Transword Research Network 37/66 (2), Fort P.O., Trvandrum-695 023, Keraa, Inda Theoretca Methods for Mcro Scae Vscous Fows, 2009: 27-72 ISBN: 978-8-7895-400-4 Edtors: Franços Feuebos and Antone Seer 6 Precse mutpoe method for cacuatng hydrodynamc nteractons between spherca partces n the Stokes fow M. L. Eke-Jeewska and E.Wajnryb Insttute of Fundamenta Technoogca Research, Posh Academy of Scences Pawnskego 5B, 02-06 Warsaw, Poand. Introducton Dynamcs of mcro-partces n water-based fuds and macroscopc propertes of such dspersve systems are of fundamenta mportance for numerous boogca and ndustra appcatons [-7]. Typcay, for such systems fud nerta s rreevant, and the mechansms of ocomoton as we as basc features of the partce dynamcs dffer sgnfcanty from those vad at the macroscopc scae. Fud moton at the mcro-scae usuay satsfes Correspondence/Reprnt request: Dr. M. L. Eke-Jeewska, Insttute of Fundamenta Technoogca Research Posh Academy of Scences, Pawnskego 5B, 02-06 Warsaw, Poand. E-ma: meke@ppt.gov.p
28 M. L. Eke-Jeewska & E.Wajnryb the statonary Stokes equatons, whch have to be suppemented by approprate boundary condtons at the partce surfaces and at the nterfaces, whch confne the fud [6, 7, 8]. Effcent and accurate methods of sovng these equatons are necessary to nvestgate Stokesan dynamcs of mcro-partces and to determne structure and effectve transport propertes of dspersve systems. For spherca non-deformabe mcro-partces, the method of nterest s the spherca-mutpoe expanson, corrected for ubrcaton [9, 0]. Its man advantage s the hgh accuracy, whch s controed by the choce of the mutpoe order of the truncaton. Moreover, the method has been mpemented numercay, and the HYDROMULTIPOLE codes have been extensvey tested and apped to many physca systems, wth varous types of partces and nterfaces confnng the fud. In ths paper, an outne of the spherca-mutpoe expanson apped to the statonary Stokes equatons s gven, based on the agorthm deveoped by Cchock, Federhof, Jones, Schmtz and coaborators. In Sec. 2, geometry of the system s specfed, as we as boundary condtons at the partce surfaces and at the nterfaces, whch confne the fud. The generazed frcton and mobty probems for the partces are formuated. The Stokes probem for the fud fow s transformed nto a set of boundary ntegra equatons for the force densty at the partce surfaces. In Sec. 3, these equatons are soved by projectng onto a compete set of mutpoe functons, and truncatng at a certan mutpoe order L, wth the detas expaned n the Appendces. For cose partces n reatve moton, the mutpoe expanson s sowy convergent wth the ncreasng L. Therefore n Sec. 4, the mutpoe expanson s corrected for ubrcaton and some estmates of the precson are gven. In Sec. 5, modfcatons needed to descrbe moton of partce congomerates are ponted out. Fnay, Sec. 6 contans exampes of appcatons. 2. Fud, partces and boundares Consder N partces mmersed n a vscous fud. Assume that the partces are non-deformabe. Imagne that externa (non-hydrodynamc) forces and torques are apped to each partce =,..., N, and there exst an ambent fud fow v (r). Each partce =,..., N has a spherca shape of radus a, and moves wth a transatona and rotatona veocty, U and. The resutng fud fow s characterzed by a very sma Reynods number Re and the fud nerta effects are neggbe [6, 7]. Here Re s the product of a partce veocty and ts radus, dvded by the fud knematc vscosty. The Stokes number Sk s aso much smaer than unty and the partce nerta s aso rreevant. Here Sk s the product of the partce veocty, ts radus, and the arger of the partce and the fud denstes, dvded by the
Mutpoe method for cacuatng hydrodynamc nteractons 29 fud dynamc vscosty []. The Pecet number s arge, Pe, and the Brownan moton s rreevant []. Here Pe s the product of a partce veocty and ts radus, dvded by the dffuson constant. Moreover, the Strouha number St s not much arger than unty, and the fud fow s statonary [3]. Here St s the rato of the characterstc frequency of the fud veocty varatons and the fud veocty, mutped by the characterstc dmenson. For such a system, the hydrodynamc frcton forces and torques exerted by the movng partces on the fud are equa to the externa forces and torques mposed on the partces. In the generazed frcton probem, the queston s what are and, f the partce transatona U j and rotatona j veoctes and the ambent fow v (r) are gven. In the generazed mobty probem,, and v (r) are known whe U j and j are searched for. Sovng one of these probems (or a mxed one) corresponds to evauaton of hydrodynamc nteractons between the partces. In ths paper, t w be outned how to appy the spherca-mutpoe method to sove the generazed frcton and mobty probems [9, 0] and evauate the fud fow. 2.. Basc equatons for the fud fow For the system specfed above, the fud veocty v and pressure p satsfy the statonary Stokes equatons [6, 7], 2 v p = 0, () v = 0, (2) where s the fud dynamc vscosty. The above set of parta dfferenta equatons has to be suppemented by the correspondng boundary condtons at the surface S of each spherca partce =,..., N, and at the nterfaces, whch confne the fud. The ambent-fow veocty v and pressure p satsfy the Stokes equatons () n the absence of partces. When the group of N partces s mmersed, t affects the surroundng fud, but there s no change far away from the partces. For an unbounded fud, the correspondng boundary condton at nfnty reads, v(r) v (r) 0, for r. (3) For a confned fud, both the ambent and the actua fows, v (r) and v(r), have to satsfy the proper boundary condtons at the nterfaces. The mutpoe method has been deveoped and apped for varous geometres: 3D
30 M. L. Eke-Jeewska & E.Wajnryb or 2D-perodc boundary condtons [2-8] and for a fud mted by one [9] or two parae fat nterfaces [20]. Such an nterface may be a hard (sod) wa [2, 22], a free surface [23, 24], or a fud-fud boundary, wth or wthout a surfactant [25, 26]. These boundary condtons are expcty sted n the next secton. A of them can be aso apped at the partce surfaces, and the mutpoe method has been deveoped to descrbe non-deformabe spherca partces made of sod, fud or gas, wth the cean surface or covered wth a surfactant [27-29]. For carty of presentaton of the basc concepts, ths paper s many focused on sod partces and the stck boundary condtons on ther surfaces, v(r) = w (r) U + (r R ), for r S, =,..., N, (4) where R stands for the poston of the center of partce. Modfcatons needed to descrbe hydrodynamc nteractons between other types of partces and nterfaces w be mentoned; the fu treatment can be found e.g. n Refs. [27-29]. 2.2. Boundary condtons The mutpoe method has been deveoped for the foowng boundary condtons at the surface I confnng the fud and at the partce surfaces S, =...N. If the fud s n contact wth a smooth sod surface, the stck (or no-sp) boundary condtons appy. The fud veocty at the surface S of a sod spherca partce s equa to ts rgd veocty, v(r) = w (r), as n Eq. (4). If the fud n a haf-space z > 0 s mted at z = 0 by a fat surface I, whch s the motoness hard wa, the stck boundary condtons at I have the form, v(r) = 0, for r = (x, y, 0). (5) The above mode can be generazed, aowng for a sp at the boundary. The mxed stck-sp boundary condtons at the partce have the form [30], n v(r) = n w (r), for r S, (6) t (v(r) w (r)) = ( /) t (r) n, for r S, (7) where n = (r R )/ r R and t are the unt vectors norma and tangenta to the partce surface S. The frst condton, Eq. (6), expresses the fact that no
Mutpoe method for cacuatng hydrodynamc nteractons 3 fud passes through the spherca surface. The second one, Eq. (7), states that the tangenta component of the force exerted by the fud on the unt surface of the sphere s proportona to the sp of the oca tangenta veocty,.e. to the dfference between the fud-fow and the partce-surface veoctes. The Cartesan components of the fud stress tensors are gven by the reaton, ( ) p. (8) The vaue of the sp parameter = 0 n Eq. (7) corresponds to the stck boundary condtons, the vaue = to the perfect sp. We consder now the boundary condtons at an nterface between two fuds wth dfferent vscostes [3]. We assume that the nterface s not deformabe owng to a very hgh surface tenson. The spherca surface of a dropet wth an nterna vscosty dfferent from the vscosty of the host fud s descrbed by the condton that the norma components of the fows v(r) and v(r) outsde and nsde the dropet are the same and equa to the norma component of the dropet veocty U, n v(r) = n U, for r S, (9) n v(r) = n U, for r S. (0) Ths condton expresses the fact that no fud passes through the dropet non-deformabe surface. Moreover, the tangenta veocty and tangenta stress are contnuous, t v(r) = t v(r), for r S, () t (r) n = t (r) n, for r S, (2) where the Cartesan components of the fud stress tensors nsde the dropet are gven by ( ) p. (3) Notce that the rotatona veocty s not a reevant varabe n the descrpton of the dropet moton, and t has to be excuded from the frcton and mobty probems, formuated n Sec. 2.3. The boundary condtons at the dropet surface can be easy modfed to descrbe a fat fud-fud nterface at z = 0, when the fud n a haf-space
32 M. L. Eke-Jeewska & E.Wajnryb z > 0 has the vscosty, and the fud on the other sde of the nterface has the vscosty, n v(r) = n v(r) = 0, for r = (x, y, 0), (4) t v(r) = t v(r), for r = (x, y, 0), (5) t (r) n = t (r) n, for r = (x, y, 0). (6) where v(r) and v(r) are the fows on the z > 0 and z < 0 sdes of the nterface. For a fud-fud nterface covered wth an ncompressbe surfactant the tangenta stress contnuty reatons (2) and (6) are repaced by the condton that the fow aong the nterface s ncompressbe, s v s =0, (7) where s s the gradent operator aong the nterface and v s s the tangenta component of the fow veocty. It s of speca nterest to consder a gas-qud nterface (free surface). The correspondng boundary condtons can be obtaned form those specfed above n the mt = 0. Across a free surface between a gas and a qud, there s no qud fow and the tangenta stress s equa to zero at the nterface. For a bubbe, n v(r) = n U, for r S, (8) t (r) n = 0, for r S (9) For a fud n a haf-space z > 0, mted by a fat free surface at z = 0, the free boundary condtons have the form, n v(r) = 0, for r = (x, y, 0), (20) t (r) n = 0, for r = (x, y, 0). (2) 2.3. Frcton and mobty of the partces Aternatvey to the boundary condtons for the (unknown) fud veocty, the effect of the suspended partces on the surroundng fud can be aso descrbed n terms of the dstrbuton of the (unknown) nduced forces
Mutpoe method for cacuatng hydrodynamc nteractons 33 f = (f,..., f N ), exerted by the partces on the fud [30-32]. For the stck boundary condtons, f (r) = ( r R a ) (r) n, where s the stress tensor and n s the unt vector norma to the partce surface, pontng nto the fud. The subsequent moments (ntegras) of the force densty are the forces, torques, stressets S exerted by the sphere on the fud, 3 3 3 d d d f () r r, ( r R ) f () r r, S ( r R ) f () r r,... (22) where the bar over a tensor denotes ts symmetrc traceess part, and the hgher moments are ndcated by the dots. For a spherca partce, the Cartesan components of the stresset are gven as, S [( ) ( )] 3., r R,,, r R, d r 2 f f (23) In the foowng, we combne the forces, torques and stressets nto = (,..., N ), = (,..., N ) and S = (S,..., S N ), respectvey. In anaogy, we represent the partce transatona and rotatona veoctes as U = (U,...,U N ) and = (,..., N ), respectvey. In a smar way, v = (v,..., v N ), = (,..., N ), g = (g,..., g N ),... denote the ambent fow veoctes, ther gradents and hgher dervatves, taken at the center of each sphere, wth v = v (R ), = 2 v (r) r=r and g, 2 [ v ( r ) v ( r )] rr. Owng to nearty of the Stokes equatons ()-(2),, and S depend neary on v U, and g, tt tr td.. v U rt rr rd.., S dt dr dd.. g : : : : : : (24) where the components pq wth p, q = t, r, d,... form the generazed frcton (or grand resstance) symmetrc tensor [7, 33, 34, 35]. Each component pq conssts of the eements pq j wth, j =...N. Each of them depends on the confguraton of a the partces. In partcuar, the eements wth p, q = t, r form the 6N 6N frcton tensor for N spherca partces, n bref denoted as,
34 tt tr. rt rr M. L. Eke-Jeewska & E.Wajnryb (25) Evauaton of the generazed frcton tensor s essenta to sove the generazed frcton probem [6, 34],.e. to determne the hydrodynamc frcton forces and torques exerted by the partces on the fud, f ther moton and the ambent fud fow are known. On the other hand, f the hydrodynamc frcton forces and torques exerted by the partces on the fow and the ambent fud fow are known, the partce moton s determned by sovng the generazed mobty probem [6, 34], tt tr td v.. U rt rr rd..., S dt dr dd.. g : : : : : : (26) wth the use of the generazed (grand) mobty tensor, whch conssts of the eements µ pq, p, q = t, r, d,... and depends on the confguraton of a the partces. The eements wth p, q = t, r ony form the mobty tensor for N spherca partces, denoted as µ, tt tr. rt rr (27) Note that µ s the nverse of,, (28) but the generazed mobty tensor s obtaned by ony a parta nverse of the generazed frcton tensor. In many appcatons, the queston s what s the partce moton under gven externa forces and torques, and an ambent fow v (r). Souton of ths probem s constructed from Eq. (24), whch s now rewrtten as, U, (29)
Mutpoe method for cacuatng hydrodynamc nteractons 35 where and are hydrodynamc forces and torques exerted by the motoness partces on the fud n the presence of an ambent fow, but the absence of externa forces, v tt tr td.... rt rr rd.. g : (30) Eq. (29) s now soved for the partce transatona and anguar veoctes, v U C, (3) g : where C s the convecton operator [36], C tt tr td... rt rr rd.. (32) Evauaton of the generazed frcton and mobty tensors s based on sovng the boundary ntegra equaton by the mutpoe expanson. These procedure w be outned n the foowng sectons. 2.4. Boundary ntegra equaton Wth the use of the nduced forces [30-32], the set of parta dfferenta equatons ()-(2) for the fud veocty and pressure can be transformed nto a set of boundary ntegra equatons for the densty of the nduced forces f. Ths procedure w be outned beow. The fud fow fed outsde the partces can be represented as [9] N j 3 vr () v () r Trr (, ) f () r d r, (33) j In the above equaton v denotes the mposed ambent fow and the ntegra term descrbes the fow generated by the nduced forces. Here
36 M. L. Eke-Jeewska & E.Wajnryb Trr (, ) s the Green functon for the Stokes fow n the presence of the boundares. It s convenent to wrte s as Trr (, ) T( rr) Trr (, ), (34) 0 where the Oseen tensor, rr ˆˆ T0 () r, (35) 8 r s the Green functon for the Stokes fow n the unbounded space. In case of the unbounded space, Trr (, ) 0. Otherwse Trr (, ) descrbes the fow refected from the nterfaces, whch confne the fud. For a fud n a haf-space z > 0, mted by a fat free surface at z = 0,.e. for the boundary condtons gven by Eqs. (8)-(9), the tensor T has the form [23, 2], Trr (, ) T( rr) P, (36) where 0 P 2 nn, (37) r Pr ( x, y, z), (38) wth r ( x, yz, ) and the unt vector n norma to the nterface pontng nto the fud. For a fud n a haf-space z > 0, mted by a hard wa at z = 0,.e. for the stck boundary condtons gven by Eq. (5), the tensor T has the form [6], Trr (, ) T( rr) 2 nt ( rr) P T( rr) P, (39) where 2 2 0 z 0 r z r 0 [ Wr ( ) r] [ Wr ( )]. (40) r
Mutpoe method for cacuatng hydrodynamc nteractons 37 To obtan the boundary ntegra equaton for the force densty f j, the ntegra representaton (33) w now be combned wth the boundary condtons. In partcuar, for the stck boundary condtons (4) on the surface of partce, N 3 j d S j w () r v () r T( r r) f () r r, r. (4) Other boundary condtons on the sphere requre more detaed anayss. In genera, we decompose the fow around partce nto two fows n out vr () v () r v (), r (42) where n v s the ncdent (reguar) and out v the scattered (snguar) part of the tota fow v(r) around the partce. The snguar fow s gven by out () ( ) () 3, 0 j d v r T r r f r r (43) and t represents the fow scattered by the consdered partce. The nduced force dstrbuton f on the surface of the partce and the n fow v ncdent to ths partce are neary reated. The reaton can be expressed n the form n d f () r Z ( r R, r R )[ v () r w ()] r r, (44) where the snge-partce frcton operator Z depends on the specfc boundary condtons ony at the partce, and s expcty obtaned by sovng the Stokes equatons for an soated partce subject to an externa fow [28, 37]. Exampes of such expct expressons are gven n Refs. [27-29]. In Eq. (44), the expresson v n w denotes the ncdent fow n the frame of reference movng wth the partce. The Stokes fow v n w s fuy determned by ts boundary vaue on the partce surface S and the condton that t s nonsnguar n the regon occuped by the partce. Thus Eq. (44) can be nterpreted as a near functona reaton between the force vector fed f and the ncdent fow v n w on the surface S. Snce a non-
38 M. L. Eke-Jeewska & E.Wajnryb zero ncdent fow aways produces a non-zero force dstrbuton f, the reaton (44) can be nverted, n () () (, ) () 3,. d S v r w r Z r R r R f r r r (45) By coectng reatons (42), (43) and (45) we obtan the expresson 3 3 d 0 d S vr () w () r Z ( r R, r R ) f () r r + T ( r r ) f () r r, r,(46) for the fow v(r) at the surface S of the partce. Usng the ntegra representaton (33) at the surface S of the partce, and appyng the boundary condton (46), we obtan the set of the boundaryntegra equatons for the nduced force denstes f [22], 3 R R d w () r v () r Z ( r, r ) f () r r+ N j [( ) ( ) (, )] ( ),. 3 j T0 r r T r r fj r d r r S (47) For the rgd spheres, the boundary condton (46) reduces to the no-sp requrement, vr ( ) w ( r ) for r S, and Z ( rr, r R ) T0 ( rr), f rr, S. In ths case, the boundary ntegra equaton (47) has the smpe form gven n Eq. (4). In the foowng secton, the method of sovng Eq. (47) w be outned. The set of the boundary ntegra equatons w be transformed nto an nfnte set of agebrac equatons for the force mutpoes. 3. Mutpoe expanson In ths secton, the basc dea of the spherca-mutpoe expanson [9, 34, 35, 38] w be outned. We appy ths expanson to N spherca partces n a fud under a genera ambent fow v ( r ). The procedure may be apped to dfferent geometres of the boundares, f the correspondng Green functon s known. The forces, torques and veoctes are projected on a basc set of mutpoe functons, and represented by the coeffcents of ths expanson (the so-caed force and veocty mutpoes), see Appendx A for the detas. Here we use the rea mutpoe vector functons u ( r R ), wth R denotng m
Mutpoe method for cacuatng hydrodynamc nteractons 39 the center of sphere. The subscrpts are the mutpoe ndces =, 2,..., m = 0,±,..,± and = 0,, 2. The defntons of a u m are gven n Ref. [2] and aso n Appendx A. The projecton casts the ntegra equaton (47) nto an nfnte set of agebrac equatons, N j m 0 2 c( m ) M( m, jm ) f( jm ), (48) whch reate the force mutpoes, 3 m r R fj r d r f ( jm ) u ( ) ( ), (49) to the veocty mutpoes c(m), whch are defned n terms of two contrbutons, c( m ) c ( m) c ( m ), (50) w where c w and c are, respectvey, the expanson coeffcents of the partce veocty w (r) = U + (rr ) at a pont r on the surface of sphere, and of the ambent fow veocty v (r), R cw um R m 0 U ( r ) ( m ) ( r ), (5) 2 v () r c ( m ) u ( rr ). (52) m 0 m Note that the ony non-vanshng cw ( m ) are those wth =, = 0,. Each of them s proportona to a component of the transatona or the anguar veocty of the sphere. Smary, the force mutpoes wth (, ) = (, 0) and (, ) = (, ) are proportona to components of the force and the torque exerted by the sphere on the fud, wth the coeffcents gven n Appendx B. The mutpoe matrx eements M ( m, jm ) n Eq. (48) are determned by the correspondng eements of Z, T 0 and T (see Appendx A for the detas),
40 M. L. Eke-Jeewska & E.Wajnryb 0 M( m, jm ) Z ( m, m ) j ( )[ T ( m, jm ) T ( m, jm )]. j (53) After truncatng of the expanson at order L,.e. negectng the terms wth, > L, equaton (48) reduces to a fnte set of near agebrac equatons. These equatons are soved for the force mutpoes by nvertng the arge matrx M formed by the coeffcents M ( m, jm ) wth, L, N L j m 0 2 f( m ) Z ( m, jm ) c( jm ), (54) where Z L = M L s caed the spherca generazed frcton (or grand resstance) matrx. The coeffcents ZL ( m, jm ) depend on the mutpoe order L of the truncaton. In Appendx A, ther propertes are dscussed and the references are gven to ther expct form for specfc types of the partces. In Appendx B, the Cartesan generazed frcton tensors pq, ntroduced n Sec. 2.3, are expressed n terms of the coeffcents ZL ( m, jm ) of the spherca generazed frcton matrx. In ths way the generazed frcton probem s soved. In partcuar, n the absence of an ambent fow, for gven transatona and anguar veoctes of the sphere, the force and the torque are determned by Eq. (54) for the L-dependent force mutpoes wth =, m = 0, ± and = 0, ony, w, L f ( m) Z ( m, jm ) c ( jm ). (55) m 0 L The mutpoe eements ZL ( m, jm ), whch enter Eq. (55), form the spherca frcton matrx, whch soves the frcton probem. On the other hand, for a gven ambent fow v, t s of nterest to evauate the force and the torque exerted by a system of motoness spheres on the fud. In the mutpoe expanson, t means that we search the force mutpoes wth =, m = 0, ± and = 0,. We denote them as f, L ( m ), wth the subscrpt specfyng the probem (the sphere s fxed and the force mutpoes are determned by the coeffcents c (m) ony). Note that a the force mutpoes depend on L, as ndcated by the second subscrpt L. They are evauated from Eq. (54), whch now takes the form, w
Mutpoe method for cacuatng hydrodynamc nteractons 4, L N L j m 0 2 f ( m) Z ( m, jm ) c ( jm ). (56) In Appendx C, t s descrbed how to project an ambent fow onto the mutpoe functons, and evauate the veocty mutpoes c (m). In many practca appcatons, the ambent fow s a combnaton of a sma number of the mutpoe functons ony, e.g. for the shear or Poseue fows. In such cases, t s possbe to truncate the expanson at such a mutpoe order L, that a the coeffcents c (jm) 0 are ncuded n Eq. (56). Assume now that the spherca partces are movng n an ambent fow, under gven externa forces and torques, whch determne f (m) for =,..., N, =, m= 0, ± and =0,. The goa s to evauate the partce transatona and rotatona veoctes. As expaned n Appendx B, these veoctes are expressed by the veocty mutpoes c w, L wth =, m = 0,± and = 0,, where the ndex L remnds the dependence on the truncaton order. By vrtue of Eqs. (50) and (54), one obtans N cw ( m) ( m, jm )[ f( jm ) f ( jm )], (57), L L, L j m 0 L wth f, L aready cacuated n (56). Here µ L (m, jm) denote coeffcents of the spherca mobty matrx, whch s the nverse of the correspondng spherca frcton matrx. The spherca-mutpoe mobty matrx can be easy transformed nto the correspondng Cartesan mobty tensor. The expct transformaton from the spherca to the Cartesan representaton s gven n Appendx B. The agorthm descrbed above has been mpemented n a numerca FORTRAN code caed HYDROMULTIPOLE, and cacuatons have been carred out wth both doube and quadrupe precson. The accuracy s controed by changng the mutpoe order L of the truncaton, and even extrapoatng to L. For systems of sod partces wth no reatve moton of cose surfaces, the method presented above s suffcent for precse cacuatons aready for very ow mutpoe order L [39]. However, f cose sod surfaces move wth respect to each other, the approprate treatment of the ubrcaton effects s needed for accurate and effcent computatons. Such a modfcaton of the agorthm w be dscussed n the next secton.
42 M. L. Eke-Jeewska & E.Wajnryb 4. Mutpoe expanson corrected for ubrcaton 4.. Lubrcaton between two cose sod surfaces If a sod sphere (abeed ) moves wth respect to another sod sphere (abeed j), amost touchng t, the fud strongy ressts the movement; the frcton force and torque dverge when the sze of the gap between the surfaces tends to zero, and the moton of the partces s fxed [6, 40]. For very sma dstances between the sphere surfaces, = R j /a 2, (58) wth R j = R R j and a = (a + a j )/2, the two-partce frcton tensors (j) have the foowng asymptotc form, ub A( j) ( j) B( j)n C( j) D( j) n ( ), (59) The constant matrces A, B, C, D are specfed expcty e.g. n Refs. [6, 40]. The ony non-zero eements of the matrx A correspond to the forces caused by the reatve moton of the spheres aong ther ne of centers. The expresson Eq. (59) for the ubrcaton snguartes s genera. It appes to the generazed frcton tensor for two partces of an arbtrary shape [4]. Hydrodynamc nteractons descrbed by Eq. (59) are caed ubrcaton nteractons [6]. They requre a speca attenton n numerca cacuatons [42]. In partcuar, for very cose spheres wth n reatve moton, they cause a very sow convergence of the mutpoe expanson wth the ncreasng mutpoe order L. Therefore the asymptotc expressons (59) have been used to construct the so-caed ubrcaton correcton [4, 43, 44] for manypartce hydrodynamc nteractons. Ths procedure w be outned n the next sectons. 4.2. Accurate frcton tensor for two spherca partces For two spheres, abeed and j, the frcton tensor (j) was frst evauated n Ref. [40], and then recacuated wth an mproved precson and generazed for varous ambent fows wth the use of severa dfferent technques, ncudng bspherca coordnates (extensvey dscussed n ths book) and the mutpoe expanson [28]. Wthn the mutpoe method, any frcton coeffcent can be represented as a sum of mutpe scatterng sequences, each proportona to a gven power k of the nverse nterpartce
Mutpoe method for cacuatng hydrodynamc nteractons 43 dstance x = 2a/R j. Next, the mutpe scatterng sequences wth the same k may be coected, to obtan the frcton coeffcent as a power seres, so that (j) = k C k (j) x k. For very cose sphere surfaces, however, t s essenta to speed up the convergence rate of the seres, n a smar way as t was proposed by Jeffrey and Onsh [40]. Ths s acheved by subtractng from the frcton tensor the correspondng asymptotc expressons (59), non-anaytc and dvergent when 0, ( j) ( j) ub ( j), (60) Then, the dfference s represented as a power seres of x, k n( j) Dk( j) x. n k 0 (6) The matrx D k dffers from C k by the seres expanson of ub. As the resut, the seres (6) s fast convergent and ts truncaton eads to a hgh accuracy of the frcton coeffcents [40]. Typcay, n the sphercamutpoe numerca codes, n = 300 has been used [0], wth the precacuated tabes of a the coeffcents of D k. Ths procedure s suffcent to reach the 2 0 5 absoute precson of the two-partce frcton coeffcents even at the contact. 4.3. Lubrcaton correcton for many-partce hydrodynamc nteractons 4.3.. Standard approach Consder frst such a system of N spherca partces, where a partce moves wth respect to another very cose partce abeed 2, and the other spheres are we-separated from each other and from spheres and 2. The tota hydrodynamc force exerted by the fud fow on sphere s practcay caused by the moton of the fud n the ubrcaton gap between the partces and 2. Indeed, n ths case the ubrcaton expresson ub (2), gven by Eq. (59), s arge and domnates a the other contrbutons to the N-partce frcton tensor (2...N). Ths ubrcaton contrbuton s ndependent of the other partces. If the sphere s surrounded by severa spheres, whch amost touch t, one may expect that the tota frcton forces and torques exerted on sphere by the fud fow are approxmated by the superposton of the two-
44 M. L. Eke-Jeewska & E.Wajnryb partce contrbutons (59) correspondng to ubrcaton gaps between the surfaces of spheres and j. Ths property has been used to construct a modfed mutpoe expanson, fast-convergent even for, and vad for a genera confguraton of N spheres [43, 44]. In ths procedure, a superposton of the two-partce frcton tensors s formed n the foowng way, sup N (... N) ( j), j (62) sup (... N) ( j) for j, (63) j where the ower ndces j abe the 6 6 tensor components of the 6N 6N tensors. The two-partce frcton tensor (j) s evauated by the procedure descrbed n Sec. 4.2, wth a hgh precson even for extremey sma dstances between the sphere surfaces. In the foowng, the arguments (...N) w be omtted wherever ths does not nterfere wth carty of the presentaton. The dea ntroduced n Refs [43, 44] s to correct the sow convergence rate of L, repacng t by another fast-convergent expresson, (64) L L L, where the ubrcaton correcton, sup (65) L sup L s defned as the dfference between the accurate parwse-addtve expressons (62)-(63) and ther mutpoe approxmatons of the order L, N sup L, (... N) L, ( j), j (66) sup Lj, (... N) Lj, ( j), for j, (67) wth L ( j ) denotng the mutpoe approxmaton wth the order L of the two-partce frcton tensor (j).
Mutpoe method for cacuatng hydrodynamc nteractons 45 Both L and L approach the same mt when L, and keep the same ong-dstance asymptotcs, because for we-separated partces the ubrcaton correcton s neggbe. The key pont of Eqs. (64)-(65) s that sup sup ( ) s fast-convergent even f some of the partces are L L L cose and move wth respect to each other. Indeed, the frst term, sup, ndependent of the mutpoe order, and the second one, L sup L, does not contan any ubrcaton snguartes and therefore s fast-convergent wth the ncreasng L. It s essenta that L s an accurate approxmaton of aready for a ow mutpoe order. Lubrcaton correcton of the other generazed frcton coeffcents s constructed by the same reasonng. Then, the corrected generazed mobty tensor foows from Eq. (26). In partcuar, correctng the L-order mobty tensor resuts n L L L L L L L L L L L [ ].... (68) The standard ubrcaton correcton aows for accurate evauaton of the partce dynamcs. However, a refned treatment s needed f the custer expanson s performed and three-partce hydrodynamc nteractons are cacuated separatey [0, 47, 48]. In partcuar, the three-partce contrbuton to the transatona sef-dffuson coeffcent s nfnte f evauated wth the standard ubrcaton correcton [0]. The spurous dvergence s caused by the ncorrect asymptotcs of the three-partce mobty µ tt s for such a confguraton n whch a snge sphere (e.g. abeed 3) s far away from the other two (wth abes and 2), wth R 3 R 2 and R 23 R 2. In ths case, tt 4 the domnant three-body contrbuton to 33 (23) scaes as /R 2. However, tt the standard ubrcaton correcton adds to 33 (23) a sma artfca term, 2 whch scaes as /R 2 and therefore s non-ntegrabe. Ths paradox w be soved n the next secton. 4.3.2. Improved ubrcaton correcton In genera, the standard ubrcaton correcton changes the tota hydrodynamc force and torque exerted on the fud by a par of spheres n reatve moton f a thrd partce s present. The ubrcaton correcton adds a very sma spurous snget contrbuton, a source of a /r fow, whch
46 M. L. Eke-Jeewska & E.Wajnryb domnates at arge dstances r f the rea fow s proportona to /r 2. Therefore n Ref. [0] an mproved ubrcaton correcton was constructed, whch does nether modfy the tota force nor the torque exerted on the fud by a par of spheres n reatve moton. The procedure descrbed n the prevous secton was repeated, but wth the frcton tensor repaced by a tensor s, whch contans the same snguar terms and satsfes the same symmetres (transatona, rotatona and Lorentz nvarance). Moreover, s apped to an arbtrary rgd moton of the par of spheres has to gve vanshng forces and torques. These condtons are satsfed f T s( j) q ( j) q, (69) and the 6 6 matrx q projects onto reatve moton of the spheres, wth q = q 2. In practce, the operator q s constructed from the requrement that c = q, apped to (U, U 2,, 2), resuts n a rgd moton of both spheres, and c = c 2. The rgd moton of the spheres s not unquey defned, athough of course t shoud be cose to the moton of both spheres. For exampe, a choce of the rgd moton s the transaton of the center of mass system, superposed wth the rotaton around the center of mass wth the anguar veocty ( + 2 )/2. In partcuar, for dentca spheres, the center of mass s ocated at (R + R 2 )/2 and transates wth veocty (U + U 2 )/2. The projecton c on ths rgd moton corresponds to the foowng operator I c = q, whch projects on reatve moton [0], q I I A A = I I A A, 2 0 0 I I 0 0 I I (70) where I s the 3 3 unt matrx, and A depends on reatve poston of the sphere centers, R = R 2 R, A R / 2, (7) wth the Cartesan components abeed by,,, and the summaton over. In genera, the fow nsde the sma gap between the sphere surfaces moves wth respect to the center-of-mass system, and f the sp s arge, a
Mutpoe method for cacuatng hydrodynamc nteractons 47 rgd moton can be taken whch s coser to the fud moton n the gap. For exampe, the operator c can project on a rgd moton whch s the average of the ndvdua rgd motons w (r) and w 2 (r) of spheres and 2, respectvey, see Eq. (4). Ths choce resuts n the operator q ndependent of the sphere rad, q I I 0 2A 2 0 = I I A. 2 0 0 I I 0 0 I I (72) Ths operator q projects on such a reatve moton that corresponds to the opposte veocty fed nsde both spheres, (w (r) w 2 (r))/2 and (w (r) w 2 (r))/2, respectvey. Moreover, nsde the gap and n the mt of a sma gap sze, the reatve veoctes of the sphere surfaces are amost opposte, and the rgd-moton veocty s amost equa to the averaged veocty of the cosest ponts of both surfaces. When the projecton q on the reatve motons s specfed, the expresson s n Eq. (69) s known. The mproved ubrcaton correcton s now constructed by the procedure descrbed n the prevous secton, but wth repaced by s. We obtan the modfed parwse-addtve expressons, s sup sup N (... N) s ( j ), (73) j sj (... N) sj ( j) for j, (74) and ther approxmatons of the order L, s N sup L, (... N) sl, ( ), j j (75) sup, (... ) sl, j ( ) for, sl j N j j (76) wth T s ( j) q ( j) q. (77) L L
48 M. L. Eke-Jeewska & E.Wajnryb Now, the corrected N-partce frcton tensor has the form, L L L (78) wth the mproved ubrcaton correcton, L s sup s sup L. (79) Let us comment now that f the mproved ubrcaton correcton s used, then the sef-dffuson coeffcent s fnte. Indeed, wth n Eq. (68) repaced by tt 4, the domnant three-body contrbuton to 33 (23) scaes as /R 2 wth no spurous extra terms. [0] Fnay, we brefy dscuss accuracy of the mutpoe expanson. Wth ths procedure, the convergence of the mutpoe expanson s fast, and truncaton at a reatvey sma L resuts n a hgh accuracy [9, 7, 45, 46]. In Ref. [9], frcton and mobty coeffcents were evauated for a number of partce confguratons and the accuracy of the resuts was estmated. For groups of rgdy movng partces, truncaton at L = 4 typcay eads to extremey hgh 0.% reatve precson of the drag coeffcents, because the coectve moton does not nvove ubrcaton nteractons. The reatve moton of partces resuts n a ower precson, whch has not been extensvey dscussed n Ref. [9]. Beow we study an exampe of a smpe partce confguraton, and we estmate the accuracy of the frcton and mobty tensors, evauated by the spherca-mutpoe method wth the standard and the mproved ubrcaton correctons. We consder two test confguratons of three dentca cose spheres, wth ther centers ocated at vertexes of an soscees rght trange. In the frst case, the smaest gap sze s equa to 0.0 dameter, and n the second case to 0.000 dameter. The frcton and mobty tensors have been evauated wth the standard and both mproved ubrcaton correctons, for the mutpoe order L = 4 and for L = 25. Reatve precson of the resuts wth L = 4 has been estmated by evauatng the dfferences of the tensor eements correspondng to L = 4 and L = 25, and cacuatng the square root of the sum of the squared dfferences, normazed by the square root of the sum of a the squared eements. The resutng accuracy s equa to 7 0 5 3 0 3. The reatve dfference between the resuts for L = 4, evauated wth dfferent ubrcaton correctons, s smaer than the above precson. The accuracy rapdy mproves f the dstance between the sphere surfaces s ncreased.
Mutpoe method for cacuatng hydrodynamc nteractons 49 5. Congomerates of partces 5.. Frcton and mobty In ths secton we consder hydrodynamc nteractons between rgd arrays of partces. We sove the frcton and mobty probems, formuated n Sec. 2.3, but now for the congomerates rather than for the ndvdua partces. We consder K congomerates of partces. Each congomerate, abeed wth k =,...,K, (80) conssts of N (k) spherca partces (n genera wth dfferent rad) abeed wth k, k k ( s) ( s) N,..., N. (8) k s s The tota number of partces s equa to N, K ( ) N k k N. (82) The poston n space of a congomerate k s defned by the poston of an ( k ) arbtrary reference pont R 0 of ths congomerate (often t s the geometrca center of ths congomerate) and the Euer anges. In genera, there are three such anges; for the congomerates wth axa symmetry (e.g. ( k ) near poymers) two anges are suffcent. The reference pont R and the correspondng Euer anges determne the postons R k of a the sphere centers n the congomerate k. Congomerates move coectvey ke rgd bodes. Therefore the moton of a congomerate k s characterzed by the transatona coectve veocty U (k) of the reference pont ( k ) 0 R and the rotatona coectve veocty (k) of ths congomerate. The transatona and rotatona veoctes U k and k of a the N (k) partces of ths congomerate foow as near functons of U (k) and (k), 0 U k ( k) ( k) ( k ) 0 U + ( R R ), (83) k
50 M. L. Eke-Jeewska & E.Wajnryb k ( k ), (84) wth the range of k and k gven by Eqs. (80) and (8). Reatons (83) and (84) can be wrtten n short, C U U C, C (85) wth the abbrevated notaton for the partce veoctes U and defned n Secton 2.3. In anaogy, we have arranged the congomerate veoctes nto 3K-dmensona vectors U C = (U (),..., U (K) ) and C = ( (),..., (K) ). The 6N 6K rectanguar matrx C can be read out expcty from Eqs. (83-84). The matrx C s a functon of the postons R k of a the N sphere centers. The tota force and (k) and tota torque (k) exerted by the congomerate k on the fud are gven as superpostons of the ndvdua forces and torques, respectvey, ( k ) k, (86) ( k ) k ( k ) 0 [ ( R R ) ], (87) k k k k wth the range of k gven by Eq. (80) and the range of the summaton gven by Eq. (8). Foowng Sec. 2.3, we use the abbrevated notaton and for the ndvdua forces and torques. We aso represent the tota forces and torques exerted by the congomerates on the fud as 3K-dmensona vectors C = ( (),..., (K) ) and C = ( (),..., (K) ). Now the reatons (86) and (87) can be rewrtten n short, C T C C. (88) From Eqs. (86-87) t foows that the 6K 6N rectanguar matrx of the near transformaton s just the transposed matrx C. In genera, the externa forces C and torques C mposed on the congomerates are controed. Ths s not
Mutpoe method for cacuatng hydrodynamc nteractons 5 the case for ndvdua partces, whch undergo reacton forces n addton to the mposed ones. Therefore for congomerates the frcton reaton reads, C C C U C C, (89) where the 6K 6K many-congomerate frcton tensor, C T C C, (90) has been obtaned from the many-partce frcton tensor, defned n Eq. (25), wth the use of Eqs. (85) and (88). As expaned n Sec. 4.3, the mutpoe agorthm used for evauatng the N-partce frcton tensor s n genera corrected for ubrcaton. Whe evauatng C form Eq. (90), the ubrcaton correcton s ncuded n ony for the pars of spheres beongng to dfferent congomerates. The pars of spheres beongng to the same congomerate move coectvey and therefore the mutpoe expanson wthout a ubrcaton correcton s fast convergent, as expaned n Sec. 4.3. Wth the ubrcaton correcton swtched off, spheres n a snge congomerate may touch each other. Fnay, we defne the congomerate mobty tensor by the reaton, C C ( ). (9) The mobty tensor aows to evauate the coectve veoctes and then ntegrate the trajectores of a the K congomerates, whch are subject to externa forces and torques, e.g. sedmentng under gravty. 5.2. Moton n ambent fow In ths secton we evauate the transatona and rotatona veoctes of congomerates whch are subject to externa forces and torques and to an externa fow. Ths task s smar to the moton of partces under externa forces, torques and an ambent fow, soved n Sec. 2.3. We frst appy C T to the.h.s. of Eq. (29), then use Eqs. (85), (88) and (90), C C C U T C C C +, (92)
52 M. L. Eke-Jeewska & E.Wajnryb and fnay use Eq. (9) to evauate the coectve veoctes of the congomerates, C C C U C C C C where C T C. C, (93) (94) Here and are the forces and torques exerted by the motoness partces on the fow, evauated from Eq. (30), where, n the absence of reatve moton of the partces, the frcton tensors are determned wthout ubrcaton correctons. On the other hand, C s cacuated wth the use of ubrcaton correcton between partces beongng to dfferent congomerates. 6. Concusons An effcent procedure [9, 0, 2] wth a controed hgh accuracy, the spherca-mutpoe method, was presented, adequate for evauatng Stokesan dynamcs of non-deformabe spherca partces suspended n a fud, or hydrodynamc resstance of movng or motoness systems of such partces under ow-reynods-number fows. Beow we present a few exampes of specfc appcatons of ths method. Dfferent versons of the mutpoe expanson were used n the terature by many authors n numerous physca contexts [4, 5, 7, 43, 45], and stng a the resuts woud requre a separate revew. Therefore we concentrate many on the resuts obtaned by the accurate spherca-mutpoe method. In ths procedure, the reatve moton of partces s corrected for ubrcaton, to acheve fast convergence wth the mutpoe order of the truncaton. The man advantage of ths agorthm, even n comparson wth the Cartesan-mutpoe formuaton [49], s that t s possbe to perform computatons wth a very hgh mutpoe order of the truncaton, controng the accuracy. Moreover, the method s appcabe to systems of varous types of the partces n a fud bounded by one or two parae fat nterfaces. Frcton and mobty probems for groups of partces n an unbounded fud were soved. Drag coeffcents of congomerates of partces have been cacuated and shown to agree wth the expermenta data [50]. Dynamcs of symmetrc confguratons of three spherca sod partces was anayzed
Mutpoe method for cacuatng hydrodynamc nteractons 53 and an attractng equbrum confguraton was found, non-exstent n the pont-partce approxmaton [5]. A mode of mechanca-contact and hydrodynamc nteractons between rough spherca partces was constructed and shown to account we for the expermentay measured reatve transaton and rotaton [52, 53]. In Ref. [27], the Stokes equatons were soved for a snge surfactant-covered drop n an arbtrary ncdent fow, and then the par hydrodynamc nteractons of surfactant covered bubbes were computed from the one-partce souton usng a mutpe-scatterng expanson. Statstca propertes of partcuate systems were aso determned. Vra expanson of suspenson effectve transport coeffcents was performed. Twopartce and three-partce contrbutons to the short-tme sef-dffuson, sedmentaton veocty and hgh-frequency vscosty were evauated [0, 47, 48, 54-58]. The short tme sef-dffuson coeffcent of a sphere n a suspenson of rgd rods was cacuated n the frst order n the rod voume fracton [59]. Two-partce correaton functon for non-brownan suspenson n a statonary state was determned and used to evauate the vra expanson of the sedmentaton coeffcent. In the statonary state, the term proportona to the voume fracton was shown to be arger than n the equbrum, owng to the excess of cose partce pars n comparson to the equbrum [60]. Dynamcs of partces cose to nterfaces was aso anayzed. The effect of a panar hard wa on the moton of partce custers under externa forces, shear or Poseue fow was determned [2, 6, 62]. Hydrodynamc nteractons between sod partces touchng a free surface and movng aong t (a quas-two-dmensona system) were evauated, and the range of vadty of the ong-dstance parwse asymptotcs and the pont-partce approxmaton was gven [24, 63]. The spherca-mutpoe method was aso used for theoretca and numerca studes of hydrodynamc nteractons of spherca partces confned between two parae panar sod was. A new effcent agorthm for evauatng many-partce frcton and mobty matrces for such a system was deveoped [20, 22]. Numerca mpementaton of ths agorthm was used to evauate the hydrodynamc frcton and mobty for a snge partce, a par of partces, and a system of many partces confned between two panar was. The resuts show that the standard snge-wa-superposton approxmaton s nsuffcent for probems when the partces are ateray separated by many nter-wa dstances [64]. In Ref. [65], the effect of confnng was on the dynamcs of a dute suspenson of nonnteractng, eongated axsymmetrc partces undergong a steady shear fow n a parae-wa rheometer was presented. It was found that the partce moton n the two-wa system quatatvey resembes the Jeffrey s orbts n the unbounded space [66], but, unke n the unbounded space, the perod of the
54 M. L. Eke-Jeewska & E.Wajnryb moton and the evouton depend on the nta poston of the partce. In Ref. [64], the effect of the was on the hydrodynamc nteractons n ambent Poseue fow n a narrow channe were studed. In Ref. [67], the crossover behavor between near-fed fow and far-fed asymptotc Hee-Shaw fow [68], was anayzed. It was shown that for a few nter-wa dstances from the partce, the fow assumes the asymptotc form. Ths factates sgnfcanty the numerca evauaton of the Green tensor for the two-wa system. In Ref. [69], the new cass of bnary trajectores that resut n cross-streamne partce mgraton n a wa bounded shear fow was dentfed. Acknowedgments Ths work was supported n part by the Posh Mnstry of Scence and Hgher Educaton grant N50 020 32/994. Appendx A. Mutpoe functons and mutpoe matrx eements In Secton 3 the boundary ntegra equaton (47) has been projected onto mutpoe functons, resutng n Eq. (48). In ths Appendx, we expan n detas how ths procedure has been carred out and we provde the expct expressons for the veocty mutpoes c(m), the force mutpoes f(m) and the matrx eements M(m, jm). The compete set of eementary fows u m has been constructed [28] n terms of the reguar sod harmoncs, [37], [70]. The reguar sod harmoncs are the foowng soutons of the Lapace equaton, m () r r Y (), rˆ (95) m where the normazed compex spherca harmoncs are gven n terms of the assocated Legendre poynomas, [7] m m Y () rˆ () P (cos ) e, (96) m n m m wth m and the normazaton coeffcents, n m 4 ( m)! (2 )( m)! 2. (97) In the numerca cacuatons, the rea spherca harmoncs are used,
Mutpoe method for cacuatng hydrodynamc nteractons 55 ( R) 0 Y 0, Y (98) m m Y, m ( R) Y ( ) Ym 2 Re ( Ym ), for m, (99) 2 Y m ( R) Ym ( ) Y, m, m Ym m 2 Im ( ), for, (00) 2 and the correspondng rea sod harmoncs, ( R) ( R) r Y rˆ m m () r (). (0) The compete set of the eementary fows of the rea sod harmoncs, and the pressure feds pm sod harmoncs [28], u m s gven n terms of gradents n terms of the rea m0 m0 ( R) m u () r (), r p 0, m m0 u () r u () r r, p m 0, (2 ) 3 2 um 2() r r um 0() r r um0() r r, 2 ( )(2 )(2 3) ( R) pm2 m (). r (02) (03) (04) The foowng scaar product of vector feds A(r) and B(r) s defned AB Ar () Br () dr, (05) wth A or B contanng the factor ( r a ), because the ntegra s restrcted to the boundary surface of a partce wth radus a. The eementary fows u m are not orthogona to each other wth the scaar product (05), therefore m the adjont basc set of functons s ntroduced accordng to the reaton
56 M. L. Eke-Jeewska & E.Wajnryb b m m mm u, (06) for a vaues of the parameter b > 0, where b () r b ( r b). (07) The adjont functons m are [28], 2 (2 )(2 3) 0() r (2 ) ˆˆ 0(), 2(2) m r rr u m r (08) 2 m() r r um0() r rˆ, (09) ( ) 2 2 2() r ˆˆ m r 0(). ( )(2 ) rr u m r (0) The force and veocty mutpoes, ntroduced n secton 3, are projectons onto the correspondng mutpoe functons u m and m, respectvey. The force mutpoes were gven n Eq. (49), f ( m ) u m ( ) f. () The veocty mutpoes, defned n Eqs. (5), (52), are now expressed n terms of, m w a m c ( m) ( ) ( ) w, (2) a m c ( m) ( ) ( ) v. (3) In the above equatons, the standard bra-ket notaton s used. Moreover, A denotes the vector fed A(r) and A() represents the vector fed A(r R ). The mutpoe expanson of the partce and ambent fow veocty feds n terms of u m was gven n Eqs. (5) and (52). In anaogy, the mutpoe expanson of the nduced force dstrbuton f s gven by
Mutpoe method for cacuatng hydrodynamc nteractons 57 a m m f () r f ( m) ( r R ) ( r R ). (4) Substtutng the mutpoe expanson (4) of f nto the boundaryntegra equaton (47), appyng the bra vector () () and usng Eq.(2) and (3), one obtans the set of agebrac equatons (48) for f(m), wth mutpoe matrx eements, a m M( m, jm ) Z ( m, m ) j ( )[ T ( m, jm ) T ( m, jm )], j 0 (5) where Z a m Z a m ( m, m ) ( ) ( ) ( ) ( ), (6) T0 a m T0 a m ( m, j m ) ( ) ( ) ( j) ( j), (7) ( m, jm ) ( ) ( ) T ( j) ( j). (8) T a m a m j j From now on we w use a shorthand notaton A for the matrces wth the eements A(m,, 0 or. jm) gven n the above equatons wth A Z T T Expressons for the above matrces can be found n the terature, see e.g. [2]. The technca dffcuty s that, hstorcay, n many papers, compex rather than rea eementary fows v m and ther adjonts w m, have been used [2, 28]. As a consequence, the expct expressons have been derved for the matrces w A w rather than for A. The transformaton between the rea and the compex basc sets of the mutpoe functons + and w + s the same as between u + and v +, [2] u 0 0 v, j( ) (9)
58 M. L. Eke-Jeewska & E.Wajnryb u u m m where m m ( ) v, m 2 Re v v m, m, j( ) j( ) 2 (20) m m ( ) v, m 2 Im v v m, m, j( ) j( ) 2 (2) j for 0, 2, ( ) for. (22) Aternatvey, usng the transformaton matrx X n the many-partce space, [2] m m j jm u ( rr ) v ( rr ) X ( jm, m ), (23) m m j jm ( rr ) w ( rr ) X ( jm, m ), (24) wth X ( jm, m) jm m [ m0 j( ) ( ) C ( m)], m 0 sgn( m),sgn( m) (25) where the dagger denotes the Hermtan adjont, and C ±,± (m) are the eements of the 2 2 untary matrx { C, ( m )},,. 2 m m ( ) ( ) In short, Eqs. (23) and (24) can be wrtten as, u v X (26). w X (27)
Mutpoe method for cacuatng hydrodynamc nteractons 59 The matrx eements transform accordng to u A u X v A v X, (28). A X w A w X (29) Wth the use of the above equaton the expressons A, appearng n (6)-(8), are evauated n terms of the correspondng w A w matrces, whch n turn can be found n the terature. A bref outne of the avaabe resuts w now be gven. We start from the matrx eements of the operator w Z w v Z v. Therefore, the eements are obtaned by nverson of the matrx v Z v. Z Z. Note that (m, m) These procedure can be carred out rgorousy because, due to the spherca symmetry, v Z v s dagona n the ndex. The task s to evauate the matrx eements of the frcton operator Z of a snge spherca partce. Ths s acheved by sovng the Stokes equatons for an soated partce subject to an externa fow. The matrx v Z v s dagona n the partce abes and j. Due to the spherca symmetry, t s aso dagona n the azmutha number m. Moreover t does not mx the even and odd ndces, whch correspond to vector and pseudovector components respectvey. Owng to these propertes, u Z u v Z v. The matrx eements of the operator Z have the form [47], z,00 0 z,02 2 Z( m, jm ) (2 a) jmm 0 z, 0,,02 0 z z,22 (30) where the coeffcents z, are specfc for gven boundary condtons mposed on the spherca partces. They can be found e.g. n Refs. [27, 28, 29].