauchy-Riemann equations is investigated, and it is shownthat the obtainedsolution for the displacement boundary-value problem for a torus is unique due to the specificproperties of the suggested general solution. The analogy between problems of elastostaticsand steady Stokes flows is demonstrated, and the developed elastic solutionis used to solvethe Stokes problem for a torus.

Keywords: Torus, toroidal coordinates, exact solution, multiply-connected regions, Stokesflows

1. Introduction

This paper investigates solutions of vector boundary-value problems of elas-ticity for bodies of a complex geometry, which can be described by thetoroidal and bispherical coordinates. Due to the complexity of the body ofmathematics accompanying these coordinate systems, we restrict ourselvesto considering only the displacement boundary-value problem of elastic-ity for a torus, which attracted our attention by the challenges related tothe multiply-connectedness of the toroidal domains. However, the generalapproach presented in this paper is equally applicable to all bodies de-scribed by the mentioned coordinate systems (i. e. torus, lens-shaped body,spindle-shaped body, and bi-spheres).

Construction of the analytical solutions to the basic elasticity problems forspatial bodies is a challenging mathematical task due to the essentially vectornature of these problems. The most powerful technique yet is the applicationof curvilinear orthogonal coordinates along with the Fourier method of sepa-ration of variables. Over the past century, there has been a significantprogressin this area, so now exact solutions are available for virtually all bodies de-

c© 2005Kluwer Academic Publishers. Printed in the Netherlands.

2 P. Krokhmal

scribed by separable coordinates. However, there exists a class of coordinatesystems that do not admit the complete separation of variables in the har-monic equation, but still allow for solving scalar and vector boundary-valueproblems of mathematical physics. These curvilinear coordinate systems be-long to the family of cyclidal coordinates [1], and the simplest of them arethe toroidal and bispherical coordinates that can be used to describe toruswith lens-shaped body and bi-spheres with spindle-shaped body respectively.The incomplete separation of variables in the Laplace equation in the cyclidecoordinates resulted in a relatively rare application of these coordinate sys-tems for solving applied and engineering problems. Mainly, the bisphericalcoordinates were used for solving problems for bi-spheres, while the toroidalcoordinates were employed most often in the boundary-value problems forhalf-space, where two circular lines were separating the boundary conditionsof different types.

The first attempt to solve the basic elasticity problems for a torus goesback to Wangerin [2]. He suggested a general approach for solving elasticityproblems for bodies of revolution, and applied it to the problems for a bi-axial ellipsoid, bi-spheres and a torus. The elasticity problems for torus andbi-spheres were reduced to infinite systems of linear algebraic equations ofdiagonal form with more than thirty diagonals. Unfortunately, in the problemfor torus the representation for harmonic functions in the toroidal regionswas chosen incorrectly. Far later, Soloviev [3] used technique of generalizedanalytical functions for solving axisymmetric problems for an elastic torusand an elastic space with a toroidal cavity. The boundary-value problemswere reduced to infinite systems with eight and twelve diagonals, which weresolved numerically. In the papers by Podil’chuk and Kirilyuk [4, 5] the basicelasticity problems for a torus were reduced to infinite systems with five andfourteen diagonals. Their solutions were obtained numerically by successivetruncation.

It was the unwieldy form of the infinite systems obtained in discussedpapers that did not allow for an analytical solution, so we saw our objectivein developing a solution that would yield equations of the simplest form. Inview of that, the most interesting results were obtained in the scope of theStokes flow problem for toroidal bodies.

Being a linearized form of the Navier-Stokes equations in the approxi-mation of low Reynolds number, the Stokes equations describe very slow(“creeping”) flows of viscous incompressible fluids [6]. The reader shouldnot be confused with addressing problems of hydrodynamics while consid-ering the elastic framework. We will demonstrate that a direct analogy existsbetween the Stokes flow problems and the elastostatics problems for incom-pressible solids. In fact, it can be shown that a steady Stokes flow can beregarded as an elastic equilibrium of a solid with Poisson ratio equal to 0.5.The simplifications of the axisymmetric Stokes model allowed Payne and Pell

The Displacement Boundary-Value Problem of Elasticity for aTorus 3

[7] to construct an analytical solution of the Stokes problem for a torus thatdid not involve infinite algebraic systems. However, the approach of [7] can-not be expanded to the general elasticity case. Another interesting approachwas suggested by Wakiya [8], who obtained a tridiagonal infinite system forthe axisymmetric Stokes problem for a torus, but still solved it numerically.

Here we present an exact solution to the displacement boundary-valueproblem for a torus. The next section introduces a general solution of elas-tostatics equations, which is a generalization of the solution for Stokes flowsuggested by Wakiya [8], and can be applied to broad class of problems incyclidal coordinates. This general solution enables one to reduce the originalboundary-value problem for a torus to a set of infinite algebraic systems withno more than three diagonals. In the fourth section we present an analyticaltechnique for solving the obtained tridiagonal algebraic systems, which canbe potentially applied to systems with greater number of diagonals. Section5 discusses the issues pertaining to the uniqueness of the obtained solutionwith respect to the double-connectedness of toroidal regions. It is shown thatthe introduced form of general solution of the elastostatics equations yieldsa single-valued solution of the displacement boundary-value problem foratorus.

As an illustration of the general approach, we have the Stokes problemfor a torus revisited in Section 6. However, this does not mean at all that thepresented technique is incapable of solving some new problems. Quite thecontrary, we want to highlight the important aspects of our approach on asimple, yet physically rich example, and at the same time use the obtainedgeneral solution to present a complete and comprehensive solution to thisclassical problem.

2. General solution of the elastostatics equation for cyclidalcoordinate systems

The displacement boundary-value problem of elasticity consists in findingthe vector of elastic displacementu, which in a homogeneous isotropic solidmust satisfy the elastostatics (Lame) equation

21− ν1− 2ν

grad divu− curl curlu = 0, (1)

subject to boundary conditions

u|S = U, (2)

where ν is the Poisson ratio,S is the surface of the solid andU is thedisplacement vector prescribed onS.

4 P. Krokhmal

A key role in the presented approach plays the general solution of theelastostatics equation (1) that we develop below. Introducing the notations

div u = −1− 2ν

2− 2νϑ and γ =

1

4(1− ν), (3)

we reduce equation (1) to a non-homogeneous harmonic equation for thedisplacement vectoru:

1u = 2γ gradϑ. (4)

Observing thatϑ has to be a harmonic function, we write the general solutionof (4) as the sum of a partial solution of the non-homogeneous equation (4)and the general solution for the corresponding homogeneous equation:

u = B+ γ rϑ, where 1B = 0, 1ϑ = 0. (5)

FunctionsB andϑ are not independent; they have to satisfy the differentialconstraint

(γ + 1)ϑ + γ r · gradϑ + div B = 0. (6)

The choice of the general solution to be used in a particular problemis usually determined by the geometry of the problem. For example, thePapkovich-Neuber general solution

u = B− 14(1−ν) grad(r · B+ B0), 1B = 0, 1B0 = 0,

works well in the elasticity problems for bodies whose surface representsaplane (e. g., half-spaces or layers), but produces unwieldy equations for bod-ies with a complex geometry, such as torus. In the following sections we willshow that the derived general solution (5)–(6) leads to the equations ofthesimplest form in the boundary-value problem (1)–(2) for a torus. In general,given the similarities between the cyclidal coordinate systems, solution (5)–(6) will yield the simplest solutions for other bodies described by cyclidalcoordinates, such as spindle- or lens-shaped body, or bi-spheres.

Solution of the elasticity problems for bodies of revolution can oftenbe simplified if the original boundary-value problem is first formulated incylindrical polar coordinates, and then solved using the special curvilinearcoordinate system that describes the body surface. In accordance tothisapproach, we write functionsB, ϑ , and vectoru as the Fourier series withrespect to polar angleϕ of the cylindrical coordinate system{r, z, ϕ}, whosez-axis coincides with the body’s axis of revolution:

B =∞∑′

k=0

{ir Brk(r, z)

cossin

kϕ + iϕBϕk(r, z)sin− cos

kϕ + izBzk(r, z)cossin

kϕ

},


ϑ =∞∑′

k=0

ϑk(r, z)cos

sinkϕ,

u =∞∑′

k=0

{ir uk(r, z)

cossin

kϕ + iϕvk(r, z)sin− cos

kϕ + izwk(r, z)cossin

kϕ

}. (7)

Hereir , iϕ, andiz are the basis vectors of the cylindrical coordinates, and theprimed summation

∑′ indicates that the first term of the sum (in our case,the term withk = 0) is halved.

Harmonicity of functionsB andϑ determines the equations to be satisfiedby their Fourier components. Noting thatBϕ0 = 0, for all k ≥ 0 we have

1k±1(Brk ± Bϕk) = 0, 1k Bzk = 0, 1kϑk = 0, (8)

where operator1k is

1k =∂2

∂r 2+

1

r

∂

∂r+∂2

∂z2−

k2

r 2.

Readily, the Fourier components ofB can be expressed as

Brk = φk+1+ ψk−1, Bϕk = φk+1− ψk−1, Bzk = χk,

whereψ−1 = 0 and functionsφk+1, ψk−1, andχk solve equations

1k+1φk+1 = 0, 1k−1ψk−1 = 0, 1kχk = 0. (9)

By equality (5), the Fourier transforms (7) of the displacement vectoru forharmonics withk ≥ 1 can be written as

uk = φk+1+ ψk−1+ γ rϑk,

vk = φk+1− ψk−1,

wk = χk + γ zϑk, (10a)

whereas in the axisymmetric case (k = 0) they are1

u0 = φ1+ γ rϑ0, w0 = χ0+ γ zϑ0. (10b)

1 We do not consider here the trivial realization of axisymmetric stress fieldin a body ofrevolution, known as thepure torsion. In this case, the general solution contains only onenon-zero component of the displacement vector:

uϕ = −1

2v0 = −

1

2φ1,

with equation (11) satisfied identically.

6 P. Krokhmal

In terms of functions (9) the differential constraint (6) translates to

γ

(1+

1

γ+ r

∂

∂r+ z

∂

∂z

)ϑk +

(∂

∂r+

k+ 1

r

)φk+1

+(∂

∂r−

k− 1

r

)ψk−1+

∂χk

∂z= 0. (11)

Now determine the boundary conditions for functions (9). Writing theFourier expansion for vectorU in the same way as (7)

u|S=∞∑′

k=0

{ir Uk(r, z)

cossin

kϕ + iϕVk(r, z)sin− cos

kϕ + izWk(r, z)cossin

kϕ

}, (12)

then, substituting the expressions (10) and (7) foru in (12), after a minorrearrangement we find that functions (9) at the boundary of a solid mustsatisfy conditions

γϑ0

∣∣∣S=(−

1

rφ1+

1

rU0

)∣∣∣S, χ0

∣∣∣S=(z

rφ1−

z

rU0+W0

)∣∣∣S, (13a)

and fork ≥ 1,

γϑk|S =(−

2

rφk+1+

1

r(Uk + Vk)

)∣∣∣∣S

,

ψk−1|S = (φk+1− Vk)|S ,

χk|S =(2

z

rφk+1−

z

r(Uk + Vk)+Wk

)∣∣∣S. (13b)

Thus, to find the Fourier components (10) of the displacement vectoru, onehas to solve the boundary-value problem (13) for functionsφk+1, ψk−1, χk,andϑk, subject to the differential constraint (11). Note that (11) is by nomeans an equation to be solved: the general form of the functions it containsis already predetermined by equations (8)–(9). However, the functionsφk+1,ψk−1, χk, andϑk are not independent, thus having them satisfy the constraint(11) eliminates the arbitrariness in the solution.

In what follows, we will apply the toroidal coordinates to solve theboundary-value problem (13) whereS is a toroidal surface.

3. Exact solution of the boundary-value problem in toroidalcoordinates

The previous section demonstrated how the displacement boundary-valueproblem (1)–(2) for a body of revolution is reduced to the boundary-valueproblem (13), (11) for four functions (9), (8). The differential constraint (11)


to be satisfied by these functions complements three equations of the bound-ary conditions (13), making a complete system for determining functionsφk+1, ψk−1, χk, andϑk.2

A successful solution of a three-dimensional boundary-value problemcan be achieved when an orthogonal and separable system of curvilinearcoordinates exists, whose coordinate surfaces fit the surface of the bound-ary conditions. For problems involving toroidal bodies, such system is thetoroidal coordinates3, which belong to the family of the cyclidal coordinates[1]. Being a conjugate system of revolution, the toroidal coordinates{ξ, η, ϕ}relate to the cylindrical coordinates{r, z, ϕ}, whereϕ is the polar anglecommon for both systems, as

r =c sinhξ

coshξ − cosη, z=

c sinη

coshξ − cosη, 0≤ ξ <∞, −π < η ≤ π, (14)

wherec is the metric parameter. The coordinate surfacesξ = ξ0 = constrepresent toroids; orthogonal to them are surfacesη = η0 = const thatgenerate spherical segments leaning on the circler = c in planez = 0(Fig. 1). In the toroidal system, the origin has coordinatesξ = 0, η = π ,the infinitely remote point of space isξ = 0, η = 0, whereas points withξ = ∞ correspond to the circler = c, z = 0. Given the radiia andb of theoutermost rim of the torus and the torus opening, the parametersc andξ0 canbe found as

c =√

ab, ξ0 = 2 arctanh√

b/a.

Figure 1. Torus and the toroidal coordiantes.

The distinguishing property of the cyclidal coordinate systems, in-cluding the toroidal coordinates, comparing to other separable coordinatesystems (spherical, elliptical etc.), is the incomplete separation of vari-ables in the Laplace equation and equations of type (9). A solution

2 In the axisymmetric case (k = 0), there are three functions (φ1, χ0, andϑ0) and twoboundary conditions (13a).

3 The toroidal coordinates, as well as other cyclidal coordinate systems,are not completelyseparable, see below.

8 P. Krokhmal

of the harmonic equation in a cyclidal system{ξ1, ξ2, ξ3} has the formR(ξ1, ξ2, ξ3)X1(ξ1)X2(ξ2)X3(ξ3), whereR(ξ1, ξ2, ξ3) is themodulating fac-tor [1] that must be introduced to make possible the separation of eigen-functions Xi (ξi ). The incomplete separation of variables in the cyclidalcoordinates can significantly complicate solution even of the basic prob-lems of the potential theory (see, for example, paper by Lebedev [9] on theproblems of the theory of potential for torus of elliptical cross-section).

In the toroidal and bi-spherical coordinate systems, the modulating factorhas the form ofRiemannian radical

√coshξ − cosη, and the eigenfunctions

Xi (ξi ) are the trigonometric/hyperbolic functions and Legendre functions.The separation parameters for toroidal regions are chosen such that thegeneral solutions of equations (9) and (8) have the form

ϑk=1

c

√coshξ− cosη

∞∑′

n=0

(An,k cosnη+ An,k sinnη)Lkn− 1

2(coshξ),

φk+1=√

coshξ− cosη∞∑′

n=0

(Bn,k cosnη+ Bn,k sinnη)Lk+1n− 1

2(coshξ),

ψk−1=√


n=0

(Dn,k cosnη+ Dn,k sinnη)Lk−1n− 1

2(coshξ),

χk=√


n=0

(Cn,k sinnη−Cn,k cosnη)Lkn− 1

2(coshξ), (15)

whereLkn− 1

2(coshξ) denote the Legendre functions of semi-integer index of

the first or the second kind,Pkn− 1

2(coshξ) and Qk

n− 12(coshξ) (see, for in-

stance, [10, 11]). The following integral representations are availablefor theconsidered Legendre functions [11]:

Pkn− 1

2(coshξ) =

Ŵ(n+ 1/2+ k)

πŴ(n+ 1/2)

π∫

0

coskτ dτ

(coshξ + sinhξ cosτ)1/2−n(16a)

Qkn− 1

2(coshξ) =

(−1)k

2√

2πŴ(k+ 1/2)

π∫

−π

sinhkξ cosnτ dτ

(coshξ − cosτ)k+1/2(16b)

The asymptotic behavior of the Legendre functions with argument approach-ing infinity or unity [11] dictates that functionsQk

n− 12(coshξ) must be used

in the expansions (15) for the inner problems, andPkn− 1

2(coshξ) – for outer

problems for a torus.The constants of integrationAn,k, . . . , Dn,k, and An,k, . . . , Dn,k in the

functions (15) have to be determined from the boundary conditions (12) and


equation (11). Let first consider the equality (11). It is easy to verify that eachsummand in the differential constraint (11) satisfies to equation1k(·) = 0,i. e.,

1k

(ϑk +

ϑk

γ+ r

∂ϑk

∂r+ z

∂ϑk

∂z

)= 0, . . . , 1k

(∂χk

∂z

)= 0,

and, consequently, the left-hand side of (11) is a solution to the equation1k(·) = 0. Therefore, on inserting representations (15) in equation (11), ittransforms to√

coshξ− cosη∞∑

n=0

({. . .} cosnη + {. . .} sinnη

)Lk

n− 12(coshξ) = 0, (17)

where the terms in braces contain linear combinations of coefficients (15),tilded and untilded separately. Evidently, for equation (17) to hold for allvalues ofξ andη, the terms in braces must be equal to zero. This yields twoseparate infinite sets of linear algebraic equations with respect to the tildedand untilded coefficients (15). To shorten the presentation, we considerhereonly the untilded constants:

γ((1/2+1/γ )A0,k + (k+1/2)A1,k

)− (k+1/2)2B0,k

−(k+1/2)(k+3/2)B1,k − D1,k + D0,k + (k+1/2)C1,k = 0,

γ((n+k+1/2)An+1,k + (1+2/γ )An,k − (n−k−1/2)An−1,k

)

−(n+k+3/2)(n+k+1/2)Bn+1,k + 2(n+k+1/2)(n−k−1/2)Bn,k

−(n−k−1/2)(n−k−3/2)Bn−1,k − Dn+1,k + 2Dn,k − Dn−1,k

+(n+k+1/2)Cn+1,k − 2nCn,k + (n−k−1/2)Cn−1,k = 0, n ≥ 1,(18)

whereCk0 = 0. The system for the tilded constants has the same form but

does not contain the first equation forn = 0.Now make use of the boundary conditions (13). Note that equalities (13)

present the functionsϑk, ψk−1, andχk at the boundaryξ = ξ0 in terms of theboundary values of functionφk+1. Consequently, substitution of the expan-sions (15) into the boundary conditions (13) yields expression for constantsAn,k,Cn,k, Dn,k in terms of some new unknownsxn,k, which are nothing butthe normalized coefficientsBn,k of functionφk+1:4

γ A0,k = −20,k coshξ0 x0,k + 20,kx1,k + 0,k p0,k,

γ An,k = −2n,k coshξ0 xn,k + n,k(xn+1,k + xn−1,k)+ n,k pn,k, n ≥ 1,

Bn,k = λn,k xn,k, n ≥ 0,

Cn,k = n,k(xn−1,k − xn+1,k)+ n,k sn,k, n ≥ 1,

Dn,k = γn,k xn,k + γn,k qn,k, n ≥ 0. (19)

4 The corresponding formulas for the tilded constants are identical to above ones exceptfor the casen = 0.

10 P. Krokhmal

Herepn,k, qn,k, andsn,k are the Fourier coefficients of the following functions:

√coshξ0− cosη (Uk + εkVk) = δk

∞∑′

n=0

(pn,k cosnη + pn,k sinnη),

−εkVk√

coshξ0− cosη= εk

∞∑′

n=0

(qn,k cosnη + qn,k sinnη),

Wk sinhξ0− (Uk + εkVk) sinη√

coshξ0− cosη= δk

∞∑′

n=0

(sn,k sinnη − sn,k cosnη), (20)

andn,k, γn,k, andλn,k have the form

n,k =δk

sinhξ0 Lkn− 1

2(coshξ0)

,

γn,k =εk

Lk−1n− 1

2(coshξ0)

,

λn,k =1

Lk+1n− 1

2(coshξ0)

. (21)

Coefficientsδk andεk have been introduced to the formulas (20)–(21) to makethem accomodate, along with the general asymmetric instancesk ≥ 1, theaxisymmetric casek = 0:

δ0 = 1/2, ε0 = 0, δk = εk = 1, k ≥ 1.

Finally, by plugging expressions (19)–(21) into the derived equations (18),for each harmonicsk ≥ 0 we obtain an infinite system of algebraic equationswith respect to variablesxn,k:

an,k xn+1,k − bn,k xn,k + cn,k xn−1,k = dn,k, n ≥ 0, c0,k = 0. (22)

Coefficients of the first equation(n = 0) in (22) equal to

a0,k = −(2k+ 1)1,k coshξ0+ (1+ 2/γ )0,k

−(k+ 1/2)(k+ 3/2)λ1,k − γ1,k,

b0,k = (1+ 2/γ )0,k coshξ0− (2k+ 1)1,k + (k+ 1/2)2λ0,k − γ0,k,

d0,k = γ1,k q1,k − γ0,k q0,k − (1/2+ 1/γ )0,k p0,k

−(k+ 1/2)1,k p1,k − (k+ 1/2)1,k s1,k, (23)

while for n ≥ 1 they are

an,k = −2(n+ k+ 1/2)n+1,k coshξ0+ (2n+ 1+ 2/γ )n,k


−(n+ k+ 1/2)(n+ k+ 3/2)λn+1,k − γn+1,k,

bn,k = 2(1+ 2/γ )n,k coshξ0− 2(n+ k+ 1/2)n+1,k

+2(n− k− 1/2)n−1,k − 2(n+ k+ 1/2)(n− k− 1/2)λn,k − 2γn,k,

cn,k = 2(n− k− 1/2)n−1,k coshξ0+ (−2n+ 1+ 2/γ )n,k

−(n− k− 1/2)(n− k− 3/2)λn−1,k − γn−1,k,

dn,k = γn+1,k qn+1,k − 2γn,k qn,k + γn−1,k qn−1,k

−(n+ k+ 1/2)n+1,k pn+1,k − (1+ 2/γ )n,k pn,k

+(n− k− 1/2)n−1,k pn−1,k − (n+k+1/2)n+1,k sn+1,k

+2nn,k sn,k − (n−k−1/2)n−1,k sn−1,k. (24)

The infinite system for variablesxn,k associated with tilded constants in (15)has the form

an,k xn+1,k − bn,k xn,k + cn,k xn−1,k = dn,k, n ≥ 1, x0,k = 0. (25)

wheredn,k, n ≥ 1 have the form similar todn,k in (24):

dn,k = γn+1,k qn+1,k − 2γn,k qn,k + γn−1,k qn−1,k − (n+ k+ 1/2)n+1,k pn+1,k

−(1+ 2/γ )n,k pn,k + (n− k− 1/2)n−1,k pn−1,k

−(n+k+1/2)n+1,k sn+1,k + 2nn,k sn,k − (n−k−1/2)n−1,k sn−1,k,

provided thatq0,k = p0,k = 0.Reduction of the displacement boundary-value problem for a torus to the

tridiagonal infinite systems (22), (25) represents an exact solution of theboundary-value problem (1)–(2) in terms of solvability of the basic potentialtheory problems in toroidal regions. Indeed, it can be demonstrated that theNeumann problem for equation1k(·) = 0 in a toroidal region reduces toequations of type (22), implying that harmonic function in a toroidal regioncannot be determined simpler than by a set of tridiagonal infinite systems,unless its value is specified at the whole boundary.5

Given that the displacement boundary-value of elasticity cannot be re-duced to pure Dirichlet problems for harmonic functions, the obtainedtridiagonal infinite systems of algebraic equations (22), (25) qualify as anexact solution of the boundary-value problem (1)–(2) for a torus.

We strengthen this result by presenting analytical techniques for solv-ing the tridiagonal systems of type (22), thereby finalizing the constructionof exact analytical solution of the displacement boundary-value problemofelasticity for a torus.

5 The Dirichlet problem for a torus has an exact solution, i.e. the coefficients of the har-monic function are determined directly as the Fourier transforms of its boundary value. TheNeumann problem for a torus features a similar solution in the axisymmetric caseonly, due tothe fact that fork = 0 the corresponding tridiagonal system can be split into two bidiagonalones.

12 P. Krokhmal

4. Solution of infinite algebraic systems of diagonal form

An analytical technique for solving systems of type (22) with three variablesin a row was suggested by Kutsenko and Ulitko [12]. According to [12], theoriginal tridiagonal infinite system of algebraic equations

an xn+1− bn xn + cn xn−1 = dn, c0 = 0, n ≥ 0, (26)

can be reduced to successive solving of two bidiagonal systems

an

1− βnxn+1− xn = Yn, n ≥ 0,

Yn −cn

1− βnYn−1 =

dn

1− βn, n ≥ 0, (27)

wherean = an/bn, cn = cn/bn, dn = dn/bn, andβn is a finite continuousfraction:

βn =cnan−1

1− βn−1, β0 = 0, or βn =

cnan−1

1−cn−1an−2

1− · · ·. . .

1−c1a0

1− 0.

The above bidiagonal systems have analytical solutions

xn = −Yn −∞∑

k=n+1

Yk

k−1∏

s=n

as

1− βs, n ≥ 0,

Yn =dn

1− βn+

n−1∑

k=0

dk

1− βk

n∏

s=k+1

cs

1− βs, n ≥ 0.

We present a different approach that can be applied to systems with greaternumber of variables in a row. To illustrate the idea on the tridiagonal system(26) let try a solution of (22) in the form

xn = αn

∞∑

j=n

y j , n ≥ 0, (28)

wherey j are the new unknowns, and the coefficientsαn are undetermined sofar. Substituting (28) into (26) we have

(anαn+1− bnαn + cnαn−1)

∞∑

j=n+1

y j

−(bnαn − cnαn−1)yn + cnαn−1yn−1 = dn, n ≥ 0.


If one requires coefficientsαkn to satisfy the conditions

anαn+1− bnαn + cnαn−1 = 0, n ≥ 0, c0 = 0, (29)

thenyn will be determined from the bidiagonal system

anαn+1yn − cnαn−1yn−1 = −dn, n ≥ 0, c0 = 0, (30)

whose exact solution is

yn = −1

anαn+1αn

{αn dn +

n−1∑

j=0

α j d j

n−1∏

i= j

ci+1

ai

}, n ≥ 0. (31)

To obtain a solution of the homogeneous equations (29) that involves onlyone arbitrary constant, we rewrite (29) in the form

a0α1 = b0α∗0,

a1α2− b1α1 = −c1α∗0,

anαn+1− bnαn + cnαn−1 = 0, n ≥ 2, (32)

whereα∗0 = const6= 0. Then solution of (29) can be obtained by recursion:

α−1 = 0, α0 = α∗0, αn =1

an−1(bn−1αn−1− cn−1αn−2), n ≥ 1. (33)

Note that constantα∗0 may be assigned arbitrary non-zero value, since itcancels out after substituting (33) into the expressions (31).

The formal proof of convergence for the presented solutions requires theuse of theories of continued fractions and sequence transformations, and isbeyond the scope of this paper. Nevertheless, convergence in formulas (28)–(33) can be verified in every particular case by the analysis of the coefficientsof system (26). For example, for the problems involving torus exterior, such asthe Stokes problem for a torus considered in Section 6, coefficients of systems(22), (25) converge exponentially to zero, making their solutions converge aswell.

5. On the uniqueness of the solutions for the boundary-value problemsof elasticity in toroidal regions

Besides incomplete separation of variables in the toroidal coordinates, thedisplacement boundary-value problem for a torus features one more challengein constructing the solution.

The problem is that certain classes of vector boundary-value problemsmay not be uniquely solved in multiply-connected regions. In particular,

14 P. Krokhmal

vector boundary-value problems that involve the Cauchy-Riemann equa-tions require additional analysis of the single-valuedness of their solutions.To such class of problems belong the boundary-value problems of elastic-ity, where the elastostatics equation (1) “contains” the so-calledgeneralizedCauchy-Riemann equations.

To show that solving of the elastostatics equations always involves solvingequations of Cauchy-Riemann type, we write (1) as two consecutive systemsof the fundamental equations for vector fields

{curlωωω = −gradϑ,divωωω = 0,

{curlu = ωωω,div u = − 1−2ν

2(1−ν)ϑ.(34)

Each pair of equations in (34) restores a vector field by its vorticity and di-vergence. Along with the Lame equation, systems (34) represent an invariantform of the equations for equilibrium of an elastic medium, but in contrast to(1), they emphasize the structure of the displacement and vorticity fields inan elastic solid.

By representing, similarly to (7), the vorticity vectorωωω by its Fourier serieswith componentsωrk, ωϕk, andωzk, from the first system of (34) one canderive two sets of the generalized Cauchy-Riemann equations

(∂

∂r∓

k∓ 1

r

)�±k = −

∂2±k∂z

,

∂�±k∂z

=(∂

∂r±

k

r

)2±k , (35)

where functions�±k and2±k contain the Fourier components of vorticityωωωand “dilatation”ϑ :

�±k = ωrk ± ωϕk, 2±k = ωzk± ϑk.

Note that at infinity (r, z→ ∞) both “+” and “−” systems (35) becomeidentical to the classical Cauchy-Riemann equations governing the real andimaginary parts of an analytic function of complex variable [1]. This observa-tion makes clear the analogy between (35) and the regular Cauchy-Riemannequations, and supports the conjecture that solutions of both types of systemsshould reveal similar properties. Namely, in the same way as analytic functionmust satisfy some predetermined constraints to be single-valued in a multiple-connected region, functions�±k and2±k have to satisfy certain conditions tobe a unique solution to the equations (35).6

6 The single-valuedness of the solutions of equations of type (35) is closelyconnected tothe so-calledgeneralized Dirichlet problemfor analytic functions in complex plane [13, 14].This problem consists in finding a harmonic function by its boundary value,subject to the


To show this, first observe that functions�±k , 2±k in (35) must satisfyequations

1k∓1�±k = 0, 1k2

±k = 0,

and therefore can be presented in the toroidal coordinates as

�±k =√

coshξ − cosη∞∑′

n=0

(X±n,k sinnη − X±n,k cosnη

)Lk∓1

n− 12(coshξ),

2±k =√


n=0

(Y±n,k cosnη + Y±n,k sinnη

)Lk

n− 12(coshξ). (36)

Then, to ensure the single-valuedness of functions (35) one must require theircoefficients to satisfy the following conditions:

∞∑′

n=0

Y+n,k = 0,∞∑′

n=0

Y−n,k

k∏

j=1

[n2− ( j − 1/2)2

]= 0, (37)

(as before, we make our arguments for the untilded coefficients).There areno constraints for series X±n,k. The physical reasonings behind relations (37)can be easily demonstrated for the torus exterior. Let the functions�±k beexpressed in terms of2±k from equations (35) in the following way:

r∓k+1�±k =∫

L

r∓k+1

{−∂2±k∂z

dr +(∂2±k∂r±

k

r2±k

)dz

}, (38)

whereL is an arbitrary curve in the cross-sectional half-plane{r ≥ 0 | ϕ =const}. If one wants the functions�±k determined by (38) to be single-valuedin a double-connected region then integrals (38) must vanish at arbitraryclosed contourL lying in the region. TakingL as a circleξ = ξ1 enclosingthe torus and doing the integration we obtain the relations (37).

Another way to derive the constraints (37) is to use the recursive relationsbetween the Legendre functions of adjacent indices [11]. Since these relationsare identical for both functionsPk

n− 12(coshξ) and Qk

n− 12(coshξ), equalities

(37) must hold in the interior as well as in the exterior of a torus.Similarly, it can be shown thatcoefficients Y±n,k are always uniquely deter-

mined by known constants X±n,k. This “asymmetry” in the solvability of the

condition that its conjugated by the Cauchy-Riemann conditions counterpart must be single-valued in the same region. In [13, 14] it was demonstrated that the solutionof the generalizedDirichlet problem for a planar single-connected region exists and is unique. However, for theexistence of the solution in multiply-connected planar regions some predetermined constraintsmust be imposed on the boundary value of the sought harmonic function. Evidently, this resultshould also hold for the generalized Cauchy-Riemann equations (35).

16 P. Krokhmal

generalized Cauchy-Riemann equations in toroidal domains is caused by thepresence of the Riemannian radical in the expressions (36) for functions�±kand2±k .

Now we show that the introduced form (5) of the general solution of theelastostatics equations ensures the uniqueness of the solution of the boundary-value problem (1)–(2) in the toroidal regions. Indeed, in terms of functions(15) conditions (37) become

(kγ + 1)∞∑′

n=0

An,k = (2k2+ k)∞∑′

n=0

Bn,k,

(1− kγ )∞∑′

n=0

An,k

k∏

j=1

[n2− ( j − 1/2)2

]

= (2k2− k)∞∑′

n=0

Dn,k

k−1∏

j=1

[n2− ( j − 1/2)2

]. (39)

Then, on summing equations (18) multiplied by factors 1 andk∏

j=1

[n2 −

( j − 1/2)2]

respectively, we find that conditions (39) turn into identities if

constantsAn,k, Bn,k, Cn,k, andDn,k satisfy equations (18).Let us emphasize again that the identical fulfillment of (37), and con-

sequently, the uniqueness of the obtained solution of the displacementboundary-value problem for a torus are ensured by the specific formof thegeneral solution (5)–(6). Other forms of general solution may require addi-tional argumentation for satisfying the conditions of type (37). An example ofgeneral solution that yields a non-unique solution of a vector boundary-valueproblem in a toroidal region is presented in the next section within the scopeof the Stokes problem for a torus.

6. Example: The Stokes problem for a torus

The comprehensiveness of the developed solution for the displacementboundary-value problem for a torus makes the author reluctant to illustratingthe approach by putting some numbers into the general formulas. Ultimately,such an example will be no more than an exercise in developing the Fourierexpansions (2), (20) for the boundary conditions, with subsequent pluggingthe calculated Fourier coefficients into the right-hand side of equations (22).Instead, we invite the reader to revisit a classical problem of axisymmetricmotion of a rigid torus in an unbounded Stokes flow. We will show how theelastic solutions may be applied to the problems of viscous incompressible


flows, and discuss two different approaches to solving the Stokes problem fora torus. One of them uses the presented elasticity solution, while the other willilluminate the ideas of Section 5. We will also demonstrate the relationshipbetween these two approaches, and how one of them can be used to derive anelegant solution for the other.

The most significant contribution to the Stokes problem for a torus can befound, in our opinion, in the papers by Pell and Payne [7] and Wakiya [8]that feature the two mentioned approaches to the construction of the solution.However, it was the unanswered questions left in both these papers that,infact, stimulated our initial interest to the presented work. Here we present acompletesolution to the Stokes problem for a torus, and demonstrate how thequite different approaches of [7] and [8] relate to each other.

6.1. MOTION OF A TORUS IN A VISCOUSSTOKES FLOW

The term “Stokes problem” with respect to a particular body usually ad-dresses the problem of a slow, steady motion of that body in a viscousincompressible fluid, or, equivalently, a problem of steady viscous flow aboutthe body [6]. The name attributes to G. G. Stokes, who first formulated andsolved the problem of motion of a rigid sphere in a viscous fluid [15].

We will employ the general elasticity solution developed above to solvethe Stokes problem for a torus. Given that the Stokes problem is essentiallyahydromechanics problem, this kind of approach can be justified by means oftheanalogybetween the boundary-value problems of elastostatics and thoseof steady Stokes flows of viscous fluid, that we present below.

The Stokes equations

curl curlu = −1

µgradp,

div u = 0, (40)

describeslowflows of viscous incompressible fluids [6]. Hereu is the veloc-ity vector, p is pressure, andµ is the dynamic viscositycoefficient. Equations(40) represent a linearized form of the general Navier-Stokes equations in theapproximation of low Reynolds number [6]. The second equation of (40) isthecontinuityequation that prescribes the fluid incompressibility. Writing theStokes equations in the form similar to (34):

{curlωωω = − 1

µgradp,

divωωω = 0,

{curlu = ωωω,div u = 0,

(41)

and setting in (34)ν = 1/2 (i.e., medium is incompressible) andϑ = p/µ,we observe that both Lame equation (1) and Stokes equations (40) reduce tothe same system of equations for vector fields (41). Hence, equilibrium ofan

18 P. Krokhmal

elastic incompressible solid and steady flow of a Stokes fluid are governedby the same equations. In the scope of the Stokes problemu andµ representthe velocity vector and dynamic viscosity coefficient, whereas in the elas-ticity framework they stand for the elastic displacement and shear moduluscorrespondingly.7

So, consider the problem of determining the velocity and pressure fieldsin a viscous incompressible flow due to a steady translation of a rigid toroidalbody along its symmetry axis, which is assumed to coincide with the axisOzof the cylindrical coordinates. Besides being a solution of the Stokes equa-tions (40),u andp must satisfy the boundary conditions that in the cylindricalcoordinates have the form [6]

ur |S = 0, uz|S = V, (42)

p|∞ = p∞, (43)

where V is the translation velocity, anS is the torus’ surface. Equalities(42) reflect the general property of viscous flows, i.e., adherence of the fluidparticles to the body’s surface. And without loss of generality, the constanthydrostatic pressure at infinityp∞ is assumed to be zero.

Below we demonstrate how the general theory considered in the previoussections can be used to construct a complete and comprehensive solution forthe Stokes problem (40), (42) for a torus. In view of the presented analogy be-tween problems of elastostatics and Stokes flows, one can obtain the solutionfor the Stokes problem for a torus by adapting the developed general solutionof the displacement boundary-value problem of elasticity for toroidal bodies.However, we will defer the application of this approach for a few pages,andfirst consider the traditional method for solving the Stokes flows problems,the so-calledstream functionapproach. After that we will show how thesetwo approaches relate to each other, and how to use one of them to constructthe solution for the other.

6.2. SOLUTION OF THE STOKES PROBLEM FOR A TORUS USING STREAM

FUNCTION

The traditional approach [6] to solving the axisymmetric Stokes equations isto represent the velocity vectoru as8

ur = −∂9

∂z, uz =

1

r

∂

∂r(r9) . (44)

7 The considered analogy between elasticity and Stokes flow problems doesnot span thedynamic problems.

8 Function9 defined by (44) differs from the from the classicalstream functionψ intro-duced in [6]:9 = −ψ/r . The choice of either function in representation of type (44) is merelya matter of convenience.


This representation satisfies identically the second equation in (40); the firstequation yields the condition to be satisfied by function9:

11119 = 0. (45)

In terms of function9 the boundary conditions (42) become

9|S =V

2

(r +

λ

r

)∣∣∣∣S

,∂9

∂n

∣∣∣∣S

=V

2

∂

∂n

(r +

λ

r

)∣∣∣∣S

, (46)

wheren is the normal to the surfaceS of the torus. The unknown constantλ attributes to the multiple-connectedness of torus surface, and has to bedetermined from the solution of the problem. For single-connected bodies,λ is always zero. It turns out that to determine the value ofλ, one has to con-sider the solvability problem for the generalized Cauchy-Riemann equations.Failure to determine the correct value ofλ leads to physically inadequateresults, as in [16].

The explicit form of function9 in the toroidal coordinates is required tosolve the boundary-value problem (46). First, observe that9, as a generalsolution of equation (45), admits the next representation by two harmonicfunctions:

9 = 81+ r P, 1181 = 0, 1P = 0. (47)

Taking into consideration that the problem at hand is an exterior problem,and omitting the coefficients that will cancel out after satisfying the boundaryconditions (46), we write the representations for functions81 and P in thetoroidal coordinates as

81 =√


n=0

n cosnηP1n− 1

2(coshξ),

P =1

c

√coshξ − cosη

∞∑′

n=0

n cosnηPn− 1

2(coshξ). (48)

Combining (48) and (47), we obtain the general form of function9 in thetoroidal coordinates:

9 =1

2√

coshξ − cosη

{g0P1

12(coshξ)+ h0coshξ P1

− 12(coshξ)

+∞∑

n=1

(gn P1

n− 32(coshξ)+ hn P1

n+ 12(coshξ)

)cosnη

}, (49)

where coefficientsgn and hn of function 9 are expressed in terms ofcoefficients of81 andP as

g0 = 20−1, h0 = 0− 20,

20 P. Krokhmal

gn =n+ 1/2

nn −n−1−

n

n, n ≥ 1,

hn =n− 1/2

nn −n+1+

n

n, n ≥ 1. (50)

Substituting the representation (49) in both equalities of the boundary condi-tions (46), we obtain a separate system of two linear algebraic equations withrespect togn andhn for eachn ≥ 0. The right-hand sides of these equationsare linear inλ (recall that the unknown constantλ enters the right-hand sidesof the boundary conditions (46) linearly), therefore coefficientsgn andhn offunction9 have the form

gn = g(1)n +λ

c2g(2)n , hn = h(1)n +

λ

c2h(2)n , (51)

where the explicit expressions forg(1,2)n and h(1,2)n are furnished in theAppendix.

The value ofλ in (51) cannot be determined from the boundary-valueconditions(46). Instead, the choice ofλ is justified by the uniqueness of thesolution of the boundary-value problem (42), i. e., the velocity vectoru aswell as pressurep must be single-valued in the flow region.

According to the discussion in the preceding section, the unique solutionof elastostatics boundary-value problems in toroidal regions is achieved byimposing conditions (37) on the components of vorticity and dilatation in themedium. In the context of the Stokes problem for a torus, these conditionsacquire a transparent physical meaning. Indeed, since the velocity compo-nentsur and uz are uniquely determined by the function9 (44), then forany value ofλ in (51) they are single-valued functions of spatial coordi-nate. However, pressurep cannot be directly determined from the solutionof the boundary-value problem (46) for function9. To find p = ϑµ, onehas to solve the generalized Cauchy-Riemann equations (35), whose uniquesolution is guaranteed by the conditions of type (37).

Given the axial symmetry of the considered problem, equations (35) re-duce to a single system governing vorticityω and pressureϑ = p/µ in thefluid:

(∂

∂r+

1

r

)ω = −

∂ϑ

∂z,

∂ω

∂z=∂ϑ

∂r. (52)

Simplifying the notations (36), we presentω = �±0 and p/µ = ϑ = 2±0 as

ω = −1

c2

√coshξ − cosη

∞∑′

n=0

ωn cosnηP1n− 1

2(coshξ),

p =µ

c2

√coshξ − cosη

∞∑

n=1

pn sinnηPn− 12(coshξ). (53)


With regard to functions (53), the uniqueness condition of type (37) reads as

∞∑′

n=0

ωn = 0. (54)

Then, from (44) it follows thatω =∂ur

∂z−∂uz

∂r= −119, and after somewhat

tedious but straightforward transformations we find

∞∑′

n=0

ωn = −3

2

∞∑

n=0

(gn + hn). (55)

Finally, relationships (54) and (55) yield the expression forλ:

λ

c2= −

∞∑

n=0

(g(1)n + h(1)n

)/ ∞∑

n=0

(g(2)n + h(2)n

). (56)

The Fourier coefficientspn of pressurep can be restored by the knownvorticity coefficientsωn using the equations (52)9:

pn = −(n− 1/2)ωn +∞∑

j=n+1

ω j = −(n− 1/2)ωn −n∑′

j=0

ω j .

Expression of type (56) was also derived in the paper by Pell and Payne [7],but their argumentation was rather vague. Section 5 of the present paperjus-tifies the condition (54) and the subsequent expression (56) from the generalviewpoint of solvability of the equations for vector fields and the generalizedCauchy-Riemann equations in the double-connected toroidal regions.

6.3. APPLICATION OF THE GENERAL ELASTICITY SOLUTION TO THE

STOKES PROBLEM

An alternative way to solve the considered Stokes problem for a torus is toemploy the general solution (10) for axisymmetric case (k = 0):

ur =1

2u0, uz =

1

2w0, (57a)

where in expressions (10) foru0 andw0 we setν = 1/2 andϑ = ϑ0/2 =p/µ:

u0 = φ1+r

µp, w0 = χ0+

z

µp. (57b)

9 Equations for coefficientsωn and pn are derived in the way similar to the derivationof equations (18). Each of the equations (52) yields an infinite system of linear algebraicequations that is bidiagonal either in terms ofωn or pn, and therefore can be solved explicitly.

22 P. Krokhmal

Functions p, φ1, and χ0 are not independent, and have to satisfy thedifferential constraint (11) withk = 0:

(3+ r

∂

∂r+ z

∂

∂z

)p

µ+(∂

∂r+

1

r

)φ1+

∂χ0

∂z= 0, (58)

which is a rephrase of the continuity equation (40).10 The boundary condi-tions (42) in terms of functions (57) take the form

p

µ

∣∣∣∣ξ=ξ0= −

1

rφ1

∣∣∣∣ξ=ξ0

, χ0|ξ=ξ0 =(z

rφ1+ 2V

)∣∣∣ξ=ξ0

, (59)

whereξ = ξ0 is the surface of the torus. Representing the functions (57) inaccordance to (15),

2p

µ=

1

c

√coshξ− cosη

∞∑

n=1

An,0 sinnη Pn− 12(coshξ),

φ1 =√

coshξ− cosη∞∑

n=1

Bn,0 sinnη P1n− 1

2(coshξ),

χ0 = −√


n=0

Cn,0 cosnη Pn− 12(coshξ), (60)

and repeating steps (18)–(20), we come to a tridiagonal system with respectto variablesxn,0

11:

an,0 xn+1,0− bn,0 xn,0+ cn,0 xn−1,0 = dn,0, n ≥ 1, x0,0 = 0, (61)

where in the expression (24) for coefficientsan,0, bn,0, andcn,0 it must be setγ = 1/2, and the right-hand sidedn,0 contains only terms withsn,0:

sn,0 = −8√

2

πV sinhξ0 Qn− 1

2(coshξ0), n ≥ 1. (62)

The exact solution of (61)–(62) can be obtained as shown in (28)–(33).Note that by plugging in (57)–(62) value of the coefficientγ other than

1/2, we obtain the solution of elasticity problem where a rigid toroidalinclusion in an elastic space is subject to a vertical shift of magnitudeV .

10 Note that the first equation of (40) is satisfied identically by the representation (57).11 Formulas (18) to (24) were developed for the “untilded” coefficients in (15). Analogous

expressions for the “tilded” ones have almost the same form, differingin representations forn = 0.


6.4. ANALYSIS OF THE FLOW AROUND A TORUS AND NUMERICAL

RESULTS

In this subsection we present numerical results for the Stokes problem fora torus, using both solutions presented in the subsections 6.2 and 6.3. Butfirst we have to discuss the numerical procedures used for computation ofthe Legendre functions of the first kindPk

n− 12(coshξ) and expansions of type

(15).According to the asymptotic behavior of the Legendre functions [11],

functionsPkn− 1

2(coshξ) grow with n andξ asenξ . Therefore, the use of the

integral representation (16a) for calculating of the Legendre functionsmaylead to significant inaccuracies even at smalln. An acceptable degree ofaccuracy in computing the Legendre functionsP(0,1)

n− 12(coshξ), which are used

in the solution of the Stokes problem for a torus, is achieved by implementingthe following recursive scheme:

Pn+ 12(coshξ)= coshξ Pn− 1

2(coshξ)+

sinhξ

n+ 12

P1n− 1

2(coshξ), n ≥ 0,

P1n+ 1

2(coshξ)= coshξ P1

n− 12(coshξ)+ (n+ 1

2) sinhξ Pn− 12(coshξ), n ≥ 0,

where functionsP(0,1)− 1

2(coshξ) are computed using the corresponding integral

representations. Note that due to the exponential inn growth of the functionsPk

n− 12(coshξ), the coefficients in the expansions of type (48), (49), (60) must

converge at least exponentially to ensure the convergence of these series, and,as a result, such expansions also exhibit exponential convergence. Therefore,it is usually sufficient to keep about 20–25 terms in the series of type (48),(49), (60) to achieve the degree of accuracy within 10−6 −: 10−8. To computethe solution of the tridiagonal system (61) using the technique (28)–(33),wetruncated the infinite series (31) atn = 50.

Let proceed now with the analysis of the viscous flow due to translationalmotion of a rigid torus in the fluid.

First, we compute the drag force exerted on the surface of the torus, whichis the major integral characteristics of the flow about a body. The drag forceis obtained by integrating the surface tractions over the torus’ surface. Due tothe axial symmetry of the problem, the only non-zero component of the dragforce isFz:

Fz = −2√

2πµ

{−h0+

∞∑

n=1

[(n− 3

2)(n−12)gn + (n+ 1

2)(n+32)hn]}

= −2√

2πµc∞∑′

n=0

(nAn − 2Cn

). (63)

24 P. Krokhmal

The graph of the drag forceFz, normalized by the drag force of a sphere withradiusa equal to the radius of the outermost rim of the torus, is presentedon the Fig. 2a. Withξ0 approaching zero, the normalized drag force (63)reaches the limiting value of 0.935, which is the normalized drag force forthe so-called “closed torus”, i.e. torus without opening [17]. Whenξ0 →∞,and torus degenerates into a thin circular thread, the asymptotic value of thenormalized drag force equals to

|Fz|6πµV a

∼4π

3ξ0, ξ0→∞,

which is the normalized drag force for a thin annulus.

a. Drag force exerted on a torus b. Flux through a torus opening

Figure 2. Normalized drag force exerted on the torus and the flux through the torusopening

The flux through the torus opening is characterized by the value ofconstantλ (56):

Q = −2π∫ b

0(uz− V) r dr = −πVλ (64)

The graph of the fluxQ is shown on the Fig. 2b. Whenξ0 approaches infinity,Q becomes equal to the flux of an unperturbed flow through a circle of radiusa. For a closed torus (ξ0 ∼ 0), flux Q is obviously zero.

Figures 3a and 3b display the patterns of streamlines and isobars of a vis-cous flow in the vicinity of a torus with radii ratiob/a = 0.3. The streamlinesare determined as the solutions to the equation

ψ = −r9 = const,

whereψ is the Stokes stream function [6]. However, from the numericalviewpoint it is better to construct the streamlines as the solutions to theequation

dr

ur=

dz

uz− V0


In the same way, the lines of constant pressurep = const can be obtainedfrom equation

dr∂ω∂r +

ωr

=dz∂ω∂z

which is derived using the generalized Cauchy-Riemann equations (52).

a. Streamlines of the Stokes flow about a torus b. Isobars in the Stokes flow about a torus

Figure 3. Streamlines and isobars of a viscous flow in the vicinity of a torus (b/a=0.3)

As the final step in solving the Stokes problem for a torus, we demonstratethat both general solutions (44) and (57) of the Stokes equations, though beingderived in different ways, are closely related. Moreover, this relationship canbe used to obtain an explicit solution of boundary-value problem (57), (59)without solving the tridiagonal system (61).

6.5. RELATIONSHIP BETWEEN SOLUTIONS(44) AND (57)

It is possible to construct an exact explicit solution of system (61) withoutresorting to the techniques presented in Section 4. Instead, one may employthe obtained solution of the boundary-value problem (46) for function9 tosolve equations (61).

First, observe that81 in (47) andφ1 in (57), as the solutions of equation11(·) = 0, can be represented by derivatives of some harmonic functions8

andφ:

81 =∂8

∂r, φ1 =

∂φ

∂r, 18 = 1φ = 0. (65)

Conversely, if∂8/∂r and∂φ/∂r satisfy equation11(·) = 0, then8 andφare harmonic, provided that all these functions vanish at infinity.

26 P. Krokhmal

Next, from the system (52) it follows thatϑ = p/µ andω may also beexpressed as the derivatives of some harmonic functionP:

p

µ=∂ P

∂z, ω =

∂ P

∂r, 1P = 0. (66)

Then equation (58) takes the form(

3+ r∂

∂r+ z

∂

∂z

)∂ P

∂z+(∂

∂r+

1

r

)∂φ

∂r+∂χ0

∂z= 0.

By virtue of equality∂ur

∂z−∂uz

∂r= ω =

∂ P

∂rwe obtain yet another condi-

tion on functionsP, φ, andχ0:(

3+ r∂

∂r+ z

∂

∂z

)∂ P

∂r−∂2φ

∂r ∂z+∂χ0

∂r= 0.

Let multiply the last two equations bydr anddzcorrespondingly and, takinginto account thatφ is harmonic, add them:

∂χ0

∂rdr +

∂χ0

∂zdz =

{−(

3+ r∂

∂r+ z

∂

∂z

)∂ P

∂r+

∂2φ

∂r ∂z

}dr

+{−(

3+ r∂

∂r+ z

∂

∂z

)∂ P

∂z+∂2φ

∂z2

}dz.

The obtained equality is nothing but the complete differential of functionχ0,which allows us to expressχ0 explicitly in terms of functionsP andφ:

χ0 = −2P − r∂ P

∂r− z

∂ P

∂z+∂φ

∂z+ const. (67)

To ensure the proper behavior of functions (67) at infinity, the constant ofintegration in (67) must be set to zero: const= 0. On inserting the expression(67) for functionχ0 in (57), the general solution (57)–(58) takes the form

ur =1

2

(r∂ P

∂z+∂φ

∂r

), uz =

1

2

(−2P − r

∂ P

∂r+∂φ

∂z

),

which by substitution

P = −2P, φ = −2∂8

∂z, (68)

reduces to the general solution (44), (47) with function9. From (65) and thesecond equality of (68) it follows that original functions81 andφ1 satisfy

φ1 = −2∂81

∂z.


Differentiating function81 with respect toz yields the relationship betweencoefficients ofφ1 and81:

Bn,0 =1

c

{(n+ 3/2)n+1− 2nn + (n− 3/2)n−1

}, n ≥ 1.

ReplacingBn,0 in the last equality withxn,0 similarly to (19), and expressingn by gn andhn using equations (50), we obtain the exact solution of system(61) in an explicit form:

xn,0 = −1

cλn,0

{(n− 3/2)(gn + hn)+ 3

n∑

j=0

(g j + h j )

}, n ≥ 1. (69)

Note that the form of solution (69) closely resembles the formula (28) (re-call that the finite sum in (69) can be transformed to the infinite sum usingrelationships (54) and (55)).

7. Conclusions

We have presented an exact analytical solution of the displacement boundary-value problem of elasticity for a torus. The original boundary-value problemultimately was reduced to a set of infinite system of algebraic equationswith tridiagonal matrices, which are the simplest equations that can arise ina vector boundary-value problem of elasticity in toroidal regions. The lastproposition rests on two observations: (i) a boundary-value problem for har-monic function in a toroidal region cannot reduce to an infinite system withless than three diagonals, unless the value of the sought function is prescribedat the whole boundary, and (ii) the displacement boundary-value problem ofelasticity cannot be reduced to pure Dirichlet problems for harmonic func-tions. Therefore, obtained tridiagonal infinite systems can be consideredasan exact solution of the original boundary-value problem in terms of thesolvability of basic problems of the potential theory in toroidal regions. Ontop of that, we presented two analytical techniques for solving tridiagonalsystems, with one technique being potentially applicable to systems withgreater number of diagonals.

Reduction of the original displacement boundary-value problem for atorus to the equations of the simplest form was possible due to the specialform of the derived general solution of the elastostatics equation. This generalsolution presents the vector of elastic displacement as a linear combinationof a vector and scalar harmonic functions. However, these functions are notindependent and have to satisfy to an additional differential constraint. Thepresence of the differential constraint to be satisfied by the harmonic func-tions makes the introduced general solution different from other types of

28 P. Krokhmal

general solution of the elastostatics equation. Although we have concentratedonly on the problem for a torus, the developed general approach is applicableto problems for other bodies described by cyclidal coordinates (lens-shapedbody, spindle-shaped body, and bi-spheres) and will produce similar results.

However, it has been shown that double-connectedness of the torus maylead to non-unique solutions of certain vector boundary-value problems,such as elasticity problems that involve the generalized Cauchy-Riemannequations. In Section 5 we have determined conditions that ensure single-valuedness of the solutions of the generalized Cauchy-Riemann equations,and have shown that these conditions are satisfied identically for the intro-duced form of the general solution.

In the considered example we have presented a complete solution of theStokes problem for a torus, using both the traditional stream function ap-proach and the developed elastic solution. It has been shown that, in contrastto the presented approach, the stream function approach does not yieldasingle-valued solution, and requires the obtained uniqueness conditions forthe generalized Cauchy-Riemann equations to be satisfied. Finally, we havedemonstrated a direct relation between both types of general solution for theStokes equations, and used this relation to derive an explicit solution for thetridiagonal system obtained by application of the elastic solution to the Stokesproblem for a torus.

Acknowledgements

The author is grateful to Prof. Andrei F. Ulitko for many useful discussionsand suggestions concerning this work.

References

1. P. M. Morse and H. Feshbach,Methods of Theoretical Physics.New York: McGraw-Hill (1953) 997 pp.

2. A. Wangerin, Ueber das Problem des Gleichgewichts elastischer Rotationskorper.Arch.der Math. und Phys.55 (1873) 113–146.

3. Yu. I. Soloviev, The Axisymmetric Problem of Elasticity for a Torus anda Space withToroidal Cave.Mekh. Tverd. Tela3 (1969) 99–105 [in Russian].

4. Yu. N. Podil’chuk and V. S. Kirilyuk, Nonaxially Symmetric Deformationof a Torus.Soviet Appl. Mech.19 (1983) 743–748.

5. V. S. Kirilyuk, Stress Concentration in an Isotropic Medium with an Elastic ToroidalInhomogeneity.Soviet Appl. Mech.24 (1988) 11–14.

6. J. Happel and H. Brenner,Low Reynolds Number Hydrodynamics.New York: PrenticeHall (1965) 556 pp.

7. W. H. Pell and L. E. Payne, On Stokes flow about a torus.Mathematika7 (1960) 78–92.8. S. J. Wakiya, On the Exact Solution of the Stokes Equations for a Torus. J. Phys. Soc.

Jap.37 (1974) 780–783.


9. N. Lebedev, The functions associated with a ring of oval cross-section. TechnicalPhysics1 (1937) 3–24.

10. E. W. Hobson,Spherical and Ellipsoidal Harmonics.Cambridge: University Press(1931) 500 pp.

11. Bateman, H., and A. Erdelyi.Higher Transcendental Funcitons. Vol. 1.New York:McGraw-Hill (1953) 282 pp.

12. G. V. Kutsenko and A. F. Ulitko, An Exact Solution of the Axisymmetric Problem ofthe Theory of Elasticity for a Hollow Ellipsoid of Revolution.Soviet Appl. Mech.11(1975) 3–8.

13. F. D. Gakhov,Boundary value problems.New York: Pergamon Press (1966) 564 pp.14. N. I. Muskhelishvili,Singular integral equations.Groningen: Noordhoff (1953) 447 pp.15. G. G. Stokes,Mathematical and physical papers. Vol. 1.Cambridge: University Press

(1880) 352 pp.16. Ghosh, S. On the steady motion of a viscous liquid due to translation of a tore parallel

to its axis.Bull. of the Calcutta Math. Soc.18 (1927) 185–194.17. H. Takagi, Slow Viscous Flow Due to the Motion of a Closed Torus.J. Phys. Soc. Jap.

35 (1973) 1225–1227.

Appendix. Exact solution of the boundary-value problem for function 9

Coefficientsg(1,2)0 andh(1,2)0 of (51) have the following form:

g(1)0 =2B∗

ϒ0

{2P1

2

(Q1

12− coshξ0 Q1

− 12

)

− coshξ0

(3P− 1

2Q1

12+ P1

− 12

Q 12

)− 3

cosh2 ξ0

sinhξ0

},

g(2)0 =2B∗

ϒ0

{−

2

3P1

2

(Q1

12+ 3 coshξ0 Q1

− 12

)

+ coshξ0

(P− 1

2Q1

12+ 3P1

− 12

Q 12

)− 3

cosh2 ξ0

sinhξ0

},

h(1)0 =2B∗

ϒ0

{1

sinhξ0− 2P1

2Q1

12+ 3 coshξ0

(P1

12

Q− 12+ P1

2Q1− 1

2

)},

h(2)0 =2B∗

ϒ0

{−

1

sinhξ0− 2P1

12

Q 12+ 3 coshξ0

(P1

12

Q− 12+ P1

2Q1− 1

2

)},

whereB∗ =√

2

πcV and

ϒ0 = 2P112

P12− 3 coshξ0

(P1

12

P− 12+ P1

2P1− 1

2

).

30 P. Krokhmal

Forn ≥ 1 the expressions forg(1,2)n andh(1,2)n are

g(1)n =B∗

nϒn

{−(n+ 1

2)(n+32)

sinhξ0+ Gn

}

h(1)n =B∗

nϒn

{−(n− 1

2)(n−12)

sinhξ0+ Hn

}

g(2)n =4B∗

3n(n− 12)(n−

32)ϒn

{−(n− 1

2)(n−32)

sinhξ0+ Gn

}

h(2)n =4B∗

3n(n+ 12)(n+

32)ϒn

{−(n+ 1

2)(n+32)

sinhξ0+ Hn

}

where

Gn = (n− 12)(n−

32)P

1n+ 1

2Qn− 3

2− (n+ 1

2)(n+32)Pn+ 1

2Q1

n− 32,

Hn = (n+ 12)(n+

32)P

1n− 3

2Qn+ 1

2− (n− 1

2)(n−32)Pn− 3

2Q1

n+ 12,

ϒn = (n+ 12)(n+

32)Pn+ 1

2P1

n− 32− (n− 1

2)(n−32)P

1n+ 1

2Pn− 3

2.

In the above formulas all Legendre functions are of argument coshξ0.

exact solution of the displacement boundary-value problem of elasticity...

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