Retarded hydrodynamic interaction between two spheres immersed in a viscous incompressible fluid
aa r X i v : . [ phy s i c s . f l u - dyn ] A p r Retarded hydrodynamic interaction between two spheresimmersed in a viscous incompressible fluid
B. U. Felderhof ∗ Institut f¨ur Theorie der Statistischen PhysikRWTH Aachen UniversityTemplergraben 5552056 AachenGermany (Dated: April 9, 2019)
Abstract
Retarded or frequency-dependent hydrodynamic interactions are relevant for velocity relaxationof colloidal particles immersed in a fluid, sufficiently close that their flow patterns interfere. The in-teractions are also important for periodic motions, such as occur in swimming. Analytic expressionsare derived for the set of scalar mobility functions of a pair of spheres. Mutual hydrodynamic inter-actions are evaluated in one-propagator approximation, characterized by a single Green functionacting between the two spheres. Self-mobility functions are evaluated in a two-propagator approx-imation, characterized by a single reflection between the two spheres. The approximations shouldyield accurate results for intermediate and long distances between the spheres. Both translationsand rotations are considered. For motions perpendicular to the line of centers there is translation-rotation coupling. Extensive use is made of Fax´en theorems which yield the hydrodynamic forceand torque acting on a sphere in an incident oscillating flow.
PACS numbers: 47.57.-s, 47.63.mf, 47.57.J-, 83.30.Pp ∗ Electronic address: [email protected] . INTRODUCTION In many problems of physics and chemistry the dynamics of particles suspended in afluid is influenced by hydrodynamic interactions. In principle these can be evaluated on thebasis of the Navier-Stokes equations for the fluid flow. Often the fluid can be assumed tobe incompressible.On the slow time-scale of diffusion inertia of the fluid can be neglected. In this limitthe dynamics of the fluid can be derived from the Stokes equations of low Reynolds numberhydrodynamics [1], and the hydrodynamic interactions are embodied by a friction matrixdependent on the instantaneous configuration of particles. Equivalently one can considerthe mobility matrix, which is the inverse of the friction matrix.In particular, for two spheres with no-slip boundary conditions Batchelor [2] derived thefirst few terms in an expansion of the mobility matrix in powers of the inverse distancebetween centers. In early work we derived some more terms of the expansion for mixed slip-stick boundary conditions [3], and considered also permeable spheres and rotational motions[4]. Schmitz and Felderhof extended the expansion further in algebraic form [5]. Later theexpansion was expressed in a recursive scheme which allowed accurate numerical calculationof the pair hydrodynamic interactions in the Stokes limit [6]-[8]. At present numericalschemes exist which allow calculation of the steady state hydrodynamic interactions betweenmany particles [9],[10].On the time-scale of velocity relaxation and Brownian motion one must go beyond thesteady state calculations, and must consider retarded or frequency-dependent hydrodynamicinteractions. The Stokes limit corresponds to zero frequency. In the following we consideroscillatory translations and rotations of two spheres. Velocity relaxation is summarized by a12 ×
12 frequency-dependent admittance matrix, which can be expressed in terms of a massmatrix and a friction matrix. We aim at an approximate analytic calculation of the set ofscalar mobility functions, the elements of the mobility matrix, as functions of distance andfrequency.It follows from the reciprocal theorem that the admittance matrix, the friction matrix,and the mobility matrix are all symmetrical [8]. The symmetry leads to a set of reciprocityrelations between the scalar mobility functions. Each of the matrices can be expressed viaa multiple scattering expansion as a sum of contributions describing successive scatteringbetween the two spheres [11]. This allows decomposition of each matrix as a sum of n -propagator contributions, where n counts the number of Green functions acting between thetwo spheres. It follows from the symmetry of the single sphere friction operator [12] andthe symmetry of the Green function that the individual n -propagator contributions are alsosymmetric.Our approximation to the mobility matrix consists of the one-propagator contribution,which is calculated exactly, and a low order multipole approximation to the two-propagatorcontribution. In both calculations use is made of Fax´en type theorems for the hydrodynamicforce and torque exerted on a single sphere by an incident oscillatory flow [8],[13],[14].The one-propagator contribution to the translational-translational ( tt ) part of the mo-bility matrix is calculated from the primary flow of the first sphere at given frequency, asfound by Stokes [15], and from the corresponding force exerted on the second sphere, asfound from the Fax´en type theorem. We show explicitly that this contribution satisfies thereciprocity relation. In an earlier calculation by Ardekani and Rangel [16] for two equalspheres an approximate form of the Fax´en type theorem, due to Maxey and Riley [17], was2sed, so that their result is not exact. If their calculation is extended to unequal spheres thereciprocity relation is not satisfied. The same criticism applies to the calculation of Jungand Schmid of the tt hydrodynamic interaction in a compressible fluid [18].We consider both translations and rotations of the two spheres. We calculate also the one-propagator contribution to the rotational-translational ( rt ) part, the translational-rotational( tr ) part, and the rotational-rotational ( rr ) part of the one-propagator mobility matrix.For the latter two contributions the primary flow is that of a sphere performing rotatoryoscillations. All these contributions properly satisfy the reciprocity relations.The calculation of the two-propagator contribution to the self-mobility functions is tech-nically demanding. It involves hydrodynamic scattering from a moving sphere [12]. Theexpressions for the self-mobility functions are correspondingly more complicated than thosefor the mutual functions. Since the calculation is limited to low multipole orders of thesecondary flow, the reciprocity relation between the rt and tr functions is satisfied onlyapproximately.The calculated mobility functions should be accurate at all frequencies and at intermedi-ate and long distance between centers. At short distance higher order terms in the multiplescattering expansion must be taken into account. The corresponding two-center multipoleexpansion can be formulated on the basis of exact addition theorems [19], but the algo-rithm is quite elaborate and requires numerical evaluation [20],[21]. Clercx [21] providedplots of the scalar mobilities for some selected short distances as functions of frequency.Hermanns formulated the multiple scattering expansion for the admittance matrix [22], andinvestigated in particular the behavior of the velocity relaxation functions at long time [23].In Sec. II of this article we discuss the linear dynamics of two unequal spheres immersedin a viscous incompressible fluid. In Sec. III we consider earlier approximations to the tt part of the mobility matrix. In Sec. IV we derive the analytical expressions for the mutualmobility functions in one-propagator approximation. In Secs. V, VI and VII we derive theanalytical expressions for the self-mobility functions in two-propagator approximation. InSec. VIII we discuss results for both mutual and self-mobility functions. In Sec. IX wecalculate the mutual functions also for mixed slip-stick boundary conditions. We finish thearticle with conclusions in Sec. X. II. LINEAR RESPONSE OF TWO SPHERES
We consider two spheres labeled
A, B of radii a and b centered at positions R A and R B and immersed in a viscous incompressible fluid of shear viscosity η and mass density ρ . Thespheres are subjected to applied forces E A ( t ) , E B ( t ) and torques N A ( t ) , N B ( t ) whichoscillate at frequency ω . At the surface of each sphere the fluid flow velocity satisfies theno-slip boundary condition. The resulting flow velocity u ( r , t ) and pressure p ( r , t ) of thefluid are assumed to satisfy the Navier-Stokes equations ρ (cid:20) ∂ u ∂t + u · ∇ u (cid:21) = η ∇ u − ∇ p, ∇ · u = 0 . (2.1)3he flow velocity tends to zero at infinity and the pressure tends to the ambient value p .The equations of motion for the two spheres read m A d U A dt = E A + K A , m B d U B dt = E B + K B ,I A d Ω A dt = N A + T A , I B d Ω B dt = N B + T B , (2.2)where m A , m B are the masses, U A , U B are the translational velocities, and K A , K B arethe forces exerted by the fluid on the two spheres. Furthermore I A , I B are the momentsof inertia, Ω A , Ω B are the rotational velocities, and T A , T B are the torques exerted by thefluid on the two spheres. We assume the spheres to be uniform with mass densities ρ A and ρ B , so that m A = 4 πρ A a / , m B = 4 πρ B b / I A = m A a , I B = m B b .For small amplitude of the applied forces and torques the spheres perform small os-cillations about rest positions R A and R B . The relative position vector is denoted as R = R B − R A . One may linearize the Navier-Stokes equations by omitting the inertialterm ρ u · ∇ u and calculate the hydrodynamic forces K A , K B and torques T A , T B fromthe linearized equations. In principle the calculation can be performed exactly in terms ofa multiple scattering expansion, and leads to a linear relation between sphere translationaland rotational velocities and applied forces and torques. In complex notation the relationreads U Aω = Y ttAA ( R , ω ) · E Aω + Y ttAB ( R , ω ) · E Bω + Y trAA ( R , ω ) · N Aω + Y trAB ( R , ω ) · N Bω , U Bω = Y ttBA ( R , ω ) · E Aω + Y ttBB ( R , ω ) · E Bω + Y trBA ( R , ω ) · N Aω + Y trBB ( R , ω ) · N Bω , Ω Aω = Y rtAA ( R , ω ) · E Aω + Y rtAB ( R , ω ) · E Bω + Y rrAA ( R , ω ) · N Aω + Y rrAB ( R , ω ) · N Bω , Ω Bω = Y rtBA ( R , ω ) · E Aω + Y rtBB ( R , ω ) · E Bω + Y rrBA ( R , ω ) · N Aω + Y rrBB ( R , ω ) · N Bω . (2.3)According to linear response theory of statistical physics the complete 12 ×
12 admittancematrix Y ( R , ω ) = Y ttAA ( R , ω ) Y trAA ( R , ω ) Y ttAB ( R , ω ) Y trAB ( R , ω ) Y rtAA ( R , ω ) Y rrAA ( R , ω ) Y rtAB ( R , ω ) Y rrAB ( R , ω ) Y ttBA ( R , ω ) Y trBA ( R , ω ) Y ttBB ( R , ω ) Y trBB ( R , ω ) Y rtBA ( R , ω ) Y rrBA ( R , ω ) Y rtBB ( R , ω ) Y rrBB ( R , ω ) (2.4)is symmetric [24].In the theory of Brownian motion the 12 ×
12 velocity correlation matrix C ( R , t ) isrelated to the admittance matrix Y ( R , ω ) by the fluctuation-dissipation theorem [25]. Withone-sided Fourier transform ˆ C ( R , ω ) = Z ∞ e iωt C ( R , t ) dt (2.5)this reads ˆ C ( R , ω ) = k B T Y ( R , ω ) , (2.6)with Boltzmann’s constant k B and absolute temperature T .In the linear theory the equations of motion Eq. (2.2) become − iωm A U Aω = E Aω + K Aω , − iωm B U Bω = E Bω + K Bω , − iωI A Ω Aω = N Aω + T Aω , − iωI B Ω Bω = N Bω + T Bω , (2.7)4ith hydrodynamic forces and torques related to the velocities by friction tensors as K Aω = − ζ ttAA ( R , ω ) · U Aω − ζ ttAB ( R , ω ) · U Bω − ζ trAA ( R , ω ) · Ω Aω − ζ trAB ( R , ω ) · Ω Bω , K Bω = − ζ ttBA ( R , ω ) · U Aω − ζ ttBB ( R , ω ) · U Bω − ζ trBA ( R , ω ) · Ω Aω − ζ trBB ( R , ω ) · Ω Bω , T Aω = − ζ rtAA ( R , ω ) · U Aω − ζ rtAB ( R , ω ) · U Bω − ζ rrAA ( R , ω ) · Ω Aω − ζ rrAB ( R , ω ) · Ω Bω , T Bω = − ζ rtBA ( R , ω ) · U Aω − ζ rtBB ( R , ω ) · U Bω − ζ rrBA ( R , ω ) · Ω Aω − ζ rrBB ( R , ω ) · Ω Bω . (2.8)The 12 ×
12 friction matrix defined by these linear relations is symmetric. The symmetry maybe derived as a consequence of the reciprocal theorem of linear hydrodynamics [4],[8]. Thesymmetry also follows from the multiple scattering expansion combined with the symmetryof the two single sphere friction operators [12] and the symmetry of the Green function.We note that the friction matrix follows from the solution of the linearized Navier-Stokesequations for specified translational and rotational velocities of the two spheres under thecondition that the flow disturbance tends to zero at infinity. The friction matrix is indepen-dent of the mass density of the two spheres.The translational friction tensors defined in Eq. (2.8) include high frequency behaviorproportional to − iω . Subtracting this we obtain ζ ttij ( R , ω ) = − iω m aij ( R ) + ζ tt ′ ij ( R , ω ) , ( i, j ) = ( A, B ) , (2.9)with added mass tensors m aij ( R ) and remaining friction tensors ζ tt ′ ij ( R , ω ).We write Eq. (2.8) in abbreviated form as (cid:18) KT (cid:19) = − (cid:18) ζ tt ζ tr ζ rt ζ rr (cid:19) · (cid:18) UΩ (cid:19) , (2.10)with 6 × ζ mn . The inverse relation (cid:18) UΩ (cid:19) = − (cid:18) µ tt µ tr µ rt µ rr (cid:19) · (cid:18) KT (cid:19) , (2.11)defines the 6 × µ mn . The 12 ×
12 mobility matrix µ is symmetric.Cichocki and Felderhof [26] derived the low frequency expansion µ ttij ( R , ω ) = µ ttij ( R , − α πη I + O ( α ) , µ trij ( R , ω ) = µ trij ( R ,
0) + O ( α ) , µ rtij ( R , ω ) = µ rtij ( R ,
0) + O ( α ) , µ rrij ( R , ω ) = µ rrij ( R ,
0) + O ( α ) , ( i, j ) = ( A, B ) , (2.12)with variable α defined by α = ( − iωρ/η ) / , Re α > , (2.13)and with unit tensor I .Our strategy in the following will be to calculate an approximation to the mobility ma-trix by finding approximate values for the sphere velocities for simple flow situations withspecified forces and torques. The calculation will be performed analytically. By inversionthe calculation also yields an approximation to the friction matrix ζ = µ − . Hence we can5nd an approximation to the admittance matrix Y = ( − iω m + ζ ) − , where m is the massmatrix, which is diagonal with elements m A , m B , I A , I B .By isotropy the various mobility tensors take the form µ ttij ( R , ω ) = α ttij ( R, ω ) ˆ R ˆ R + β ttij ( R, ω )( I − ˆ R ˆ R ) , µ trij ( R , ω ) = β trij ( R, ω ) ǫ · ˆ R , µ rtij ( R , ω ) = β rtij ( R, ω ) ǫ · ˆ R , µ rrij ( R , ω ) = α rrij ( R, ω ) ˆ R ˆ R + β rrij ( R, ω )( I − ˆ R ˆ R ) , ( i, j ) = ( A, B ) , (2.14)where ˆ R = R /R , I is the unit tensor, and ǫ is the Levi-Civita tensor. The scalar functionssatisfy the reciprocity relations α ttij ( R, ω ) = α ttji ( R, ω ) β ttij ( R, ω ) = β ttji ( R, ω ) ,β trij ( R, ω ) = − β rtji ( R, ω ) ,α rrij ( R, ω ) = α rrji ( R, ω ) β rrij ( R, ω ) = β rrji ( R, ω ) , ( i, j ) = ( A, B ) , (2.15)as follows from the symmetry of the 12 ×
12 mobility matrix µ .It is convenient to use Cartesian coordinates with the origin at R A and z axis in thedirection of ˆ R . In this representation it is evident that translational motions along the lineof centers and rotations about this line decouple from translational motions perpendicular tothe line and rotations about an axis perpendicular to the line of centers. We call these twotypes of motion longitudinal and transverse, and denote them by the symbols k and ⊥ . Bypermutation of rows and columns the mobility matrix can be rearranged in the decomposedform µ D = µ ⊥ µ −⊥
00 0 µ k , (2.16)where the transverse matrix µ ⊥ = β ttAA β ttAB β trAA β trAB β ttAB β ttBB β trAB β trBB β trAA β trAB β rrAA β rrAB β trAB β trBB β rrAB β rrBB , (2.17)corresponds to translational motions and forces in the x direction and rotational motionsand torques in the y direction. We have used the reciprocity relations Eq. (2.15). Thesecond matrix in Eq. (2.16) is given by µ −⊥ = β ttAA β ttAB − β trAA − β trAB β ttAB β ttBB − β trAB − β trBB − β trAA − β trAB β rrAA β rrAB − β trAB − β trBB β rrAB β rrBB , (2.18)and corresponds to translational motions and forces in the y direction and rotational motionsand torques in the x direction. It can be transformed to µ ⊥ by referring rotational motionsand torques to the negative x direction, corresponding to axial symmetry. The longitudinalmobility matrix µ k = α ttAA α ttAB α ttAB α ttBB α rrAA α rrAB α rrAB α rrBB , (2.19)6orresponds to translational motions and forces and rotational motions and torques in the z direction. It is evident that for longitudinal motions translations and rotations decouple.The expressions in Eqs. (2.17-19) show that only a limited number of scalar functions needbe calculated. III. APPROXIMATE CALCULATIONS
The mobility matrix µ ( R , ω ) or the friction matrix ζ ( R , ω ), may be calculated in terms ofa multiple scattering expansion, but the calculation is complex. In the literature final resultsare often presented in graphical form [20]-[22]. The response of a single sphere to an appliedforce and an incident flow is described by a multipolar friction matrix [12]. Propagationbetween spheres is described by the Green function of the linearized Navier-Stokes equations η [ ∇ v − α v ] − ∇ p = 0 , ∇ · v = 0 , ( r > . (3.1)The Green function reads G ( r , ω ) = G k ( r, ω )ˆ r ˆ r + G ⊥ ( r, ω )( I − ˆ r ˆ r ) , (3.2)with ˆ r = r /r and longitudinal and transverse parts G k ( r, ω ) = 12 πηα r [1 − (1 + αr ) e − αr ] ,G ⊥ ( r, ω ) = − πηα r [1 − (1 + αr + α r ) e − αr ] . (3.3)Van Saarloos and Mazur [27] performed the multiple scattering expansion on the basis ofa spatial Fourier transform and derived the first few terms in an expansion of the mobilitymatrix in powers of inverse distance R − . The tt mutual interaction, presented in Eq. (4.3)of their article, does not satisfy the reciprocity relation Eq. (2.15), and is therefore notacceptable. Their expression for the self-interaction does not depend on mutual distance.Pie´nkowska [28] derived an approximate expression for the mutual friction tensor, validat low frequency, which satisfies reciprocity. Later she derived a more detailed expression,which again satisfies reciprocity [29]. Her expression is closely related to the Green function.Henderson et al. [30] considered equal spheres and only the longitudinal part of thefriction tensor, but their expression is easily generalized to the complete tensor and tounequal spheres. The mutual friction tensor is approximated as ζ ttAB ( R , ω ) = ζ ttBA ( R , ω ) ≈ − (6 πη ) ab G ( R , ω ) , (3.4)and the single sphere admittances as Y tA k = Y tA ⊥ ≈ πηa (1 + αa ) , (3.5)and similarly for B . We recall that the frequency-dependent translational friction coefficientof a single sphere is given by ζ tA ( ω ) = 6 πηa (cid:0) αa + 19 α a (cid:1) , (3.6)7here the last term corresponds to the added mass. Hence the single sphere admittance is Y tA = [ − iω ( m A + 12 m fA ) + 6 πηa (1 + αa )] − , (3.7)where m fA = 4 πρa / A . Clearly the high frequencybehavior is not well described by the approximation Eq. (3.5). Henderson et al. [30] findgood agreement with their experiments in the low frequency regime. A better approximationwould be obtained by use of Eq. (3.7) instead of Eq. (3.5).Bonet Avalos et al. studied the role of time-dependent hydrodynamic interactions onthe dynamics of polymers in solution [31]. They made a point approximation in which themutual mobility tensor is given by µ ttAB ( R , ω ) = µ ttBA ( R , ω ) ≈ G ( R , ω ) . (3.8)The approximation satisfies reciprocity, but is clearly rather drastic. Ardekani and Rangel[16] in their first approach, and Tatsumi and Yamamoto [32] made the same approximation.The approximation was recommended for use in two-point passive microrheology by C´ordobaet al. [33]. In their second approach Ardekani and Rangel [16] attempted to derive a finite-size correction for the translational friction functions, but they did not take proper accountof the expression for the Fax´en type expression for the force on a sphere in a non-uniformflow, as discussed below.The calculations below are based on an approximation in which the mutual terms involveonly a single propagator between spheres and the self terms of the mobility matrix areapproximated by their single sphere expressions corrected by at most one reflection from theother sphere. Finite-size corrections are taken into account. Later we shall compare withthe point approximation Eq. (3.8). IV. MUTUAL MOBILITY FUNCTIONS
Our calculation is based on the method of reflections and amounts to an approximate so-lution of the linearized Navier-Stokes equations as a boundary value problem. We employedthe same method earlier to calculate an approximate mobility matrix at zero frequency [3].We start from a given set of translational and rotational velocities U A, , U B, , Ω A, , Ω B, .To lowest order this corresponds to a set of hydrodynamic forces K A, , K B, and torques T A, , T B, , as given by the single sphere translational and rotational friction coefficients ζ tA ( ω ) = 6 πηaA ( αa ) , ζ tB ( ω ) = 6 πηbA ( αb ) ,ζ rA ( ω ) = 8 πηa B ( αa )1 + αa , ζ rB ( ω ) = 8 πηb B ( αb )1 + αb , (4.1)with abbreviations A ( λ ) and B ( λ ) given in the Appendix. Higher order corrections U A,j , U B,j , Ω A,j , Ω B,j ( j = 1 , , ... ) are calculated from the condition that in each step of thecalculation the corresponding forces and torques vanish. This is achieved by the applicationof Fax´en type theorems.Fax´en’s theorem for the hydrodynamic force acting on a sphere of radius a with centerat the origin and subject to an incident flow v ( r , ω ) , p ( r , ω ), which is a solution of thelinearized Navier-Stokes equations, reads [8] K A = 6 πηa (cid:2) B ( αa ) v + B ( αa ) a ∇ v (cid:3)(cid:12)(cid:12) r = − ζ tA ( ω ) U A , (4.2)8ith the further abbreviation B ( λ ) given in the Appendix. In the original formulation ofthe Fax´en theorem by Mazur and Bedeaux [34] the force exerted by the fluid on a singlesphere is expressed in terms of surface and volume averages of the incident flow field. Thismakes clear that only 1 mN and 1 mP multipole components of the incident flow, in thenotation of Felderhof and Jones [12], contribute to the force.The analogous theorem for the hydrodynamic torque reads [13],[14] T A = 4 πηa e αa αa ( ∇ × v ) (cid:12)(cid:12) r = − ζ rA ( ω ) Ω A . (4.3)The equivalent form which we derived [35],[36] is more complicated, but makes clear thatonly the three 1 mM multipole components of the incident flow contribute to the torque.Consider first the zero order situation where only U A, = U A, e z and K A, differ fromzero. The corresponding zero order flow pattern is [15],[12] v UAω, ( r ) = U A, (cid:20) B ( αa )2 α a u A ( r ) + 12 αa v A ( r , α ) (cid:21) , (4.4)where u A ( r ) is a potential flow and v A ( r , α ) is a viscous flow. Explicitly u A ( r ) = − (cid:18) ar (cid:19) B z (ˆ r ) , v A ( r , α ) = 2 π e αa [2 k ( αr ) A z (ˆ r ) + k ( αr ) B z (ˆ r )] , (4.5)with modified spherical Bessel functions [37] k l ( z ), and vector spherical harmonics given by A z (ˆ r ) = e z , B z (ˆ r ) = e z − e z · ˆ r )ˆ r . (4.6)The flows u A and v A satisfy the equations ∇ u A = 0 , ∇ · u A = 0 , ∇ v A − α v A = 0 , ∇ · v A = 0 , ( r > . (4.7)In the notation of Felderhof and Jones [12] u A ( r ) = 2 √ π α a v − P ( r ) , v A ( r ) = 4 √ π e αa α v − N ( r ) , (4.8)with functions v − lmN ( r ) = 2 απl ( l + 1)(2 l + 1) [( l + 1) k l − ( αr ) A lm (ˆ r ) + k l +1 ( αr ) B lm (ˆ r )] , v − lmP ( r ) = − α (2 l + 1) r − l − B lm (ˆ r ) , (4.9)with vector spherical harmonic A lm (ˆ r ) and B lm (ˆ r ). It is checked by use of Eq. (4.8) thatthe flow pattern in Eq. (4.4) satisfies the no-slip boundary condition v UAω, ( r ) (cid:12)(cid:12) r = a = U A, e z . (4.10)9o first order sphere B moves with a velocity U B, = U B, e z such that it exerts no forceon the fluid. By use of the Fax´en theorem Eq. (4.2) U B, = ζ tA ( ω ) α ttBA, ( R, ω ) U A, , (4.11)with α ttBA, ( R, ω ) = B ( αa ) B ( αb ) − (1 + αR ) e α ( a + b − R ) πηα R A ( αa ) A ( αb ) . (4.12)Since K A, = − ζ tA ( ω ) U A, , this may be identified with the first order scalar mobility function.The reciprocity relation α ttBA ( R, ω ) = α ttAB ( R, ω ) is obviously satisfied in the approximation.At low frequency Eq. (4.12) becomes α ttAB ( R, ω ) = α ttBA ( R, ω ) = 14 πηR − a + b πηR − α πη + O ( α ) . (4.13)The zero frequency limit agrees to terms of order R − with earlier results [3][4]. By use ofthe Fax´en theorem Eq. (4.3) one finds that in this situation the rotational velocity of sphere B vanishes, so that α rtBA, ( R, ω ) = 0.By the same derivation for motion in the x direction one finds the transverse mobilityfunction β ttBA, ( R, ω ) = − B ( αa ) B ( αb ) + A ( αR ) e α ( a + b − R ) πηα R A ( αa ) A ( αb ) , (4.14)with abbreviation A ( λ ) given in the Appendix. At low frequency this becomes β ttAB ( R, ω ) = β ttBA ( R, ω ) = 18 πηR + a + b πηR − α πη + O ( α ) . (4.15)The zero frequency limit agrees to terms of order R − with earlier results [3]-[5]. One seesalso that Eq. (3.8) is an approximation to Eqs. (4.12) and (4.14) valid for αa << αb << B about the y axis one finds in this situationby use of the Fax´en theorem Eq. (4.3) the mobility function β rtBA ( R, ω ) = 18 πηR A ( αa ) B ( αb ) (1 + αR ) e α ( a + b − R ) . (4.16)At low frequency this becomes β rtBA ( R, ω ) = 18 πηR + O ( α ) . (4.17)The zero frequency limit agrees to terms of order R − with earlier results [4],[5].Next we consider a situation where in zero order sphere A rotates about the z axis withangular velocity Ω A, corresponding to hydrodynamic torque T A, . The flow pattern aboutsphere A for rotation in the z direction in the absence of incident flow is given by v Ω Aω, ( r ) = Ω A, a k ( αr ) k ( αa ) C z (ˆ r ) , (4.18)10ith vector spherical harmonic C z (ˆ r ) = e z × e r . The flow pattern satisfies the no-slipboundary condition v Ω Aω, ( r ) (cid:12)(cid:12) r = a = Ω A, a e z × e r , (4.19)and the equations ∇ v Ω Aω, − α v Ω Aω, = 0 , ∇ · v Ω Aω, = 0 . (4.20)Calculating the resulting first order rotational velocity of sphere B by use of the Fax´entheorem Eq. (4.3) we find the scalar mobility function α rrBA, ( R, ω ) = (1 + αR ) e α ( a + b − R ) πηR B ( αa ) B ( αb ) . (4.21)This clearly satisfies reciprocity α rrAB ( R, ω ) = α rrBA ( R, ω ). At low frequency α rrAB ( R, ω ) = 18 πηR + O ( α ) . (4.22)The zero frequency limit agrees to terms of order R − with earlier results [4],[5]. By useof the Fax´en theorem Eq. (4.2) one finds that in this situation the translational velocity ofsphere B vanishes, so that α trBA, ( R, ω ) = 0.By the same derivation for rotation about the x direction one finds the transverse mobilityfunction β rrAB, ( R, ω ) = − A ( αR ) e α ( a + b − R ) πηR B ( αa ) B ( αb ) . (4.23)At low frequency this becomes β rrAB ( R, ω ) = β rrBA ( R, ω ) = − πηR + O ( α ) . (4.24)The zero frequency limit agrees to terms of order R − with earlier results [4][5]. Calculatingthe function β trBA, ( r, ω ) in this situation one finds β trBA ( R, ω ) = − πηR B ( αa ) A ( αb ) (1 + αR ) e α ( a + b − R ) , (4.25)quite similar to Eq. (4.16), and in accordance with the reciprocity relation Eq. (2.15). V. LONGITUDINAL SELF-MOBILITY FUNCTIONS
In the calculation of the self-mobility functions we carry the method of reflections oneorder higher. We consider first the self-mobility function α ttAA ( R, ω ). In lowest approximationwe calculate this from the velocity of sphere A caused by the primary flow field reflectedonce from sphere B . In the theory of Felderhof and Jones [12] the reflection is described ashydrodynamic scattering by the moving sphere B . We shall take account of multipoles ofthe incident flow on sphere B only up to order l = 2.Thus we expand the zero order flow v UAω, ( r ), centered at R A = , in terms of multipoleflows centered at R B as v UAω, ( r ) = c +10 N v +10 N ( r − R B ) + c +10 P v +10 P ( r − R B )+ c +20 N v +20 N ( r − R B ) + c +20 P v +20 P ( r − R B ) + ..., (5.1)11ith superposition coefficients which can be calculated from the flow given by Eq. (4.4) andwith functions v + lmN ( r ) = 12 l + 1 (cid:2) ( l + 1) g l − ( αr ) A lm (ˆ r ) + lg l +1 ( αr ) B lm (ˆ r ) (cid:3) , v + lmP ( r ) = r l − A lm (ˆ r ) . (5.2)As shown in Eqs. (4.4-9) the flow v UAω, ( r ) is a linear combination of outgoing waves v − N ( r ) and v − P ( r ). More generally, singular solutions of this type can be expanded as theaddition theorems [19],[38] v − l N ( r ) = X l ′ S + − ( R ; l ′ N, l N ) v + l ′ N ( r < ) , v − l P ( r ) = X l ′ S + − ( R ; l ′ P, l P ) v + l ′ P ( r < ) , (5.3)where r < = r − R e z = r − R B . The coefficient functions S + − can be found [19],[38] from theelaborate results derived by Langbein [39]. The coefficients c +10 N and c +10 P in Eq. (5.1) canbe expressed in terms of functions S + − ( R ; l N, N ) and S + − ( R ; l P, P ), respectively.For the low orders under consideration we prefer a more pedestrian approach. We useinstead the Taylor expansion of the function v UAω, ( r ) in powers of R − R B , with each termprolonged into a regular solution of the linearized Navier-Stokes equations Eq. (3.1), as v UAω, ( r ) = ∞ X n =0 v ( n ) UAB , (5.4)where n indicates the power of r − R B in the Taylor expanson. The first term is v (0) UAB = c +10 N v +10 N ( r − R B ) + c +10 P v +10 P ( r − R B ) . (5.5)At r = R B we have from Eq. (4.4) v UAω, ( R B ) = U (0) B e z , U (0) B = U A, aα R (cid:2) B ( αa ) − (1 + αR ) e α ( a − R ) (cid:3) . (5.6)Comparing this with Eq. (5.5) we find the relation U (0) B = c +10 N √ π + c +10 P r π . (5.7)It follows from the Fax´en theorem Eq. (4.2) that B ( αb ) U (0) B e z + B ( αb ) b ( ∇ v (0) UAB ) (cid:12)(cid:12)(cid:12)(cid:12) r = R B = A ( αb ) U B, e z , (5.8)with U B, given by Eq. (4.11). Only the first term on the right in Eq. (5.5) contributes tothe second term in this equation. Substituting we obtain the value of c +10 N as c +10 N = − √ π U A, aα R (1 + αR ) e α ( a − R ) , (5.9)12nd from Eq. (5.7) we find c +10 P = r π U A, aα R B ( αa ) . (5.10)The first line in Eq. (5.1) generates the scattered wave v − B ( r ) = c +10 N (cid:2) χ NN v − N ( r − R B ) + χ NP v − P ( r − R B ) (cid:3) + (cid:0) c +10 P − r π U B, (cid:1)(cid:2) χ NP v − N ( r − R B ) + χ P P v − P ( r − R B ) (cid:3) , (5.11)with scattering coefficients χ NN = 1 − e αb α , χ NP = − be αb , χ P P = − α b k ( αb ) k ( αb ) , (5.12)and outgoing waves of types 10 N and 10 P given by Eq. (4.9). In the second line of Eq.(5.11) we took account [12] of the fact that sphere B is moving with velocity U B, .For the coefficients in the second line of Eq. (5.1) we find c +20 N = 0 , c +20 P = 9 r π U A, aα R (cid:20) e α ( a − R ) B ( αR ) − B ( αa ) (cid:21) . (5.13)The corresponding scattered wave is v − B ( r ) = c +20 P (cid:2) χ NP v − N ( r − R B ) + χ P P v − P ( r − R B ) (cid:3) , (5.14)with scattering coefficients χ NP = − π bαk ( αb ) , χ P P = − α b k ( αb ) k ( αb ) , (5.15)and outgoing waves of types 20 N and 20 P , given by Eq. (4.9).The scattered waves act back on sphere A and the corresponding velocity of sphere A ,as its moves in this flow with zero force and torque, can be evaluated with the aid of Fax´entheorems Eqs. (4.2) and (4.3), as above. We verify from Eq. (4.3) that the rotationalvelocity of sphere A vanishes, corresponding to α rtAA ( R, ω ) = 0. In our approximation thelongitudinal translational self-mobility function is given by α ttAA ( R, ω ) = 16 πηaA ( αa ) + α ttAA, ( R, ω ) + α ttAA, ( R, ω ) , (5.16)where the first term is the single sphere contribution, the second term follows from Eq.(5.11), and the third term follows from Eq. (5.14). The explicit expression for the mobilityfunction α ttAA, ( R, ω ) reads α ttAA, ( R, ω ) = (cid:20) − α b B ( αa ) B ( αb ) − A ( αb )(1 + αR ) e α ( a − R ) + 8 α b B ( αa )(1 + αR ) e α ( a + b − R ) + (9 − αb + α b )(1 + αR ) e α ( a + b − R ) (cid:21)(cid:30)(cid:18) πηα R A ( αa ) A ( αb ) (cid:19) . (5.17)13t low frequency this has the expansion α ttAA, ( R, ω ) = b πηR + O ( α ) . (5.18)We note that if the term with U B, in Eq. (5.11) were omitted, corresponding to a fixedsphere B , the function would have a contribution proportional to R − .The explicit expression for the mobility function α ttAA, ( R, ω ) reads α ttAA, ( R, ω ) = − b πηα R (1 + αb ) A ( αa ) (cid:18) e α ( a − R ) B ( αR ) − B ( αa ) (cid:19) × (cid:18) e α ( a + b − R ) B ( αR ) − B ( αa ) A ( αb ) (cid:19) , (5.19)with abbreviation A ( λ ) given in the Appendix. At low frequency this has the expansion α ttAA, ( R, ω ) = 18 πηa (cid:20) − ab R + 10 a b + 3 ab R − a b + 3 a b R (cid:21) + O ( α ) . (5.20)The zero frequency limit of the function defined in Eq. (5.16) agrees to terms of order R − with earlier results [4][5], which took account of higher order terms in the equivalent of Eq.(5.1). The term linear in α , arising from the first term on the right in Eq. (5.16), agreeswith the exact relation Eq. (2.12).Finally we calculate an approximation to the rotational self-mobility functions, corre-sponding to a torque in the z direction exerted on sphere A . The flow pattern of sphere A rotating with angular velocity Ω A, about the z axis is given by Eq. (4.18).From the Fax´en theorem Eq. (4.2), as applied to v Ω Aω, for the sphere centered at R B ,we see that the resulting translational velocity of sphere B vanishes, since R = R e z and e z × R = . This confirms that in the longitudinal case there is no translation-rotationcoupling.We express the Taylor expansion of the function v Ω Aω, ( r ) in powers of R − R B , witheach term prolonged into a regular solution of the linearized Navier-Stokes equations Eq.(3.1), as v Ω Aω, ( r ) = ∞ X n =0 v ( n )Ω AB . (5.21)In order to find a low order approximation to the self-mobility function α rrAA ( R, ω ) we cal-culate the rotational velocity of sphere A caused by the flow field reflected from sphere B .The lowest order term in Eq. (5.21) vanishes, since R = R e z and e z × e z = 0. We expandthe zero order flow v Ω Aω, ( r ) in terms of incident multipole flows centered at R B as v Ω Aω, ( r ) = c +10 M v +10 M ( r − R B ) + c +20 M v +20 M ( r − R B ) + ..., (5.22)with functions v + lmM ( r ) = g l ( αr ) C lm (ˆ r ) , (5.23)where g l ( z ) is the regular modified spherical Bessel function. The corresponding outgoingwaves are defined by v − lmM ( r ) = 2 απl ( l + 1) k l ( αr ) C lm (ˆ r ) . (5.24)14he function v Ω Aω, ( r ) in Eq. (4.18) is proportional to v − M ( r ), and in analogy to Eq. (5.3)there is an addition theorem of the form v − l M ( r ) = X l ′ S + − ( R ; l ′ M, l M ) v + l ′ M ( r < ) . (5.25)For the coefficients of low order in Eq. (5.22) one finds straightforwardly in the same wayas before c +10 M = 2 √ π Ω A, a αR αR αa e α ( a − R ) ,c +20 M = − √ π Ω A, a α R B ( αR )1 + αa e α ( a − R ) . (5.26)The first term in Eq. (5.22) yields the scalar mobility function α rrBA, ( R, ω ) given by Eq.(4.21). This corresponds to the first order angular velocity Ω B, of sphere B , given byΩ B, = a R αR αa e α ( a + b − R ) B ( αb ) Ω A, . (5.27)The flow generated by sphere B is given by v − B ( r ) = (cid:0) c +10 M − r π bg ( αb ) Ω B, (cid:1) χ MM v − M ( r − R B )+ χ MM c +20 M v − M ( r − R B ) , (5.28)with scattering coefficients χ lMM = − πl ( l + 1)2 α g l ( αb ) k l ( αb ) . (5.29)In the first line of Eq. (5.28) we took account of the fact [12] that sphere B is rotating withangular velocity Ω B, .The scattered wave acts back on sphere A and the corresponding rotational velocity ofsphere A , as its moves in this flow with zero torque, can be evaluated with the aid of Fax´en’stheorem Eq. (4.3) as above. As in Eq. (5.16) α rrAA ( R, ω ) = 1 + αa πηa B ( αa ) + α rrAA, ( R, ω ) + α rrAA, ( R, ω ) . (5.30)The function α rrAA, ( R, ω ) is given by α rrAA, ( R, ω ) = P ( αb )16 πηα R B ( αa ) B ( αb ) (1 + αR ) e α ( a − R ) , (5.31)with abbreviation P ( λ ) given in the Appendix. At low frequency the function behaves as α rrAA, ( R, ω ) = − b πηR α + O ( α ) . (5.32)This vanishes at zero frequency. The function α rrAA, ( R, ω ) is given by α rrAA, ( R, ω ) = 4516 πηα R P ( αb ) B ( αa ) B ( αb ) B ( αR ) e α ( a − R ) , (5.33)15t low frequency this has the expansion α rrAA, ( R, ω ) = − b πηR + O ( α ) . (5.34)The zero frequency limit agrees to terms of order R − with earlier results [4],[5]. VI. TRANSVERSE SELF-MOBILITY FUNCTIONS
The calculation of the transverse self-mobility functions is more complicated because oftranslation-rotation coupling, but in principle we can follow the same scheme as in Sec.V. In this section we derive approximations to the self-mobility functions β ttAA ( R, ω ) and β rtAA ( R, ω ). We consider the situation where to zero order sphere A moves with velocity U A, in the x direction.The zero order flow v UAω, ( r ) is given by Eqs. (4.4-6) with on the right hand side thesubscript z replaced by x . Correspondingly the functions in Eq. (4.8) are replaced by u A ( r ) = √ π α a [ v − − P ( r ) − v − P ( r )] , v A ( r ) = 2 √ π e αa α [ v − − N ( r ) − v − N ( r )] . (6.1)It is checked that the flow pattern v UAω, ( r ) satisfies the no-slip boundary condition v UAω, ( r ) (cid:12)(cid:12) r = a = U A, e x . (6.2)To first order sphere B translates and rotates with zero force and torque in the flow pattern v UAω, ( r ). Sphere B translates in the x direction and rotates about the y direction. From theFax´en theorems Eqs. (4.2) and (4.3) we find for the first order translational and rotationalvelocities of sphere BU B, = − U A, a α R A ( αb ) (cid:20) B ( αa ) B ( αb ) − A ( αR ) e α ( a + b − R ) (cid:21) , Ω B, = − U A, a R B ( αb ) (1 + αR ) e α ( a + b − R ) , (6.3)The velocities U B, and Ω B, were taken into account in Eqs. (4.14) and (4.16).At r = R B we have v UAω, ( R B ) = U (0) B e x , U (0) B = U A, a α R (cid:2) − B ( αa ) + A ( αR ) e α ( a − R ) (cid:3) . (6.4)Again we use an expansion of the form Eq. (5.4). The first term is v (0) UAB = c +11 N v +11 N ( r − R B ) + c +1 − N v +1 − N ( r − R B )+ c +11 P v +11 P ( r − R B ) + c +1 − P v +1 − P ( r − R B ) , (6.5)with functions v + lmN ( r ) and v + lmP ( r ) defined in Eq. (5.2).16omparing Eqs. (6.4) and (6.5) we find the relations U (0) B = − c +11 N r π − c +11 P r π ,c +1 − N = − c +11 N , c +1 − P = − c +11 P . (6.6)The Fax´en theorem Eq. (4.2) must hold in the form B ( αb ) U (0) B e x + B ( αb ) b ( ∇ v (0) UAB ) (cid:12)(cid:12)(cid:12)(cid:12) r = R B = A ( αb ) U B, e x . (6.7)Only the first line on the right in Eq. (6.5) contributes to the second term in this equation.Substituting we obtain the value of c +11 N as c +11 N = − r π U A, aα R A ( αR ) e α ( a − R ) . (6.8)From Eq. (6.6) we find c +11 P = r π U A, aα R B ( αa ) . (6.9)The corresponding scattered wave is v − B ( r ) = c +11 N (cid:2) χ NN v − N ( r − R B ) + χ NP v − P ( r − R B ) (cid:3) + (cid:0) c +11 P − r π U B, (cid:1)(cid:2) χ NP v − N ( r − R B ) + χ P P v − P ( r − R B ) (cid:3) + c +1 − N (cid:2) χ NN v − − N ( r − R B ) + χ NP v − − P ( r − R B ) (cid:3) + (cid:0) c +1 − P − r π U B, − (cid:1)(cid:2) χ NP v − − N ( r − R B ) + χ P P v − − P ( r − R B ) (cid:3) , (6.10)where U B, ± are spherical components of U B, e x . The scattered wave acts back on sphere A ,and we can calculate its translational and rotational velocity by use of the Fax´en theoremsEqs. (4.2) and (4.3). The translational velocity yields a contribution to the mobility function β ttAA ( R, ω ) given by β ttAA, ( R, ω ) = (cid:20) α b B ( αa ) B ( αb ) − α b B ( αa ) A ( αR ) e α ( a + b − R ) − (9 − αb + α b ) A ( αR ) e α ( a + b − R ) + 9 A ( αb ) A ( αR ) e α ( a − R ) (cid:21)(cid:30) (cid:20) πηα R A ( αa ) A ( αb ) (cid:21) . (6.11)The function behaves at low frequency as β ttAA, ( R, ω ) = − b πηR + O ( α ) . (6.12)17he rotational velocity yields a contribution to the mobility function β rtAA ( R, ω ) given by β rtAA, ( R, ω ) = 1 + αR πηα R A ( αa ) B ( αa ) A ( αb ) (cid:20) (cid:0) e αb − (cid:1) A ( αb ) A ( αR ) e α ( a − R ) + 4 α b B ( αa ) e α ( a + b − R ) − αbA ( αR ) e α ( a + b − R ) (cid:21) . (6.13)The function vanishes at low frequency as α .The next term in the expansion of the form Eq. (5.4) is v (1) UAB = c +11 M v +11 M ( r − R B ) + c +1 − M v +1 − M ( r − R B )+ c +21 P v +21 P ( r − R B ) + c +2 − P v +2 − P ( r − R B ) . (6.14)The coefficients in Eq. (6.14) are found to be c +1 ± M = − i r π U A, a (1 + αR ) e α ( a − R ) αR ,c +2 ± P = ∓ r π U A, aα R (cid:2) B ( αa ) − B ( αR ) e α ( a − R ) (cid:3) , (6.15)with abbreviation B ( λ ) given in the Appendix. From the theorem Eq. (4.2) we see thatthe flow v (1) UAB does not generate a translational velocity. From the theorem Eq. (4.3) wecalculate the rotational velocity of sphere B , as it rotates in this flow with zero torque. Thisyields the mobility function given by Eq. (4.16).The spherical components of the angular velocity of sphere B areΩ B, ± = − ia √ U A, R B ( αb ) (1 + αR ) e α ( a + b − R ) , Ω B, = 0 . (6.16)The flow generated by sphere B is v − B ( r ) = χ MM (cid:20)(cid:18) c +11 M − r π bg ( αb ) Ω B, (cid:19) v − M ( r − R B )+ (cid:18) c +1 − M − r π bg ( αb ) Ω B, − (cid:19) v − − M ( r − R B ) (cid:21) + χ NP (cid:2) c +21 P v − N ( r − R B ) + c +2 − P v − − N ( r − R B ) (cid:3) + χ P P (cid:2) c +21 P v − P ( r − R B ) + c +2 − P v − − P ( r − R B ) (cid:3) , (6.17)with outgoing waves given by Eqs. (4.9) and (5.24).The wave in Eq. (6.17) acts back on sphere A and we can calculate its resulting trans-lational and rotational velocity by use of the theorems in Eqs. (4.2) and (4.3). From thetranslational velocity we get a contribution to the mobility function β ttAA ( R, ω ) given by β ttAA, ( R, ω ) = − P ( αb )16 πηα R A ( αa ) B ( αb ) (1 + αR ) e α ( a − R ) − b πηα R (1 + αb ) A ( αa ) (cid:20) B ( αa ) − B ( αR ) e α ( a − R ) (cid:21) × (cid:20) B ( αa ) A ( αb ) − B ( αR ) e α ( a + b − R ) (cid:21) , (6.18)18he first term derives from the the first two lines in Eq. (6.17). At low frequency thefunction behaves as β ttAA, ( R, ω ) = − a b + 3 a b πηR + O ( α ) . (6.19)Here the first term in Eq, (6.18) does not contribute. From the rotational velocity we get acontribution to the mobility function β rtAA ( R, ω ) given by β rtAA, ( R, ω ) = − b πηα R (1 + αb ) B ( αR ) e α ( a + b − R ) A ( αa ) B ( αa ) (cid:20) B ( αa ) − B ( αR ) e α ( a − R ) (cid:21) . (6.20)At low frequency this behaves as β rtAA, ( R, ω ) = 5 a b πηR + O ( α ) . (6.21)We must also consider the second order term v (2) UAB . This turns out to be given by v (2) UAB = c +21 M v +21 M ( r − R B ) + c +2 − M v +2 − M ( r − R B )+ c +31 P v +31 P ( r − R B ) + c +3 − P v +3 − P ( r − R B ) , (6.22)with coefficients c +2 ± M = 3 ia α R r π U A B ( αR ) e α ( a − R ) ,c +3 ± P = ± aα R r π U A (cid:20) B ( αa ) − B ( αR ) e α ( a − R ) (cid:1)(cid:21) , (6.23)with abbreviation B ( λ ) given in the Appendix.The wave generated by sphere B is given by v − B ( r ) = c +21 M χ MM v − M ( r − R B ) + c +2 − M χ MM v − − M ( r − R B )+ c +31 P (cid:2) χ NP v − N ( r − R B ) + χ P P v − P ( r − R B )]+ c +3 − P (cid:2) χ NP v − − N ( r − R B ) + χ P P v − − P ( r − R B )] . (6.24)From this wave acting on sphere A located at the origin we can evaluate the resultingtranslational velocity U A, and rotational velocity Ω A, by use of the Fax´en theorems in Eqs.(4.2) and (4.3). This yields a contribution to the mobility function β ttAA, ( R, ω ) = (cid:20) αb (cid:0) B ( αa ) A ( αb ) − e α ( a + b − R ) B ( αR ) (cid:1)(cid:0) − B ( αa ) + B ( αR ) e α ( a − R ) (cid:1) + 675 P ( αb ) e α ( a − R ) R B ( αR ) (cid:21)(cid:30)(cid:2) πηα R A ( αa ) B ( αb ) (cid:3) , (6.25)with abbreviation A ( λ ) given in the Appendix. At low frequency the function has theexpansion β ttAA, ( R, ω ) = − a b − a b + 70 a b R + 15 b R − b R πηR + O ( α ) . (6.26)19he zero frequency limit, in combination with Eqs. (6.12) and (6.19), agrees to terms oforder R − with earlier results [3],[4],[5].In our approximation the mobility function β ttAA ( R, ω ) is given by β ttAA ( R, ω ) = 1 ζ tA ( ω ) + β ttAA, ( R, ω ) + β ttAA, ( R, ω ) + β ttAA, ( R, ω ) , (6.27)with the last three contributions in Eqs. (6.11), (6.18), and (6.25). The term linear in α in the expansion of this quantity in powers of frequency agrees with the exact relation Eq.(2.12), and arises only from the first term. The interaction terms decrease with distance as1 /R in the zero frequency limit.The rotational velocity Ω A, yields a contribution to the mobility function β rtAA , β rtAA, ( R, ω ) = (cid:20) R P ( αb ) B ( αR ) B ( αR ) e α ( a − R ) + 14 α b A ( αR ) (cid:0) B ( αa ) − e α ( a − R ) B ( αR ) (cid:1) e α ( a + b − R ) (cid:21)(cid:30) (cid:2) πηα R A ( αa ) B ( αa ) B ( αb ) (cid:3) . (6.28)At low frequency this has the expansion β rtAA, ( R, ω ) = − a b + 3 b R πηR + O ( α ) . (6.29)The zero frequency limit, in combination with Eq. (6.21), agrees to terms of order R − withearlier results [4],[5]. In our approximation the mobility function β rtAA ( R, ω ) is given by β rtAA ( R, ω ) = β rtAA, ( R, ω ) + β rtAA, ( R, ω ) + β rtAA, ( R, ω ) , (6.30)with the three contributions in Eqs. (6.13), (6.20), and (6.28). VII. TRANSVERSE ROTATIONAL SELF-MOBILITY FUNCTIONS
In this section we derive approximations to the self-mobility functions β trAA ( R, ω ) and β rrAA ( R, ω ). In the primary flow sphere A rotates about the x axis, with flow pattern givenby Eq. (4.18) with C z replaced by C x = e x × e r . The zero order flow v Ω Aω, ( r ) is givenby Eqs. (4.18-20) with subscript z replaced by x . We expand this into multipole flows byuse of a Taylor expansion about the point R B in a form similar to Eq. (5.21), v Ω Aω, ( r ) = ∞ X n =0 v ( n )Ω AB , (7.1)where n indicates the power of r − R B in the Taylor expanson. The first term is v (0)Ω AB = c +11 N v +11 N ( r − R B ) + c +1 − N v +1 − N ( r − R B )+ c +11 P v +11 P ( r − R B ) + c +1 − P v +1 − P ( r − R B ) . (7.2)20t r = R B we have v Ω Aω, ( R B ) = U (0) B e y , U (0) B = − a R Ω A, αR αa e α ( a − R ) . (7.3)Comparing this with Eq. (7.2) we find the relations U (0) B = − i (cid:20) c +11 N r π + c +11 P r π (cid:21) ,c +1 − N = c +11 N , c +1 − P = c +11 P . (7.4)Sphere B acquires a first order velocity U B, in the y direction with value U B, = − a R Ω A, αR (1 + αa ) A ( αb ) e α ( a + b − R ) , (7.5)as accounted for in Eq. (4.25).The Fax´en theorem Eq. (4.2) can be applied in the form B ( αb ) U (0) B e y + B ( αb ) b ( ∇ v (0)Ω AB ) (cid:12)(cid:12)(cid:12)(cid:12) r = R B = A ( αb ) U B, e y . (7.6)Only the first line on the right in Eq. (7.2) contributes to the second term in this equation.Substituting we obtain the value of c +11 N as c +11 N = 2 i r π A, α a R (1 + αR )(1 + αa ) Q ( αR ) , (7.7)with abbreviation Q ( λ ) given in the Appendix. The coefficient c +11 P is found from Eqs. (7.3)and (7.4) as c +11 P = − i r π A, a e αa (1 + αR ) (cid:2) Q ( αR ) e − αR + 4 α R (cid:3) R (1 + αa ) Q ( αR ) . (7.8)The wave v − B ( r ) scattered by sphere B has formally the same expression as Eq. (6.10).The scattered wave acts back on sphere A and we can calculate its translational and rota-tional velocity by use of the Fax´en theorems Eqs. (4.2) and (4.3). The translational velocityyields a contribution to the mobility function β trAA ( R, ω ) given by β trAA, ( R, ω ) = 1 + αR πηα R A ( αa ) B ( αa ) A ( αb ) Q ( αR ) × (cid:20) b (cid:2) A ( αR ) e α ( a + b − R ) − B ( αa ) B ( αb ) (cid:3) Q ( αR ) e α ( a + b − R ) + 2 α R A ( αb ) (cid:0) e αb − (cid:1) A ( αR ) e α (2 a − R ) − bA ( αb ) (cid:0) Q ( αR ) e − αR + 4 α R (cid:1) A ( αR ) e α (2 a + b − R ) + bA ( αb ) B ( αa ) e αa (cid:18) B ( αb ) Q ( αR ) e − αR + 4 B ( αb ) α R − e αb α R (cid:19)(cid:21) . (7.9)21he expression vanishes at zero frequency. The rotational velocity yields a contribution tothe mobility function β rrAA ( R, ω ) given by β rrAA, ( R, ω ) = (1 + αR ) e α ( a − R ) πηR A ( αb ) B ( αa ) Q ( αR ) × (cid:20) A ( αb ) (cid:18) α R e αa (cid:0) − e αb (cid:1) + 3 bQ ( αR ) e α ( a + b − R ) + 4 α bR e α ( a + b ) (cid:19) − bQ ( αR ) e α ( a +2 b − R ) (cid:21) . (7.10)The expression vanishes at zero frequency.At zero frequency the mobility functions β trAA ( R,
0) and β rrAA ( R,
0) do not vanish. There-fore we consider also the term v (1)Ω AB in the expansion Eq. (7.1). This takes the form of Eq.(6.14) with new coefficients, which are found to be c +1 ± M = ± r π A, a αR A ( αR )1 + αa ,c +2 ± P = i r π
10 Ω A, a R B ( αR )1 + αa . (7.11)The flow generated by sphere B takes the form of Eq. (6.17) with these coefficients andwith spherical components of the angular velocity of sphere B given byΩ B, ± = ± √ A, a A ( αR ) R (1 + αa ) B ( αb ) e α ( a + b − R ) , Ω B, = 0 . (7.12)The scattered wave acts back on sphere A and we can calculate its translational and rota-tional velocity by use of the Fax´en theorems Eqs. (4.2) and (4.3). The translational velocityyields a contribution to the mobility function β trAA ( R, ω ) given by β trAA, ( R, ω ) = a e α ( a − R ) πηα R A ( αa ) B ( αa )(1 + αb ) B ( αb ) × (cid:20) αb B ( αa ) B ( αb ) A ( αb ) B ( αR ) − αb B ( αb ) B ( αR ) B ( αR ) e α ( a + b − R ) − R (1 + αb )(1 + αR ) A ( αR ) P ( αb ) e α ( a − R ) (cid:21) . (7.13)At low frequency the function behaves as β trAA, ( R, ω ) = − a b + 3 b πηR + O ( α ) . (7.14)The rotational velocity yields a contribution to the mobility function β rrAA ( R, ω ) given by β rrAA, ( R, ω ) = − πηα R B ( αa ) (1 + αb ) B ( αb ) (cid:20) α b B ( αb ) B ( αR ) e α (2 a + b − R ) + (1 + αb ) P ( αb ) A ( αR ) e α ( a − R ) (cid:21) . (7.15)22t low frequency the function behaves as β rrAA, ( R, ω ) = − b πηR + O ( α ) . (7.16)We must also consider the second order term v (2)Ω AB . This turns out to be given by theexpression on the right hand side of Eq. (6.22) with coefficients c +2 ± M = ∓ r π A, a α R (1 + αa ) B ( αR ) e α ( a − R ) ,c +3 ± P = − i r π
21 Ω A, a R (1 + αa ) A ( αR ) e α ( a − R ) . (7.17)The wave generated by sphere B is given by the expression on the right hand side of Eq.(6.24) with the present values for the coefficients.From this wave acting on sphere A located at the origin we can evaluate the resultingtranslational velocity U A, and rotational velocity Ω A. by use of the Fax´en theorems in Eqs.(4.2) and (4.3). The translational velocity U A, yields a contribution to the β trAA mobilityfunction β trAA, ( R, ω ) = (cid:20) R P ( αb ) B ( αR ) B ( αR ) e α ( a − R ) − α b B ( αR ) A ( αR ) e α (2 a + b − R ) + 6 α b B ( αa ) A ( αb ) A ( αR ) e α ( a − R ) (cid:21)(cid:30) (cid:2) πηα R A ( αa ) B ( αa ) B ( αb ) (cid:3) . (7.18)At low frequency the function has the expansion β trAA, ( R, ω ) = − a b + 15 b − b R πηR + O ( α ) . (7.19)The zero frequency limit, in combination with Eq. (7.14), agrees to terms of order R − withearlier results [4],[5]. Note that the sum of Eqs. (6.21) and (6.29) is the negative of the sumof Eqs. (7.14) and (7.19) to terms of order R − .A comparison of the expression for β rtAA in Eq. (6.30) with the analogous result for β trAA which follows from Eqs. (7.9), (7.13), and (7.18) shows that in the approximation thesymmetry relation β rtAA = − β trAA is not satisfied. However, it is satisfied in low powers of α and R − . This suggests that the expansion must be carried to sufficiently high order in r − R B to verify the symmetry.The rotational velocity Ω A, yields a contribution to the β rrAA mobility function β rrAA, ( R, ω ) = e α (2 a + b − R ) πηα R B ( αa ) B ( αb ) (cid:20) e − αb P ( αb ) B ( αR ) − α b A ( αR ) (cid:21) . (7.20)At low frequency this has the expansion β rrAA, ( R, ω ) = − b πηR + O ( α ) . (7.21)The zero frequency limit, in combination with Eq. (7.16), agrees to terms of order R − with earlier results [4],[5]. The interaction terms decrease with distance as 1 /R in the zerofrequency limit. 23 III. DISCUSSION OF RESULTS
In the above we derived detailed analytic results for the scalar mobility functions of twospheres immersed in a viscous incompressible fluid, based on the approximation of a singlepropagator between the spheres for the mutual functions, and a single reflection with twopropagators for the self-functions. Finite-size corrections and sphere rotations were takeninto account. It is clear that for large and intermediate distances the mutual functionsdominate. By their nature the self-functions are analytically complicated. In practicalapplication one will often restrict oneself to the relatively simple expressions for the mutualfunctions. The expressions for the self-functions will allow an estimate of the error onecommits in doing this. The derivation presented in Sec. II shows how to find the motionof the spheres from the mobility functions. In the following we present some numericalestimates for the mobility functions.First we compare the difference of the calculated translational mutual functions with thesimple point approximation discussed in Sec. III. We consider two equal spheres of radius a at center-to-center distance R = 5 a . As typical frequency we consider the value αa = 1.From Eqs. (3.3) and (4.12) we then find the ratio α ttAB, /G k = 1 . R = 10 a the ratio is larger, and equal to 1 . β ttAB, /G ⊥ = 1 . R = 10 a the ratio is 1 . α ttAA, , and α ttAA, , given by Eqs. (5.17) and(5.19) with the single sphere value 1 /ζ tA , given by Eq. (4.1). For R = 5 a and αa = 1 wefind α ttAA, ζ tA = − . α ttAA, ζ tA = − . β ttAA, ζ tA = 0 . β ttAA, ζ tA = − . β ttAA, ζ tA = − . α rrAA, and α rrAA, , given by Eqs. (5.31) and(5.33) with the single sphere value 1 /ζ rA , given by Eq. (4.1). For R = 5 a and αa = 1 wefind α rrAA, ζ rA = − . × − and α rrAA, ζ rA = − . × − , showing that the correction termsare small. Similarly we find from Eqs. (7.10), (7.15) and (7.20) β rrAA, ζ rA = − . × − , β rrAA, ζ rA = − . × − , and β rrAA, ζ rA = − . × − . Again these values are small.It follows from Eqs. (4.16) and (4.25) that there are translation-rotation couplings oflong range, provided the frequency is not too high. At higher frequency these couplings areexponentially screened.We investigate the frequency-dependence of the tt mutual mobility functions in somemore detail. It follows from Eqs. (4.12) and (4.14) that at large distance the in-teraction is dipolar with a frequency-dependent amplitude factor given by the ratio B ( αa ) B ( αb ) / ( A ( αa ) A ( αb )). At zero frequency this takes the value unity, but at highfrequency the factor tends to the value 9, implying a significant dependence on frequency.The factor also characterizes the deviation from the simple Green function approximationEq. (3.8).We note that at zero frequency the translational mutual mobility functions are given by α ttAB, ( R,
0) = 14 πηR − a + b πηR ,β ttAB, ( R,
0) = 18 πηR + a + b πηR . (8.1)24his is just the Rotne-Prager hydrodynamic interaction [40],[41], generalized to unequalspheres. Thus the functions α ttAB, ( R, ω ) and β ttAB, ( R, ω ) may be regarded as generalizationsof the Rotne-Prager interaction to finite frequency. Similarly, Eqs. (4.21) and (4.23) aregeneralizations to non-zero frequency of the dipolar expression at zero frequency [4],[5].At fixed center-to-center distance R all mobility functions depend strongly on frequency.This must be taken into account in applications. In particular, in the determination of thedynamic modulus of viscoelastic fluids via observed motion of a pair of particles [33], it isnecessary to use an accurate approximation to the retarded hydrodynamic interactions.Finally, we discuss briefly the effect of added mass. We restrict attention to motion alongthe axis of centers. It follows from Eqs. (2.7), (2.8), and (2.14) that in this case translationaland rotational motions decouple. The friction coefficients for translational motion are givenby ζ tt k AA ( R, ω ) = α ttBB α ttAA α ttBB − α ttAB , ζ tt k AB ( R, ω ) = − α ttAB α ttAA α ttBB − α ttAB . (8.2)This shows that in the friction coefficients the mutual mobility function α ttAB ( R, ω ) entersin the form of a geometric series. At large distance R the mutual term α ttAB ( R, ω ) is smallin comparison with α ttAA ( R, ω ) and α ttBB ( R, ω ), which tend to µ tA = 1 /ζ tA and µ tB = 1 /ζ tB ,respectively. At high frequency the mutual term tends to zero asymptotically for large R as( a + b ) /R , at low frequency it tends to zero as ( a + b ) /R .At fixed R the denominators in Eq. (8.1) tend to zero at high frequency as 1 /α , andthe numerators tend to zero as 1 /α , so that the friction coefficients increase in proportionto frequency, corresponding to an added mass in the equations of motion. The added massdepends on the distance R . For transverse motions there is a similar added mass effect, butthe situation is more complicated due to the translation-rotation coupling.The added mass effect implies that at short time t = 0+ the velocity correlation function C ttAB ( t ) of the two spheres does not vanish. The velocities are correlated, in contrast to theprediction of equipartition. The effect is similar to that studied by Zwanzig and Bixon [42] fora single sphere. The effect was seen in the computer simulation of Tatsumi and Yamamoto[32]. As discussed above, the use of the approximation Eq. (3.8) leads to prediction of thecorrelation effect smaller by a factor 9 than found by use of Eqs. (4.12) and (4.14). IX. MUTUAL MOBILITY FUNCTIONS FOR MIXED SLIP-STICK BOUNDARYCONDITIONS
The derivation of Sec. IV can be extended without difficulty to the case where the fluidvelocity satisfies mixed slip-stick boundary conditions on the surface of the two spheres.First we cast the known Fax´en theorems for this case in the convenient form of Eqs. (4.2)and (4.3). Albano et al. [43] derived the Fax´en theorem for the force in terms of surfaceand volume averages of the incident flow. By use of the identities for the averages derivedby Kim and Karrila [8] the theorem can be cast in the convenient form K A = 6 πηa (cid:2) B ( αa, ξ A ) v + B ( αa, ξ A ) a ∇ v (cid:3)(cid:12)(cid:12) r = − ζ tA ( ω, ξ A ) U A , (9.1)with slip parameter ξ , as defined earlier [35], and with functions as defined in the Appendix.The value ξ = 0 corresponds to no-slip, and the value ξ = 1 / T A = 4 πηa (1 − ξ A ) e αa αa + ξ A α ( ∇ × v ) (cid:12)(cid:12) r = − ζ rA ( ω, ξ A ) Ω A . (9.2)The primary flow of sphere A moving with velocity U A, in the z direction is given by [12] v UAω, ( r ) = U A, (cid:20) B ( αa, ξ A )2(1 + ξ A αa ) α a u A ( r ) + 12 αa − ξ A ξ A αa v A ( r , α ) (cid:21) , (9.3)and the primary flow of sphere A rotating with angular velocity Ω A, about the z directionis given by [12] v Ω Aω, ( r ) = Ω A, a (1 − ξ A ) k ( αr ) k ( αa ) + ξ A e − αa C z (ˆ r ) . (9.4)The derivation proceeds as in Sec. IV. We only quote the final expressions for the mutualmobility functions in one-propagator approximation. The longitudinal tt mutual mobilityfunction is given by α ttBA, ( R, ω ) = (1 + ξ A αa )(1 + ξ B αb ) B ( αa, ξ A ) B ( αb, ξ B ) − (1 − ξ A )(1 − ξ B )(1 + αR ) e α ( a + b − R ) πηα R (1 + ξ A αa )(1 + ξ B αb ) A ( αa, ξ A ) A ( αb, ξ B ) . (9.5)The transverse tt mutual mobility function is given by β ttBA, ( R, ω ) = − (1 + ξ A αa )(1 + ξ B αb ) B ( αa, ξ A ) B ( αb, ξ B ) + (1 − ξ A )(1 − ξ B ) A ( αR ) e α ( a + b − R ) πηα R (1 + ξ A αa )(1 + ξ B αb ) A ( αa, ξ A ) A ( αb, ξ B ) . (9.6)These functions satisfy reciprocity. The term linear in α in an expansion in powers of α forboth functions is given by − α/ (6 πη ), as for no-slip. The longitudinal rt mutual mobilityfunction vanishes, and the transverse one is given by β rtBA ( R, ω ) = 1 − ξ A ξ A αa πηR A ( αa, ξ A ) B ( αb ) (1 + αR ) e α ( a + b − R ) . (9.7)The longitudinal tr mutual mobility function vanishes, and the transverse one is found tobe β trBA ( R, ω ) = − − ξ B ξ B αb πηR B ( αa ) A ( αb, ξ B ) (1 + αR ) e α ( a + b − R ) . (9.8)in agreement with the reciprocity relation Eq. (2.15).Both the longitudinal and transverse rr mutual mobility functions are found to be inde-pendent of the slip coefficients ξ A , ξ B . They are therefore given by Eqs. (4.21) and (4.23).Earlier we found this remarkable property at zero frequency [4],[5]. X. CONCLUSIONS
The analytic expressions for retarded hydrodynamic interactions between two sphereswill be useful in qualitative work in situations where the spheres are not very close. Existingschemes by which the interactions can be calculated accurately, also at short distances [20]-[22], are cumbersome and not easily implemented. In particular the simple expressions forthe mutual interactions, which we derived in Secs. IV and IX, will find practical use.26he retarded interactions are of interest in situations on the timescale of velocity relax-ation, as well as in situations where oscillating forces or torques are applied. The theoryof velocity relaxation and correlation functions can be applied in the discussion of com-puter simulations of Brownian motion [32]. Oscillating forces occur in the numerical studyof swimming of two unequal spheres [44]-[46]. Our theoretical analysis of this model tookaccount of fluid inertia only as an added mass effect [47]. It will be of interest to applythe present results to this model. The theory can also be applied to the three-sphere swim-mer [48]-[51], provided the hydrodynamic interaction can be approximated as a sum of pairinteractions.The mobility functions involving rotation are of interest in the application to ferrofluids[52]. The long range of the hydrodynamic interaction suggests that it is relevant formagnetic relaxation.
Acknowledgment
I thank Prof. R. B. Jones for a critical reading of the manuscript.27 ppendix A:
In this Appendix we list the abbreviations used in the main text, A ( λ ) = 1 + λ + 19 λ , A ( λ ) = 1 + λ + λ ,A ( λ ) = 15 + 15 λ + 6 λ + λ , A ( λ ) = 105 + 105 λ + 45 λ + 10 λ + λ ,B ( λ ) = 1 + λ + 13 λ , B ( λ ) = λ − [ e λ − B ( λ )] ,B ( λ ) = 6 + 6 λ + 3 λ + λ , B ( λ ) = 45 + 45 λ + 21 λ + 6 λ + λ ,P ( λ ) = 3 B ( λ ) − (3 − λ + λ ) e λ , Q ( λ ) = A ( λ ) − (1 − λ + λ ) e λ . (A1)These functions are relevant for no-slip boundary conditions. In Sec. IX we use relatedfunctions for mixed slip-stick boundary conditions with slip parameter ξ . These functionsare A ( λ, ξ ) = 11 + ξλ (cid:20) A ( λ ) − ξ (cid:18) λ − λ (cid:19)(cid:21) ,B ( λ, ξ ) = 11 + ξλ (cid:20) B ( λ ) − ξ (cid:18) λ − λ (cid:19)(cid:21) ,B ( λ, ξ ) = B ( λ ) + ξ (1 + λ )(1 + λ − e λ ) λ (1 + ξλ ) . (A2)The no-slip condition corresponds to slip parameter ξ = 0, and perfect slip corresponds tothe value ξ = 1 /
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