## Abstract

Exact solutions to three-dimensional Stokes flow problems for asymmetric translation and rotation of two fused rigid spheres of equal size have been obtained in toroidal coordinates. The problems have been reduced to three-contour equations for meromorphic functions from a certain class, and then the latter have been reduced to Fredholm integral equations of the second kind by the Mehler–Fock transform of order 1. For the specified class of functions, the equivalence of the corresponding three-contour and Fredholm equations has been established in the framework of Riemann boundary-value problems for analytic functions. As an illustration for the obtained solutions, the pressure has been calculated at the surface of the body for both problems, and resisting force and torque, experienced by the body in asymmetric translation and rotation, have been computed as functions of a geometrical parameter of the body.

## 1. Introduction

Stokes flows about rigid particles have been and continue to be a popular subject of research in fluid mechanics (Happel & Brenner 1983). Since the work of Stokes (1880), who was the first to solve the problem of translation of a rigid sphere in a viscous incompressible fluid under zero Reynolds number assumption and to derive the drag force, exerted on the sphere, significant progress has been made in studying three-dimensional axially symmetric and asymmetric Stokes flows about rigid bodies of revolution. While the stream function approach is widely used to analytically solve the Stokes flow problem for axially symmetric translation of bodies of revolution, in particular of *prolate* and *oblate spheroids* (Oberbeck 1876; Payne & Pell 1960; Happel & Brenner 1983), *circular disk* (Oberbeck 1876; Happel & Brenner 1983), *spherical cap* (Payne & Pell 1960; Collins 1963; Ulitko 2002), *two spheres* (Stimson & Jeffery 1926), *torus* (Pell & Payne 1960*a*; Wakiya 1974), *spindle* (Pell & Payne 1960*b*; Zabarankin & Ulitko 2006*b*) and *lens* (Payne & Pell 1960; Zabarankin & Ulitko 2006*a*), a solution to Stokes equations in the form of Dean & O'Neill (1963) has proved to be efficient for exact solution of the problem for motion of a sphere parallel to a nearby plane wall (O'Neill & Stewartson 1967) and problems for asymmetric translation and rotation of *two spheres* (Wakiya 1967; O'Neill 1969; Nir & Acrivos 1973), *torus* (Takagi 1973; Wakiya 1974; Goren & O'Neill 1980) and *spindle* (Zabarankin 2007). In the axially symmetric case, the relationship between the stream function approach and the approach due to Dean and O'Neill was established in Krokhmal (2002). The reader interested in numerical methods for the Stokes model, in particular those based on integral equations, may refer to Youngren & Acrivos (1975).

Stokes flow problems for two spheres have attracted much of the attention, dedicated to this research subject, mainly due to their application to suspending liquids (e.g. Bartok & Mason 1957; Lin *et al.* 1970; Happel & Brenner 1983; Jeffrey & Onishi 1984). It should be mentioned that the problems for the two separate spheres are solved in bispherical coordinates (Stimson & Jeffery 1926; Wakiya 1967), while those for the two touching spheres are solved in tangent-sphere coordinates (Wakiya 1971; Nir & Acrivos 1973; Takagi 1974) by the aforementioned approaches. Stokes flow problems for two fused equal spheres (also known as biconvex lens; figure 1) complement the problems for separate and touching spheres and were addressed experimentally by Bartok & Mason (1957) in connection with the presence of fused particles (during manufacture) in suspending liquids. Payne & Pell (1960) solved the Stokes flow problem for axially symmetric translation of a lens (and, consequently, of the two fused spheres) by the stream function approach in toroidal coordinates (see also Zabarankin & Ulitko 2006*a*). To the best of our knowledge, exact solutions to Stokes flow problems for asymmetric translation and rotation of the two fused spheres, even of equal size, have not been obtained. This work bridges the gap.

From a mathematical perspective, the asymmetric Stokes flow problem for the two separate spheres (Wakiya 1967; O'Neill & Majumdar 1970) and also for the torus (Wakiya 1974; Goren & O'Neill 1980) reduces to the second-order difference equation,(1.1)with respect to unknown coefficients *X*_{k}, where *a*_{k}, *b*_{k}, *c*_{k} and *F*_{k} are known functions such that *a*_{k}→0, *b*_{k}→0, *c*_{k}→0 and *F*_{k}→0 as *k*→∞. Although in the work of Wakiya (1967, 1974) and Goren & O'Neill (1980), this equation was truncated at some large *k* and solved numerically, a closed-form solution to (1.1), also known as tridiagonal infinite system of algebraic equations, can be obtained by reducing it to successive solving of two bidiagonal systems (Krokhmal 2002).

The asymmetric problem for a rigid spindle (Zabarankin 2007) and the two fused equal spheres reduces to the so-called three-contour equation,(1.2)with respect to unknown meromorphic function *X*(*s*) of complex variable *s*, where *a*(*s*), *b*(*s*), *c*(*s*) and *F*(*s*) are the known functions. Equation (1.2) also arises from diffraction problems, and it is known that (1.2) has a closed-form solution when *a*(*s*), *b*(*s*) and *c*(*s*) are periodic functions with period 1 (Antipov & Silvestrov 2004*a*,*b*). In Zabarankin (2007), we reduced the asymmetric Stokes flow problem for the rigid spindle to (1.2) with *a*(*s*)=1 and *c*(*s*)=1, and then reduced the latter to a Fredholm integral equation of the second kind by complex Fourier transform, provided that *X*(*s*) is an analytic function in the strip |Re *s*|≤1. When *a*(*s*)=1 and *c*(*s*)=1, the three-contour equation (1.2) can also be reduced to a vectorial Riemann boundary-value problem for an analytic function (Zabarankin 1999). In this case, obtaining a closed-form solution depends on factorizing a matrix of the vectorial problem and is still an open issue. Notably, in the tangent-sphere coordinates, asymmetric Stokes flow problems for the two touching spheres reduce to a system of differential equations related to neither (1.1) nor (1.2) (Wakiya 1971; Nir & Acrivos 1973).

In this work, we reduce the Stokes flow problems for asymmetric translation and rotation of the two fused rigid spheres of equal size to the three-contour equation (1.2) with *a*(*s*)=*s*+3/2 and *c*(*s*)=*s*−3/2. To reduce the latter to a Fredholm integral equation of the second kind, we use the Mehler–Fock transform of order 1, which, in the case of *a*(*s*)=*s*+3/2 and *c*(*s*)=*s*−3/2, has advantage over the Fourier transform due to special properties of the associated Legendre functions. We establish the equivalence of the obtained integral equation and the three-contour equation in the framework of Riemann boundary-value problems for analytic functions. As an illustration for the obtained solutions, we calculate the pressure at the surface of the two fused spheres in asymmetric translation and rotation and also compute the resisting force (drag) and torque, experienced by the body, for different values of the body's geometrical parameter.

The rest of the paper is organized into four sections. Section 2 formulates the Stokes flow problems for asymmetric translation and rotation of the two fused spheres of equal size. Sections 3 and 4 derive Fredholm integral equations of the second kind for the asymmetric translation and rotation, and compute resisting force and torque for the body as functions of the geometrical parameter, respectively. Section 5 concludes the paper.

## 2. Problem formulation

The Stokes equations that describe the behaviour of a viscous incompressible fluid with zero (low) Reynolds number or the so-called *Stokes' (creeping) flows* are given by(2.1)where ** u** is the fluid velocity field;

*℘*is the pressure in the fluid;

*μ*is the shear viscosity and Δ

**≡grad(div**

*u***)−curl(curl**

*u***). In (2.1), the first equation is known as Stokes' creeping flow equation, and the second is the equation of continuity (e.g. Lamb 1945; Happel & Brenner 1983).**

*u*Let *S* determine the surface of the two fused rigid spheres of equal size, and let (*x*, *y*, *z*) be a system of Cartesian coordinates with the basis (** i**,

**,**

*j***), in which the**

*k**z*-axis coincides with the axis of revolution of

*S*(figure 1

*c*). Without loss of generality, studying Stokes flows, generated by asymmetric motion of a rigid body of revolution in a quiescent fluid, can be decomposed into two problems: (i) translation of the body along the

*x*-axis with the constant velocity

*v*

_{x}and (ii) rotation of the body around the

*y*-axis with the constant angular velocity .

In the problems (i) and (ii), the *no-slip* boundary conditions for the velocity vector ** u** on body's surface

*S*are determined by(2.2)(2.3)and in both the problems, the velocity

**and pressure**

*u**℘*vanish at infinity(2.4)The problem of translation is closely related to the one for the body immersed in a uniform Stokes flow with the velocity constant at infinity: (e.g. Happel & Brenner 1983). In the latter, solves (2.1) and satisfies the boundary condition . Obviously, the vectors

**and are related by , and thus the discussed problems are equivalent.**

*u*In general, a solution to Stokes' creeping flow equation can be represented in the form of Dean & O'Neill (1963)(2.5)where is the radius vector; is an arbitrary harmonic vector, i.e. ; and the functions *℘* and must satisfy div ** u**=0, which in terms of

*℘*and , takes the form(2.6)

Let (*r*, *φ*, *z*) be a system of cylindrical coordinates with the basis (*e*_{r}, *e*_{φ}, ** k**), in which the

*z*-axis coincides with the

*z*-axis in the Cartesian coordinates. For the components, (

*u*

_{r},

*u*

_{φ},

*u*

_{z}), of the velocity vector in the cylindrical coordinates, the boundary conditions (2.2) and (2.3) are reformulated as(2.7)(2.8)It follows from (2.7) and (2.8) that the vector

**has only first harmonic with respect to the angular coordinate**

*u**φ*and, consequently, for both boundary conditions (2.7) and (2.8), the representation (2.5) reduces to(2.9)where the pressure

*℘*is determined by(2.10)and, consequently, equation (2.6) takes the form(2.11)The functions

*Θ*,

*ϒ*,

*Φ*and

*Ψ*satisfy the so-called

*k*-harmonic equations(2.12)where

The surface, *S*, of the two fused equal spheres is formed by rotating two symmetric circle arcs around the *z*-axis. Let (*ξ,η,φ*) be toroidal coordinates, in which the angular coordinate *φ*∈[0,2*π*] coincides with the one in (*r,φ,z*), and *ξ* and *η* are related to *r* and *z* by(2.13)where *c* is a metric parameter of the toroidal coordinates. In the meridional cross-section *rz*-plane, the arcs are determined by *η*=*η*_{0} and *η*=−*η*_{0} (figure 2), and the radius, *a*, of a fused sphere is given by *a*=*c*/sin *η*_{0}. For *η*_{0}=*π*/2, the surface of the two fused spheres reduces to a single sphere with *a*=*c*, and the case *η*_{0}=0 corresponds to two touching equal spheres with infinite radii.

In the toroidal coordinates, the functions *Θ*, *ϒ*, *Φ* and *Ψ*, satisfying (2.12) for the region *η*∈[−*η*_{0},*η*_{0}], are represented by Mehler–Fock integrals(2.14)(2.15)(2.16)(2.17)where is the associated Legendre function of the first kind (also known as a toroidal function) behaving as at |*τ*|→∞. For *m*=0, the superscript is omitted. In (2.14)–(2.17), the functions cos[*ηs*] and sin[*ηs*] in curly brackets mean that we use one of them depending on whether *Θ*, *ϒ*, *Φ* and *Ψ* are even or odd with respect to the coordinate *η*.

The functions *A*(*s*), *B*(*s*), *C*(*s*) and *D*(*s*) are assumed to be meromorphic in the strip |Re *s*|≤1 in which they should vanish as at |Im*s*|→∞, where *γ*>*η*_{0}.

For *Θ*, *ϒ*, *Φ* and *Ψ*, determined by (2.14)–(2.17), respectively, the corresponding derivatives in (2.11) are derived similarly to formulae (2.1) and (2.2) in the work of Zabarankin & Ulitko (2006*a*) and, thus, equation (2.11) reduces to(2.18)provided that(2.19)The conditions (2.19) follow from the assumption that *A*(*s*), *B*(*s*), *C*(*s*) and *D*(*s*) may admit simple poles in the strip |Re *s*|≤1.

Consequently, the Stokes flow problems for the asymmetric translation and rotation of the two fused equal spheres reduce to solving equation (2.18) subject to (2.19) for the boundary conditions (2.7) and (2.8), respectively. It will be shown that the conditions (2.19) hold true for *η*_{0}≤*π*/2. In §§3 and 4, problems (2.18), (2.7) and (2.18), (2.8) for *η*_{0}≤*π*/2 will be reduced to the three-contour equation (1.2) with *a*(*s*)=*s*+3/2 and *c*(*s*)=*s*−3/2, and then the latter will be reduced to a Fredholm integral equation of the second kind by the Mehler–Fock transform of order 1.

## 3. Asymmetric translation

In this section, we solve the Stokes flow problem for the asymmetric translation of the two fused equal spheres along the *x*-axis and calculate the resisting force (drag), experienced by the body.

For the functions *Θ*, *ϒ*, *Φ* and *Ψ*, the boundary conditions (2.7) can be reformulated as(3.1)whence it follows that *Θ*, *ϒ* and *Φ* are even functions with respect to *η*, and *Ψ* is the odd function of *η* and, consequently, we use cos[*η**s*] in (2.14)–(2.16) and sin[*ηs*] in (2.17).

In the further analysis, we will make use of the Mehler–Fock transform of general order *m* and the corresponding inversion formula (see Nasim (1984, p. 175) formulae (3.1) and (3.2) and also Sneddon (1972, p. 416)),(3.2)where *m* is a non-negative integer, |*σ*|<*m*+1/2, and for *m*≥1 the function is defined by

We introduce the following classes of functions.

and are the classes of meromorphic functions that admit simple poles

*s*=±1/2 and*s*=1/2 in the strips −1≤Re*s*≤1 and 0≤Re*s*≤1, respectively, and vanish as at |*s*|→∞ in the corresponding domains, where*γ*≥*η*_{0}.is the subclass of such that functions from this subclass have simple zero

*s*=0.and are the subclasses of and , respectively, such that functions from these subclasses have no poles at

*s*=±1/2 and*s*=1/2, i.e. analytic in −1≤Re*s*≤1 and 0≤Re*s*≤1, respectively.

Let a new function *X*(*s*) be related to *Θ*(*ξ,η*_{0}) by the Mehler–Fock transform of order 1(3.3)(3.4)We assume and will show that this assumption holds true for *η*_{0}≤*π*/2.

The function *A*(*s*) can be expressed from (2.14) via *X*(*s*) by the Mehler–Fock transform of order 1(3.5)where we used the relationship(3.6)

Substituting representations (2.15), (2.16) and (2.17) into (3.1) and then applying the Mehler–Fock transform of orders 0, 2 and 1 to the three equations in (3.1), respectively, we obtain(3.7)(3.8)(3.9)where in obtaining the last term in the right-hand side in (3.7), we used the representation (Sneddon 1972)

Substituting (3.5) and (3.7)–(3.9) into (2.18) subject to (2.19), we obtain the three-contour equation for *Y*(*s*)(3.10)where(3.11)(3.12)and(3.13)

In terms of *X*(*s*), the conditions (2.19) are reformulated as(3.14)(3.15)In (3.14) and (3.15), the functions and have simple zeros *s*=1/2 and *s*=−1/2, respectively, which compensate corresponding poles of *X*(*s*) and 1/cos[*πs*] within the strips 0≤Re *s*≤1 and −1≤Re *s*≤0. It is seen from (3.14) and (3.15) that for *η*_{0}≤*π*/2, *X*(*s*) cannot admit poles except those at *s*=±1/2.

Conditions for the function *Y*(*s*) follow from (3.11). For *η*_{0}≤*π*/2, the multiplier sin[2*η*_{0}*s*]−*s*sin[2*η*_{0}] in the numerator in (3.11) has only ‘generic’ zero *s*=0 in the strip |Re *s*|≤1, which is compensated by the zero *s*=0 of sin[*η*_{0}*s*]. Consequently, we conclude that *Y*(*s*), defined by (3.11), is a meromorphic function from the class , provided that .

The three-contour equation (3.10) can be reduced to a Fredholm integral equation of the second kind for the following function:(3.16)

First, we show that (3.16) establishes one-to-one correspondence between the functions *H*(*s*) and . This statement is equivalent to the fact that equation (3.16) has only zero homogeneous solution with respect to .

*The solution to the homogeneous three-contour equation* *(3.16)*, *i.e. when H*(*s*)=0, *is Y*(*s*)=*C*/cos[*πs*] *in the class* , *where C is a constant, and is the zero function in the class* .

To prove the proposition, we use the approach of Riemann boundary-value problems for analytic functions, suggested in the work of Zabarankin & Ulitko (2006*b*).

For a new function *Z*(*s*), introduced by(3.17a)(3.17b)the homogeneous equation (3.16) is reformulated as(3.18)It follows from (3.17*a*) that in both cases and .

To solve (3.18), we conformally map the complex plane *s* with the strip 0≤Re*s*≤1 into the complex plane *w* with the branch cut along the segment [−1,1] bysee figure 3. The lines *s*=i*τ*, and *s*=1+i*τ*, correspond to the upper and lower banks of the branch cut with the anticlockwise orientation as shown in figure 3. The pole at *s*=1/2 in the complex plane *s* corresponds to the pole at infinity in the complex plane *w*, and infinite points of the strip |Re *s*|≤1, i.e. |*s*|→∞, correspond to the points *w*=±1.

In the complex plane *w*, the function *Z*(*s*) becomes such that(3.19)where and are the boundary values of at the upper and lower banks of the branch cut.

The problem (3.18) reduces to a homogeneous Riemann boundary-value problem for finding the analytic function in the complex plane *w* with the branch cut along the segment [−1,1]:(3.20)where in addition, has a simple pole at infinity, i.e. grows as at |*w*|→∞, and vanishes at *w*=±1. It is known that the analytic function, satisfying (3.20) and vanishing at the endpoints *w*=±1 of the non-closed contour [−1,1], can be represented in the form (Gakhov 1966)(3.21)where is a polynomial of degree *n* to be determined from the behaviour of at |*w*|→∞. If is bounded at infinity, then obviously , and is the zero function. In our case, since has a simple pole at infinity, (3.21) reduces to , where is an arbitrary complex constant and, consequently,whence

Thus, multiplying (3.17*b*) by (*s*+1/2), we obtain(3.22)where *Y*_{1}(*s*)=(*s*^{2}−1/4)*Y*(*s*). Note that and, consequently, a homogeneous solution to (3.22) is the zero function. Indeed, to solve the homogeneous equation (3.22), we repeat arguments same as for (3.18) and obtain (3.21) for a corresponding . However, since *Y*_{1}(*s*) is analytic in the strip 0≤Re*s*≤1, the function has no pole at infinity in the complex plane *w*, i.e. it is bounded at |*w*|→∞ and, consequently, it follows from (3.21) that is the zero function. Thus, (3.22) has the unique non-homogeneous solution and, consequently, the homogeneous solution to (3.16) in the class is given by . Obviously, if , then , and the proposition is proved. ▪

Thus, the non-homogeneous solution to the three-contour equation (3.16) with respect to *Y*(*s*) in the class is unique and given in theorem 3.2.

*Given H(s), the function* *is determined from* *(3.16)* *by the regular integral*(3.23)

Applying the Mehler–Fock transform of order 1 to (3.16), we obtain(3.24)where we used the fact that for Indeed, the poles *s*=±1/2 of *Y*(*s*) in −1≤Re*s*≤0 and 0≤Re*s*≤1 are compensated by the zeros *s*=−1/2 and *s*=1/2 of and , respectively.

With (3.6), equation (3.24) reduces towhence the function *Y*(*s*) is found by inverting the Mehler–Fock transform of order 1 (see (3.2))(3.25)whereUsing the representation (Bateman & Erdelyi 1953)we obtain(3.26)where and , and for obtaining the double integral in (3.26), we calculatedThe formula (3.23) follows from (3.25) and (3.26). ▪

Consequently, based on proposition 3.1 and theorem 3.2, the three-contour equation (3.10) can be equivalently reduced to the Fredholm integral equation of the second kind for the function *H*(*s*)(3.27)Re s=0.

It follows from (3.23) and (3.27) that , and thus from (3.11), we obtain(3.28)It is seen from (3.12) and (3.13) that and as *s*→0, and and as |*s*|→∞ at Re *s*=0. Thus, based on (3.27) and (3.28), we conclude that *X*(0)=0 and as |*s*|→∞ at Re*s*=0.

Consequently, the function is determined by (3.4) with (3.28).

In the case of sphere, i.e. *η*_{0}=*π*/2, we have(3.29)and

The pressure *℘* and function *Θ* are related by (2.10). Figure 4 shows as a function of *ξ* for *η*_{0}=*π*/6, *π*/3 and *π*/2. Figure 6*a* illustrates the épure of the normalized pressure, , at the surface of the two fused spheres in the *xz*-half-plane,1 i.e. when *φ*=0, for *η*_{0}=*π*/3.

In Zabarankin (2007), we showed that for the velocity field (2.9) and boundary conditions (2.7), the resisting force (drag), experienced by a rigid body of revolution, is determined by(3.30)For the two fused equal spheres, the function *Θ* is represented by (2.14) with (3.5) and, consequently, (3.30) reduces to(3.31)where the second term in the right-hand side vanishes, since for *η*_{0}<*π*/2, the function (*s*^{2}−1/4)*X*(*s*) is analytic in the strip |Re *s*|≤1, and cos[*η*_{0}*s*] has no zeros in |Re *s*|≤1. For *η*_{0}=*π*/2, *X*(*s*) is determined by (3.29), for which the second term also vanishes.

In the case *η*_{0}→0, corresponding to two touching equal spheres with infinite radii (the radius of a sphere is equal to *c*/sin *η*_{0}), the ratio of the drag force *F*_{x} to the radius of a sphere is finite and given by . This result is obtained by solving the Stokes flow problem for the asymmetric translation of two touching spheres of finite radii in tangent-sphere coordinates (see Jeffrey & Onishi 1984).

In the case of the *axially symmetric* translation of the two fused equal spheres along the *z*-axis with the constant velocity *v*_{z}, where the *z*-axis of the cylindrical coordinates (*r*, *φ*, *z*) coincides with body's axis of revolution, the vector ** u** solves (2.1), (2.4) and the boundary condition . In this problem,

**is independent of the angular coordinate**

*u**φ*and can be represented in terms of a stream function (Pell & Payne 1960

*b*; Zabarankin & Ulitko 2006

*a*). In this case, the resisting force has the component in the direction

**only and is computed in explicit form (Zabarankin & Ulitko 2006**

*k**a*)(3.32)

In particular, the normalized resisting force, , exerted on the two touching equal spheres in the axially symmetric translation, is given bywhere *a* is the radius of a touching sphere (see also Happel & Brenner 1983).

Table 1 compares the normalized resisting forces and for the two fused equal spheres for the axially symmetric and asymmetric translations, respectively, where *c*/sin *η*_{0} is the radius of a fused sphere.

## 4. Asymmetric rotation

In this section, we solve the Stokes flow problem for the asymmetric rotation of the two fused equal spheres around the *y*-axis and calculate the resisting torque, experienced by the body.

The boundary conditions (2.8) can be reformulated for the functions *Θ*, *ϒ*, *Ψ* and *Φ* as(4.1)whence it follows that *Θ*, *ϒ* and *Φ* are odd functions with respect to *η*, and *Ψ* is an even function of *η*. Consequently, representing the functions *Θ*, *ϒ*, *Φ* and *Ψ* by Mehler–Fock integrals (2.14)–(2.17), respectively, we use sin[*ηs*] in (2.14)–(2.16) and −cos[*ηs*] in (2.17).

Let *X*(*s*) be introduced by (3.3) and assumed to be from the class , i.e. a meromorphic function admitting only simple poles *s*=±1/2 in the strip |Re *s*|≤1 and having simple zero *s*=0. Then, the function *A*(*s*) can be expressed via *X*(*s*) from (2.14) similarly to formula (3.5)(4.2)

As in the case of the asymmetric translation, the functions *B*(*s*), *C*(*s*) and *D*(*s*) can be represented via *X*(*s*) by substituting (2.15), (2.16) and (2.17) into the boundary conditions (4.1) and applying the Mehler–Fock transform of orders 0, 2 and 1, respectively,(4.3)(4.4)(4.5)where in obtaining the last terms in the right-hand sides in (4.3) and (4.5), we used the representations

With formulae (4.2)–(4.5), equation (2.18) subject to (2.19) reduces to (3.10) with the following *Y*(*s*), *K*(*s*) and :(4.6)(4.7)and(4.8)

In terms of *X*(*s*), the conditions (2.19) take the form(4.9)(4.10)It follows from (4.9) and (4.10) that for *η*_{0}<*π*/2, simple poles *s*=±1/2 of *X*(*s*) in |Re *s*|≤1 are compensated by the corresponding zeros of and . For *η*_{0}=*π*/2, (4.8) reduces to the zero function and, consequently, in this case, *X*(*s*)≡0.

Conditions for the function *Y*(*s*) follow from (4.6). For *η*_{0}<*π*/2, the multiplier sin[2*η*_{0}*s*]+*s* sin[2*η*_{0}] in the numerator in (4.6) has only ‘generic’ zero *s*=0 in the strip |Re *s*|≤1, which is compensated by the zero *s*=0 of sin[*η*_{0}*s*]. Consequently, we conclude that *Y*(*s*), defined by (4.6), is a meromorphic function from , provided that .

For *η*_{0}>*π*/2, the function sin[2*η*_{0}*s*]+*s* sin[2*η*_{0}] has zeros in the strip |Re *s*|≤1. This means that in order for *X*(*s*) to satisfy (4.9) and (4.10), *Y*(*s*) must have zeros coinciding with those for sin[2*η*_{0}*s*]+*s* sin[2*η*_{0}] in |Re *s*|≤1. Interestingly, the expression sin[2*η*_{0}*s*]+*s* sin[2*η*_{0}] defines the determinant in the Stokes flow problem for the axially symmetric translation of the two fused rigid spheres of equal size (Zabarankin & Ulitko 2006*a*).

Similarly to the case of the asymmetric translation, we introduce the function *H*(*s*) by (3.16). Since *Y*(*s*) belongs to the class , it can be represented at Re *s*=0 via *H*(*s*) by (3.23) and, consequently, the three-contour equation (3.10) reduces to the Fredholm integral equation of the second kind (3.27). In this case, the function *X*(*s*) is expressed via *H*(*s*) similarly to (3.28)(4.11)Thus, the function *Θ* is determined by (3.4) with (4.11).

It follows from (4.7) and (4.8) that and as *s*→0, and and as |*s*|→∞ at Re *s*=0. Consequently, from (3.27) and (4.11), we obtain *X*(0)=0 and as |*s*|→∞ at Re *s*=0.

In the case of sphere, i.e. *η*_{0}=*π*/2, we have and, consequently, *X*_{sphere}(*s*)≡0 and *Θ*_{sphere}≡0, and the functions *ϒ*_{sphere}, *Φ*_{sphere} and *Ψ*_{sphere} take the form

As in the problem of the asymmetric translation, the pressure *℘* and function *Θ* are related by (2.10). Figure 5 illustrates as a function of *ξ* for *η*_{0}=*π*/6, 5*π*/12 and *π*/3. Figure 6*b* shows the épure of the normalized pressure, , at the surface of the two fused spheres in the *xz*-half-plane, i.e. when *φ*=0, for *η*_{0}=*π*/3.

In Zabarankin (2007), we showed that for the velocity field (2.9) and boundary conditions (2.8), the resisting torque, experienced by a rigid body of revolution, is determined by(4.12)For the two fused equal spheres, the function *Ψ* is represented by (2.17) with (4.5) and, consequently, (4.12) reduces towhere

The Stokes flow problem for the asymmetric rotation of the two touching equal spheres with finite radii is solved in tangent-sphere coordinates (see Wakiya 1971). The resisting torque, experienced by the body in this case, is determined by , where *a* is the radius of a touching sphere.

In the case of the *axially symmetric* rotation of a body of revolution around the *z*-axis with the angular velocity , the vector ** u** that solves (2.1) and (2.4) can be represented in the form , where the

*z*-axis of the cylindrical coordinates (

*r,φ,z*) coincides with body's axis of revolution. Then, the function satisfies and the boundary condition (Kanwal 1961). In this case, the resisting torque has the component in the direction

**only, and for an arbitrary body of revolution,**

*k**T*

_{z}takes the formwhich for the two fused equal spheres reduces to .

In particular, the normalized resisting torque, exerted on the two touching equal spheres in the axially symmetric rotation, is given bywhere *ζ*(.) is the zeta function, and *a* is the radius of a touching sphere (Takagi 1974).

Table 1 compares normalized torques and for the two fused equal spheres for the axially symmetric and asymmetric rotations, respectively, where is the radius of a fused sphere.

## 5. Conclusions

We have obtained exact solutions to the three-dimensional Stokes flow problems for asymmetric translation and rotation of the two fused rigid spheres of equal size. Representing the velocity field in the form of Dean and O'Neill and expressing *k*-harmonic functions by Mehler–Fock integrals in toroidal coordinates, we have reduced both problems to the three-contour equation (3.10) for the meromorphic function *Y*(*s*), related to the density in the Mehler–Fock integral that determines the pressure. The equation has the coefficients *s*+3/2 and *s*−3/2 at *Y*(*s*+1) and *Y*(*s*−1), respectively, and holds under the condition *η*_{0}≤*π*/2 that guarantees *Y*(*s*) to have no poles in the strip |Re *μ*|≤1 except *s*=±1/2. This condition follows from the fact that for *η*_{0}≤*π*/2, the functions cos[*η*_{0}*s*], sin[2*η*_{0}*s*]−*s* sin[2*η*_{0}] and sin[2*η*_{0}*s*]+*s* sin[2*η*_{0}] in (3.11) and (4.6) have no zeros in |Re *μ*|≤1.

Then, we have reduced equation (3.10) to the Fredholm integral equation of the second kind (3.27) by the Mehler–Fock transform of order 1 and have established the equivalence of the three-contour and Fredholm equations for in the framework of Riemann boundary-value problems for analytic functions. In the case of sphere, i.e. *η*_{0}=*π*/2, the Fredholm equation has a closed-form solution for the asymmetric translation and has zero solution for the asymmetric rotation.

When *η*_{0}>*π*/2, the functions cos[*η*_{0}*s*] and sin[2*η*_{0}*s*]+*s* sin[2*η*_{0}] have simple zeros in |Re *s*|≤1 (the function sin[2*η*_{0}*s*]−*s* sin[2*η*_{0}] has zeros in |Re *s*|≤1 for *η*_{0}≥2.247). Consequently, to satisfy ‘analyticity’ conditions (3.14), (3.15) and (4.9), (4.10), the function *Y*(*s*) must have zeros coinciding with those for the aforementioned functions. Solving the three-contour equation (3.10) under these conditions is still an open issue.

As an illustration for the obtained solutions to the Stokes flow problems, we have computed the resisting force and torque for the asymmetric translation and rotation of the two fused equal spheres for various values of *η*_{0}(*η*_{0}≤*π*/2), and compared them with those for the axially symmetric translation and rotation of the body, respectively. As expected, the resisting force and torque for the asymmetric motions are greater than those for the corresponding axially symmetric motions. Also, the obtained resisting force and torque for the two fused equal spheres are in accordance with those for the two touching equal spheres (see table 1). The results of this work could be used for studying properties of suspending liquids with fused particles.

## Acknowledgments

We are grateful to the anonymous referees for their valuable comments and suggestions, which helped to improve the quality of the paper.

## Footnotes

↵In the case of the asymmetric translation along the

*x*-axis, the épure is symmetric with respect to the*x*-axis; however, because for*η*_{0}=*π*/3, the parts of the épure for*z*≥0 and*z*≤0 overlap near the*x*-axis, figure 6*a*shows only the part for*z*≥0.- Received March 13, 2007.
- Accepted May 16, 2007.

- © 2007 The Royal Society

## References

## Notice of correction

The third line of the final paragraph is now in its correct form.

A detailed erratum will appear at the end of volume. 31 July 2007