# Resistance Functions for Two Spheres in Axisymmetric Flow—Part I: Stream Function Theory

Authors: Thanaa El Naqeeb; Rudi Schmitz
Journal: Journal of Applied Mathematics
 (2011)

---

## Abstract

We consider low-Reynolds-number axisymmetric flow about two spheres using a novel, biharmonic stream function. This enables us to calculate analytically not only the forces, but also the dipole moments (stresslets and pressure moments) and the associated resistance functions. In this paper the basics properties of axisymmetric flow and the stream function are discussed. Explicit series expansions, obtained by separation in bispherical coordinates, will be presented in a follow-up paper.


---

## Body

## 1. Introduction

The grand resistance and mobility tensors describe the hydrodynamic interaction between rigid bodies suspended in a fluid medium and play an all-important role in colloidal science [1–7]. More specifically, they express the linear relationship between the Cartesian force multipole moments exerted by the particles on the fluid and the gradients of the ambient flow velocity taken at the particle centers. Both tensors depend, in general, on the positions and orientations of all the suspended particles. However, in the special case of just two spherical bodies, owing to O(2)-symmetry about the line connecting the particle centers, the full tensors can be reduced to a set of scalar resistance and mobility functions [8–15]. Accurate knowledge of the two-body resistance functions is essential to overcome certain contact singularities—also referred to as lubrication singularities—that dominate the many-body tensors when a pair of particles comes close to touching [16, 17].The calculation of the 2-body resistance functions is based on the solution of the Stokes boundary value problem for stationary, low-Reynolds number flow about two spheres. In this case there exists a set of curvilinear coordinates that is adapted to the physical boundaries at hand—the bispherical coordinates—and it seems natural to try to solve the problem analytically by separation in these coordinates [18–21]. However, separation of the Stokes equations in bispherical coordinates succeeds only up to a modulation factor [22], which complicates matching of the general solution to the boundary data. In most cases, it turns out that the matching problem can be reduced to a linear recursion problem that involves an infinite tridiagonal matrix. Although this cannot, in general, be solved analytically, the numerical treatment of such a tridiagonal recursion scheme [23–28] is much more efficient than recursions based on the twin-multipole expansion [9, 12–14].Further progress can be made if the flow is axisymmetric, which happens when the spheres translate along or rotate around the line connecting their centers and when the ambient flow is axially symmetric about that line. A subset of resistance functions, including those with the strongest contact singularities, is determined by this kind of flow problems [13, 15]. It follows from the Stokes equations that for axisymmetric flow, the azimuthal velocity component is decoupled from the remaining flow variables, namely, the pressure and the radial and axial velocity components. Accordingly, there are two types of axisymmetric flow problems, problem I involving only the azimuthal velocity and problem II involving the other flow variables. Problem I can be reduced to a harmonic equation for the azimuthal velocity with Dirichlet boundary conditions. This was solved by separation in bispherical coordinates almost hundred years ago by Jeffery [29], who calculated, in particular, the torques acting on two spheres that rotate with given angular velocities around the line of centers.Problem II is more difficult, since it involves coupled equations among three variables. However, owing to the incompressibility, the two nonvanishing velocity components, that is, radial and axial velocities, can be expressed as derivatives of a scalar potential, the so-called stream function, whose introduction can be traced back to Stokes [30]. This stream function satisfies a fourth-order differential equation that can again be separated in bispherical coordinates. In this way, Stimson and Jeffrey calculated the forces on two spheres moving with equal velocities along the line of centers [31]. Many years later, Brenner [32] and Maude [33] solved the same problem for spheres moving with opposite velocities along the line of centers. The latter case, where two spheres approach each other, is the one with the strongest contact singularity. It is possible to extract the asymptotic behavior of the forces at close proximity from the series expansion in bispherical coordinates [34, 35], and the results agree with those obtained from a singular perturbation theory in stretched cylindrical coordinates [36, 37]. Singular perturbation theory, valid for two spheres near touching, has also been used to calculate the forces and their dipole moments in the presence of an external, axisymmetric rate-of-strain flow [14, 38, 39]. Moreover, the twin-multipole expansions, valid for large distances, are well-known for these cases [12–14]. Yet, the corresponding series expansions in bispherical coordinates, valid for all distances, have not been published so far.In this and a follow-up paper [40], we generalize and complete previous studies of axisymmetric two-sphere-flow problems in several ways. First, we suggest a slight conceptual improvement by considering from the beginning an inhomogeneous external flow, as generated, for example, by means of a localized external volume-force density. This allows us to identify easily the contributions from each sphere without being obliged to assume artificial deformations, as in [13, 38, 41]. Second, and more important, we introduce a novel stream function that differs from the classical one [30–33, 37] but has the advantage of being a biharmonic function. Its series expansion in bispherical coordinates matches the corresponding expansions of the flow fields better than the old stream function. In particular, the biharmonic stream function is expressed in associated Legendre functions instead of Gegenbauer polynomials. Third, with the new stream function, we are able to calculate analytically not only the forces, but also their dipole moments (stresslets and pressure moments). And we will show that this can be accomplished without full knowledge of the pressure (which would require solving a tridiagonal recursion scheme). This signifies some progress, as it was claimed earlier in the literature [38] that the stresslets cannot be calculated on the basis of the stream function alone. In this paper, we present the theory of the biharmonic stream function for axisymmetric flow, while in the follow-up paper, we derive the series expansions for the forces and dipole moments in bispherical coordinates.


## 2. Stokes Equations

We consider two spheres immersed in an unbounded, simple fluid with shear viscosityη. The spheres—in the following labeled by Greek indices ν,μ∈{1,2}—have radii aν and are centered on the 3-axis of a Cartesian frame at positions Xν=Zνe3. To exclude overlap, we assume that |Z1-Z2|>a1+a2. The dynamics of the fluid is described in terms of the local deviation from the thermal pressure, p(x), and the flow velocity u(x) which obey the linear Stokes equations for stationary, low-Reynolds number flow with stick boundary conditions [8, 42]
(2.1)∇⋅u=0∇p-ηΔu=fext(x∈B0),(p,u)⟶0(‖x‖⟶∞),u=Uν+Ων×(x-Xν)(x∈∂Bν,ν=1,2).
Here, B0 denotes the fluid region and ∂Bν the surface of particle ν. Moreover, fext is an external force density acting on the fluid, while Uν and Ων are the translational and rotational velocities of particle ν, respectively. We assume fext to have a compact support that is not overlapping with the particles. Equations (2.1) then pose a Dirichlet boundary value problem for (p,u) whose solution—for fixed geometry and given sources (fext,U,Ω)—is uniquely determined [8, 43].We are interested here in the forces and dipole moments exerted by the particles on the fluid. These are defined by (we use Latin indicesi,j,…∈{1,2,3} to label Cartesian components and the summation convention over repeated upper and lower Latin indices)(2.2)Fiμ=∫∂BμdSFiμ(x),Fijμ=aμ∫∂BμdSFiμ(x)Nj(x),(μ=1,2),
where the surface-force density [8, 42](2.3)Fμ=N⋅[p1-2η(∇u)]∂Bμ+(x∈∂Bμ)
is determined by the solution of (2.1). Here, N is the normal field on ∂Bμ (directed outwards), (∇u) is the symmetrized velocity gradient, and the subscript ∂Bμ+ indicates analytic continuation from the fluid regime onto the particle surface. It is useful to split the dipole moments into a trace, a skew-symmetric, and a symmetric-traceless part, according to(2.4)Fijμ=Qμδij-12εijTmμm+Sijμ,
where (for an arbitrary second-rank tensor with components Bij, we use the notations B(ij)=(1/2)(Bij+Bji),B[ij]=(1/2)(Bij-Bji),B〈ij〉=B(ij)-(1/3)δklBklδij)(2.5)Qμ=13δmnFmnμTiμ=-εimnF[mn]μijμ=F〈ij〉μ
denote the pressure moment [14], the torque, and the (deviatoric) stresslet [11, 44], respectively.We decompose the flow(p,u) in the form(2.6)p(x)=pext(x)+q(x),u(x)=uext(x)+v(x),(x∈B0),
where (pext,uext) is the flow caused by fext in the absence of the particles and called the external flow. The remainder (q,v) will be referred to as the excess flow. It satisfies the Dirichlet problem(2.7)∇⋅v=0,∇q-ηΔv=0,(x∈B0),(q,v)⟶0,(‖x‖⟶∞),v=uν,(x∈∂Bν,ν=1,2),
with applied velocities(2.8)uν(x)=Uν+Ων×(x-Xν)-uext(x),(x∈∂Bν,ν=1,2).
Inserting (2.6) in (2.3), one obtains a decomposition of the surface-force densities of the form(2.9)Fμ=Fextμ+Gμ,Gμ=N⋅[q1-2η(∇v)]∂Bμ+,
where Fextμ is defined as in (2.3), with  (p,u) replaced by (pext,uext). Notice that the external flow is well defined inside the particles, where it obeys the homogeneous Stokes equations (since the support of fext lies outside the particles). The contribution from Fextμ to the force moments (2.2) can therefore be evaluated by converting the surface integrals into volume integrals over the interiors Bμ of the particles. Expanding then (pext,uext) about the particle centers,  Xμ, and using isotropy, it is easy to calculate these integrals exactly.Assuming that the external flow varies slowly on the length scale of the particle diameters, we expand the applied velocities (2.8) up to first order about the particle centers. This yields(2.10)uνk(x)=Uνk+aν[-εklWνnn+Eνkl]Nl(x)+⋯(x∈∂Bν,ν=1,2),
where the expansion coefficients(2.11)Uνk=Uνk-uextk|XνWνk=Ωνk-12εmnk∂[muextn]|XνEνkl=-∂(luextk)|Xν
are referred to as the (effective) translational and rotational velocities and the local rate of strain, respectively. We also put(2.12)Pν=-pext|Xν.
It follows from incompressibility (cf. the first equation in (2.1)) that the local rates of strain are traceless,  δmnℰνmn=0. Notice that by admitting inhomogeneous external flows, we can avoid conceptual problems of earlier approaches which were based on a uniform shear flow and which, therefore, either could not always resolve the individual contributions of the spheres [11], or, in order to be able to do so, had to assume artificial deformations [13, 38, 41].Upon inserting (2.9) in (2.2) and (2.5), and evaluating the contributions from Fextμ in the manner outlined above, one obtains
(2.13)Fiμ=Giμ,Qμ=-43πaμ3Pμ+13δmnGmnμ,Tiμ=-εimnG[mn]μ,Sijμ=83πηaμ3Eμij+G〈ij〉μ,
where 𝒢iμ and 𝒢ijμ are defined as in (2.2), with Fiμ(x) replaced by Giμ(x). On the right-hand side of  𝒮ijμ, we have neglected—consistent with the expansion (2.10)—a second-order term proportional to aμ2Δℰμij≪ℰμij.


## 3. Axisymmetric Flow

Axisymmetric scalar- and vector-valued fields are of the generic form(3.1)q(x)=q(s,z),v(x)=vs(s,z)es(φ)+vφ(s,z)eφ(φ)+vz(s,z)ez,
where (s,φ,z) denote cylindrical coordinates about the 3-axis and (es(φ),eφ(φ),ez) the associated unit vectors along the coordinate curves. Explicitly,(3.2)x(s,φ,z)=(scos⁡φssin⁡φz)(s≥0,φ∈R2π,z∈R),es(φ)=(cos⁡φsin⁡φ0),eφ(φ)=(-sin⁡φcos⁡φ0),ez=(001).
Notice in (3.1) that the scalar and the vector components are independent of the azimuthal angle φ. Since the Stokes operator (u1,u2)↦(q,v), defined by the Dirchlet problem (2.7), is invariant under rotations about the 3-axis, it follows [15] that the flow (q,v) is axisymmetric if and only if the applied velocities (2.8) are axisymmetric. Expressing (2.10) in cylindrical coordinates one finds that this is the case if and only if the local velocity gradients (2.11) assume the special form(3.3)(Uνj)=Uν(001)(Wνj)=Wν(001)(Eνjl)=Eν16(-1000-10002),
where we have used that trEν=0. Inserting (3.3) in (2.10), one obtains(3.4)uν(x)=-16sEνes(φ)+sWνeφ(φ)+[Uν+23(z-Zν)Eν]ez((s,z)∈∂B̃ν),
where (s,z)∈∂B̃ν means x(s,φ,z)∈∂Bν for some (and thus all) φ∈[0,2π). Likewise, we will use the notation (s,z)∈B̃0.Upon inserting (3.1) and (3.4) into (2.7), one obtains two Dirichlet problems: one for vφ and one for (q,vs,vz). For brevity, we refer to these as problem I and problem II, respectively. Introducing the Laplace operators(3.5)Δm=1s∂ss∂s-m2s2+∂z∂z(m=0,1),
problem I reads(3.6)Δ1vφ=0,((s,z)∈B̃0),vφ⟶0,(s2+z2⟶∞),vφ=sWν,((s,z)∈∂B̃ν,ν=1,2),
while problem II reads(3.7)1s∂ssvs+∂zvz=0,∂sq-ηΔ1vs=0,∂zq-ηΔ0vz=0,((s,z)∈B̃0),(q,vs,vz)⟶0,(s2+z2⟶∞),vs=-16sEν,vz=Uν+23(z-Zν)Eν,((s,z)∈∂B̃ν,ν=1,2).These two problems are completely decoupled, since (3.6) is determined by the 𝒲ν, while (3.7) is determined by the 𝒰ν and ℰν. Also, from the general uniqueness theorem [8, 43], it follows that both problems have at most one solution.In the following, we parameterize the surfaces∂Bν in terms of the azimuthal angle φ and a polar angle ϑ according to(3.8)∂Bν={x=(sν(ϑ)cos⁡φsν(ϑ)sin⁡φzν(ϑ))∣ϑ∈[0,π],φ∈[0,2π)}(ν=1,2),
where the mapping ϑ↦(sν(ϑ),zν(ϑ))∈∂B̃ν parameterizes the solutions of the surface constraint(3.9)s2+(z-Zν)2=aν2.
This mapping is assumed to be smooth and nonsingular, with żν(ϑ)≠0 for all ϑ∈(0,π). Then, sgnżν=±1 denotes the orientation of the parameterization, and dS=a|żν(ϑ)|dϑdφ is the surface element. The parameterization will be made explicit in the following paper [40], where we introduce bispherical coordinates; for the present purpose, it is sufficient stay general.Since the mapping(q,v)↦(G1,G2), defined by the constitutive equation (2.9), is invariant under rotations about the 3-axis, it follows [15] that the surface-force densities Gμ(x) caused by axisymmetric flow (q,v) are again axisymmetric(3.10)Gμ(x)=Gsμ(ϑ)es(φ)+Gφμ(ϑ)eφ(φ)+Gzμ(ϑ)ez.Calculating the associated moments, defined as in (2.2), one can easily perform the azimuthal integrations to obtain the generic form(3.11)(Giμ)=(00G3μ)(Gijμ)=(G11μ-G21μ0G21μG11μ000G33μ),
where the nonvanishing elements involve polar integrations over the components of (3.10). Inserting these in (2.13), one finds(3.12)(Fiμ)=Fμ(001),Qμ=-43πaμ3Pμ+ΔQμ,(Sijμ)=Sμ16(-1000-10002),(Tiμ)=Tμ(001),Sμ=83πηaμ3Eμ+ΔSμ,
with(3.13)Fμ=G3μ=2πaμ(sgn⁡żμ)∫0πdϑżμGzμ,Tμ=2G21μ=2πaμ(sgn⁡żμ)∫0πdϑżμsμGφμ,ΔQμ=13(2G11μ+G33μ)=23πaμ(sgn⁡żμ)∫0πdϑżμ[(zμ-Zμ)Gzμ+sμGsμ],ΔSμ=23(G33μ-G11μ)=83πaμ(sgn⁡żμ)∫0πdϑżμ[(zμ-Zμ)Gzμ-12sμGsμ].
To determine the components of (3.10), we insert (3.1) in (2.9) and use the incompressibility condition. This yields(3.14)Gsμ(ϑ)=1aμsq(s,z)-ηaμ[(∂N-1)vs(s,z)+∂Tvz(s,z)],Gφμ(ϑ)=-ηaμ(∂N-1)vφ(s,z),Gzμ(ϑ)=1aμ(z-Zμ)q(s,z)+ηaμ[(∂T+z-Zμs)vs(s,z)-∂Nvz(s,z)],
where(3.15)∂N=s∂s+(z-Zμ)∂z,∂T=(z-Zμ)∂s-s∂z
denote the derivatives in the normal and tangential directions, respectively, and (s,z) is to be put equal to (sμ(ϑ),zμ(ϑ)) after the differentiations have been carried out. The expressions (3.14) must now be inserted in (3.13). To simplify the integrals we use the boundary conditions on ∂Bμ, as specified in (3.6) and (3.7), and the relations(3.16)ddϑ=ṡ∂s+ż∂zsṡ+(z-Zμ)ż=0ż∂T=-sddϑ,
which follow from (3.9) and (3.15). Employing the notation(3.17)qμ(ϑ)=q(s,z)|(s,z)=(sμ(ϑ),zμ(ϑ))(∂Nvz)μ(ϑ)=∂Nvz(s,z)|(s,z)=(sμ(ϑ),zμ(ϑ)),
and so forth, and using that(3.18)(sμ(0),zμ(0))=(0,Zμ-asgn⁡żμ)(sμ(π),zμ(π))=(0,Zμ+asgn⁡żμ),
we finally obtain(3.19)Fμ=2πη(sgn⁡żμ)∫0πdϑ[12ηsμ2q̇μ-żμ(∂Nvz)μ],Tμ=83πηaμ3Wμ-2πη(sgn⁡żμ)∫0πdϑżμsμ(∂Nvφ)μ,ΔQμ=23πaμ3[qμ(0)+qμ(π)]-23πη(sgn⁡żμ)∫0πdϑ{aμ2η(zμ-Zμ)q̇μ+żμ[(zμ-Zμ)(∂Nvz)μ+sμ(∂Nvs)μ]},ΔSμ=-43πηaμ3Eμ+83πη(sgn⁡żμ)×∫0πdϑ{12η(zμ-Zμ)sμ2q̇μ-żμ[(zμ-Zμ)(∂Nvz)μ-12sμ(∂Nvs)μ]}.Notice from (3.7) and (3.16) that(3.20)1ηq̇μ=ṡμ(Δ1vs)μ+żμ(Δ0vz)μ.From (3.19), it follows immediately that for axisymmetric flow, the 𝒯μ are determined via vφ by the 𝒲ν (problem I, (3.6)), while the (ℱμ,𝒮μ,Δ𝒬μ) are determined via (q,vs,vz) by the (𝒰ν,ℰν) (problem II, (3.7)). It is convenient to quantify the resulting linear relationships between the force moments and the applied velocity gradients in the scaled form [15](3.21)T̃μ=∑ν=1,2σTWμνW̃ν,(F̃μS̃μΔQ̃μ)=∑ν=1,2(σFUμνσSUμνσQUμνσFEμνσSEμνσQEμν)(ŨνẼν)(μ=1,2),
where the σ··μν are dimensionless resistance functions and(3.22)F̃μ=(6πηaμ)-1/2FμŨν=(6πηaν)1/2Uν,T̃μ=(8πηaμ3)-1/2TμW̃ν=(8πηaν3)1/2Wν,S̃μ=(203πηaμ3)-1/2μẼν=(203πηaν3)1/2Eν,ΔQ̃μ=(203πηaμ3)-1/2ΔQμ.
The scaling (3.22) is based on the one-sphere results [13](3.23)Fi=6πηaUi,Ti=8πηa3Wi,Fij=203πηa3Eij,
and differs from earlier scalings [11–13]. An advantage of the present choice is that our resistance functions satisfy σFUμν,σTWμν,σSEμν→δμν as the distance between the particles goes to infinity. From Lorentz-reciprocity [8, 9], it follows that σSUμν=σFEνμ.


## 4. Stream Function

Since problem I is already a scalar one and needs no further treatment at this point, we now focus on problem II and the calculation of(ℱμ,𝒮μ,Δ𝒬μ). To solve (3.7), we make the ansatz(4.1)vs=∂zΨ,vz=-1s∂ssΨ,
where Ψ(s,z) is a scalar stream function. Obviously, (4.1) satisfies the condition of incompressibility, (1/s)∂ssvs+∂zvz=0. Our stream function differs by a factor s from the classical stream function introduced by Stokes [30] and used hitherto [31–33, 36, 37] but has the advantage of being a biharmonic function. In fact, using the identity(4.2)∂s1s∂ss=1s∂ss∂s-1s2,
one obtains from (3.5), (3.7), and(4.1)(4.3)∂sq=η∂zΔ1Ψ,∂zq=-ηΔ01s∂ssΨ=-η1s∂ssΔ1Ψ,
which leads to(4.4)0=∂z∂sq-∂s∂zq=η[∂z∂z+∂s1s∂ss]Δ1Ψ=ηΔ1Δ1Ψ.
To satisfy the boundary condition (q,vs,vz)→0 as s2+z2→∞, it is sufficient to require that Ψ stay bounded at infinity.Next, we turn to the boundary conditions on the particle surfaces. Since the applied velocitiesuν satisfy the homogenous Stokes equations inside particles, they can be expressed in terms of stream functions, too. We denote these applied stream functions by Φν, (ν=1,2), and determine them from the conditions(4.5)uνs=∂zΦν=-16sEν,uνz=-1s∂ssΦν=Uν+23(z-Zν)Eν((s,z)∈B̃ν,ν=1,2),Φνregular(s⟶0),
which follow from (3.4) and (4.1). The solutions are(4.6)Φν(s,z)=-12sUν-16s(z-Zν)Eν((s,z)∈B̃ν,ν=1,2).It is easy to show that the boundary conditionv=uν on ∂Bν is satisfied, if(4.7)Ψ=Φν∂NΨ=∂NΦν(x∈∂Bν).The proof is based on the identities(4.8)ddϑ(Ψ-Φν)=-ṡ(vz-uνz)+ż(vs-uνs)-ṡs(Ψ-Φν)∂N(Ψ-Φν)=(z-Zν)(vs-uνs)-s(vz-uνz)-(Ψ-Φν)(ϑ∈(0,π)),
which follow from (3.15), (3.16), (4.1), and (4.5). Now, if (4.7) holds, these identities reduce to the homogeneous 2×2-system(4.9)(ż-ṡz-Zν-s)(vs-uνsvz-uνz)=(00),
which has a nonzero determinant. Hence, (4.9) admits only the trivial solution, v-uν=0.To summarizethe flow velocity(vs,vz)can be expressed by (4.1) in terms of a scalar stream functionΨ(s,z)that satisfies the biharmonic two-body Dirichlet problem(4.10)Δ1Δ1Ψ=0((s,z)∈B̃0),Ψbounded(s2+z2⟶∞),Ψ=Φν∂NΨ=∂NΦν((s,z)∈∂B̃ν,ν=1,2),where the applied stream functionsΦνare given by (4.6).The pressure at a given point(s,z)∈B̃0 can, in principle, be calculated from the stream function by integration of (4.3) along any curve in B̃0 that starts at infinity and ends in the point (s,z). However, for all but a few special points, these integrals cannot be performed analytically [40].To establish the existence of a stream function as specified by (4.10), we consider two harmonic functions, ψ1 and ψ2, that vanish at infinity,(4.11)Δ1ψi=0ψi→(1s2+z2)s2+z2⟶∞(i=1,2).
It is known from potential theory that such functions exist in bispherical geometry, and this is, of course, explicitly confirmed by the solution of problem I. Following [31], we put(4.12)Ψ(s,z)=ψ1(s,z)+zψ2(s,z),
and obtain(4.13)Δ1Ψ=2∂zψ2Δ1Δ1Ψ=0,Ψ⟶(zs2+z2)(s2+z2⟶∞).
Thus Ψ, as given by (4.12), is biharmonic and stays bounded at infinity. Moreover, the ansatz (4.12), with two independent harmonic functions, is general enough to allow specification of Ψ  and ∂NΨ on the boundaries, as will be confirmed explicitly in [40].Finally, we express the force moments(ℱμ,𝒮μ,Δ𝒬μ) in the stream function by inserting (3.20) and (4.1) into (3.19). We quote the auxiliary formulae, valid on ∂B̃μ,(4.14)1ηdqdϑ=-ż1s(∂N+1)Δ1Ψż∂Nvz=-żsΔ1Ψ+ddϑsvs,ż(z-Zμ)∂Nvz=-ż(z-Zμ)sΔ1Ψ-żsvs+ddϑ(⋯),żs∂Nvs=ż(z-Zμ)sΔ1Ψ+żsvs+ddϑ(⋯),
where (s,z)=(sμ(ϑ),zμ(ϑ)) and the unquoted expressions (⋯) tend to zero for s→0 and, therefore, do not contribute to the integrals in (3.19). The final results are(4.15)Fμ=-πη(sgn⁡żμ)∫0πdϑżμsμ((∂N-1)Δ1Ψ)μ,Sμ=23πη(sgn⁡żμ)∫0πdϑṡμsμ2((∂N-2)Δ1Ψ)μ,ΔQμ=23πaμ3[qμ(0)+qμ(π)]-2πη3aμ2(sgn⁡żμ)∫0πdϑṡμ((∂N+1)Δ1Ψ)μ.
With (4.15), we have succeeded to express the forces and stresslets in the stream function alone. The pressure moments require, in addition, the quantities qμ(0) and qμ(π), that is, the pressure at the poles of the spheres. As will be shown [40], the poles belong to the few points, where the pressure can be determined analytically in terms of the stream function.The further evaluations in bispherical coordinates and the associated series expansions for the resistance functions will be presented in [40].



---

*Source: 318907-2011-12-15.xml*