An O(N) algorithm for Stokes and Laplace interactions of particles

Transcription

1 Syracue Unverty From the SelectedWork of Ahok S. Sangan 1996 An O(N) algorthm for Stoke and Laplace nteracton of partcle Ahok S. Sangan, Syracue Unverty Guobao Mo, Syracue Unverty Avalable at:

2 An O(N) algorthm for Stoke and Laplace nteracton of partcle Ahok S. Sangan and Guobao Mo Ctaton: Phy. Flud 8, 1990 (1996); do: / Vew onlne: Vew Table of Content: Publhed by the Amercan Inttute of Phyc. Related Artcle Cloud of partcle n a perodc hear flow Phy. Flud 24, (2012) The dynamc of a vecle n a wall-bound hear flow Phy. Flud 23, (2011) A tudy of thermal counterflow ung partcle trackng velocmetry Phy. Flud 23, (2011) Partcle accumulaton on perodc orbt by repeated free urface collon Phy. Flud 23, (2011) Drag force of a partcle movng axymmetrcally n open or cloed cavte J. Chem. Phy. 135, (2011) Addtonal nformaton on Phy. Flud Journal Homepage: Journal Informaton: Top download: Informaton for Author:

3 ARTICLES An O(N) algorthm for Stoke and Laplace nteracton of partcle Ahok S. Sangan a) and Guobao Mo Department of Chemcal Engneerng and Materal Scence, Syracue Unverty, Syracue, New York Receved 27 December 1995; accepted 12 Aprl 1996 A method for computng Laplace and Stoke nteracton among N phercal partcle arbtrarly placed n a unt cell of a perodc array decrbed. The method baed on an algorthm by Greengard and Rokhln J. Comput. Phy. 73, for rapdly ummng the Laplace nteracton among partcle by organzng the partcle nto a number of dfferent group of varyng ze. The far-feld nduced by each group of partcle expreed by a multpole expanon technque nto an equvalent feld wth t ngularte at the center of the group. The reultng computatonal effort ncreae only lnearly wth N. The method appled to a number of problem n upenon mechanc wth the goal of aeng the effcency and the potental uefulne of the method n tudyng dynamc of large ytem. It hown that reaonably accurate reult for the nteracton force are obtaned n mot cae even wth relatvely low-order multpole expanon Amercan Inttute of Phyc. S I. INTRODUCTION a Electronc mal: aangan@malbox.yr.edu Numercal mulaton of moton of partcle through a upendng flud provde valuable nght nto the complex nterrelatonhp between the mcrocale phyc, the mcrotructure, and the macrocopc behavor of upenon. However, the problem of determnng hydrodynamc nteracton among many partcle computatonally ntenve wth mot of the extng method for mulaton utable only for a relatvely mall number of nteractng partcle, typcally of O(100). Whle th adequate for many problem, there are alo large number of problem for whch t derable to mulate ytem contanng much greater number of partcle. For example, the unform tate of mall Reynold number, fnte Stoke number, ga-old fludzed bed known to be untable for certan range of t parameter the volume fracton of the partcle and the Stoke number reultng n the formaton of large bubble or regon devod of partcle. Large-cale mulaton are needed to undertand n detal the mechanm reponble for thee macrocopc ntablte. Smlarly, problem nvolvng concentrated fber upenon wth nl 3 of O( ) requre large-cale mulaton nvolvng thouand of fber n order that the box ze ued n the mulaton doe not gnfcantly affect the behavor of uch upenon. Here, n the number denty of fber and l the length of fber. Moreover, recent expermental and numercal work on edmentng fber ugget that the unform tate of uch upenon untable reultng n the formaton of cluter. 1 Large-cale mulaton are needed to determne the cluter ze dtrbuton and the reultng properte of the edmentng fber upenon. Large-cale mulaton are alo needed n the tudy of upenon wth gnfcant wall effect, polydpere upenon, or for upenon n whch the hydrodynamc nteracton are expected to be creened at dtance large compared to the ze of the partcle. Two maor dffculte n computng hydrodynamc nteracton among partcle n Stoke mall Reynold number flow are: the long-range, multpartcle nature of nteracton; and the lubrcaton effect arng from a relatve moton of partcle n cloe proxmty to each other. Thee are explaned n more detal below. The velocty dturbance caued by a partcle wth a net nonzero force actng on t decay only a 1/r, r beng the dtance from the center of the partcle, and therefore t not poble to ue an arbtrary cut-off radu for truncatng the hydrodynamc nteracton among partcle. In other word, one mut compute the nteracton among all the partcle n the upenon. The velocty nduced by a partcle generally expreed n term of a dtrbuton of hydrodynamc force denty actng along t urface. The multpartcle nature of the nteracton are due to the fact that th force denty unknown and to be determned a a part of the oluton by olvng for the force denty on all the partcle multaneouly. Th dfferent, for example, from the problem of computng Coulombc nteracton among pece wth known charge for whch the nteracton are alo longranged but, becaue the charge on the ndvdual pece known, the nteracton are par-addtve. A a conequence, no mple par-addtve approxmaton can be made n computng hydrodynamc nteracton. When two partcle n cloe proxmty approach toward each other wth an O(1) relatve velocty, the flud n the gap between the partcle mut queeze out radally from the narrow gap between the partcle. Th reult n a radal velocty of O( 1/2 ) n the gap regon of thckne and a force denty of O( 2 ) localzed to an O() urface area of each partcle. Th known a the lubrcaton effect. See, for 1990 Phy. Flud 8 (8), Augut /96/8(8)/1990/21/$ Amercan Inttute of Phyc

4 example, Happel and Brenner 2 or Km and Karrla 3 for detal. Snce the lubrcaton force denty hghly localzed to a relatvely mall area of the the urface of the partcle, the conventonal numercal technque, uch a the boundary ntegral technque n whch the urface of the partcle dcretzed nto a number of urface element ee Pozrkd 4 for detal, become mpractcal for large ytem a the number of dcretzed element needed for reolvng the lubrcaton effect become prohbtvely large a the two partcle approach each other. To overcome the above two dffculte, Brady and Bo 5 deved an ngenou cheme n whch the manypartcle retvty matrx, whch gve the force denty on the partcle gven ther velocte, expreed a a um of far-feld approxmaton to the many-partcle moblty matrx nvere and the par retvty tenor. The former account for the long-range, multpartcle nature of the nteracton whle the latter account for the lubrcaton force between par of partcle whch contrbute n a par-addtve manner to the retvty tenor. Th method alo ued by Ladd 6 who howed that the approxmaton deved by Brady and Bo can be ytematcally mproved by ncludng hgherorder approxmaton to the far-feld moblty matrx. The man advantage of the method over the conventonal boundary ntegral method that relatvely few unknown typcally 11 to 26 per partcle are needed for determnng manypartcle nteracton wth an accuracy that adequate for many dynamc mulaton problem. 5,6 Unfortunately, the method requre nvertng a far-feld moblty matrx wth at leat (11N) 2 element, the computatonal effort for whch grow a N 2 a the ytem ze ncreae, N beng the number of partcle n the ytem. Th lmt the computaton to N of no more than few hundred. Alternate method that do not requre nvertng the moblty matrx have been propoed by Mo and Sangan, 7 Sangan and Mo, 8 and Cchok et al. 9 Cchok et al. employed the ame dea a Brady and Bo to account for the lubrcaton effect but avoded the matrx nveron wth the help of a utable tranformaton of the equaton governng the multpole. In the preent tudy we ue the method propoed by Sangan and Mo. Accordng to th method, the force denty on the partcle decompoed frt nto a lubrcaton force denty whch localzed to the gap regon between the cloely paced partcle and a regular force denty whch dtrbuted on the entre urface of the partcle. The velocty due to the latter expreed n term of force multpole at the center of the partcle whle that due to the former approxmated n term of a force dpole at the center of the gap between the partcle. Th method thu account for both the long-range, multpartcle nature of the nteracton and the lubrcaton effect. Applcaton of the boundary condton on the urface of the partcle lead to a ytem of lnear equaton of the form A xb, where x a vector of tranlatonal and rotatonal velocty of the partcle and the nduced force multpole, A an O(NN) matrx and b a vector that depend on the mpoed flow. In Sangan and Mo, 8 each element of the matrx A wa evaluated eparately and the reultng equaton were olved ubequently to determne the force multpole and the velocte of the partcle. The accuracy of the method wa hown to be comparable to that of the method of Brady and Bo. 5 However, nce each element of A wa evaluated eparately, the method alo requred O(N 2 ) computaton, and, conequently, no gnfcant computatonal avng reulted even though t avoded the computaton of the moblty matrx nvere. For large ytem, t wll be advantageou to deve cheme n whch the computatonal effort ncreae much more lowly wth N. The oluton of the et of lnear equaton A xb typcally obtaned by teratve method when N large. In order that th can be accomplhed wth only an O(N) computatonal effort, one mut be able to compute A x for a gven x n an O(N) tme. Th the man obectve of the preent nvetgaton. Our method baed on a fat ummaton technque baed on herarchal groupng of partcle developed for computng Coulombc and gravtatonal nteracton. There are everal way of dong th ee, for example, Apple, 10 Barne and Hut, 11 and Greengard and Rokhln. 12 Here, we hall follow the approach outlned by Greengard and Rokhln. 12 Thee nvetgator and the other co-worker of Greengard have developed an algorthm for computng Laplace and Coulombc nteracton n the twoa well a three-dmenonal pace 13,14 and for the elatc nteracton n the two-dmenonal pace. 15 The feld created by a group of partcle far from a gven partcle expreed n term of multpole at the center of the group a decrbed n more detal later n th paper. Snce the feld repreented by a group of partcle wth a fxed number of multpole become accurate when the dtance from the center of the group large compared wth the lnear dmenon of the group ze, we need a herarchy of group n whch the feld felt by a gven partcle evaluated by ung maller group of partcle that are relatvely cloe to the partcle and larger group of partcle that are further away from t. The method decrbed by Greengard and Rokhln for olvng Laplace equaton tart wth a dcretzaton of the boundary ntegral and th make t omewhat neffcent for treatng upenon problem n whch the lubrcaton force are gnfcant. Although the computatonal effort cale lnearly wth N, the number of dcretzaton element per partcle wll be prohbtvely large when the lubrcaton effect are gnfcant. However, by combnng ther technque of rapdly ummng the nteracton wth the method of Sangan and Mo, 8 n whch the number of unknown per partcle mall due to explct treatment of the lubrcaton effect, t hould be poble to decreae the overall computatonal effort gnfcantly. Alo, a we hall ee, the extenon of the method to um Stokean nteracton nontrval. The method requre developng approprate expreon for the far-feld and near-feld repreentaton of the feld nduced by a group of partcle. Greengard and Rokhln gave thee expreon for the Laplace equaton and the preent tudy derve mlar relaton for the Stoke equaton. The method appled to everal problem to ae the effcency and the potental uefulne of the algorthm. We hould perhap menton here about an O(N) algorthm baed on the lattce-boltzmann ga technque that already ext for the tudy of hydrodynamc nteracton n Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo 1991

5 upenon. The flud contnuum n Stoke nteracton replaced by a lattce-boltzmann ga wth approprate rule for t molecule to exchange ther poton and momentum. It found that wth utable rule for th exchange n the bulk and at the nterface between the partcle and the molecule of the lattce-boltzmann ga, t poble to mmc the behavor of rgd partcle upended n a Naver Stoke flow. A method baed on th dea ha been extenvely teted n two recent paper by Ladd. 16,17 Ladd ha been able to carry out Stokean dynamc mulaton of upenon wth N of O(10 4 ) ung th technque. In addton to beng O(N) n computaton, the method ha the advantage of beng able to treat both the nonzero Reynold number flow pat fxed partcle and the upenon of ubmcron zed partcle for whch the Brownan force are gnfcant. Th method, however, tll n t early tage of development wth t accuracy and effcency for large N ytem unteted and unchallenged by the other drect approache baed on olvng partal dfferental equaton arng from the contnuum approxmaton. It hoped that n the leat the method developed here may erve a a check and an alternate to the lattce-boltzmann ga baed algorthm for monodpere upenon of rgd partcle. Furthermore, nce the ze of the lattce typcally governed by the mallet dmenon of the partcle, t appear that the method of ummng nteracton by herarchal groupng wll be far more effcent n dealng wth the upenon of lender fber or polydpere upenon. Alo, nce n general, t a nontrval tak to determne the approprate rule for the exchange of momentum at the nterface to mmc boundary condton other than the no-lp condton, t expected that the method decrbed n th paper wll be more readly adapted to the upenon of charged partcle, 18 drop or bubble. 7 Note that for hghly deformable partcle and lender fber, the nteracton can be computed ung the ntegral equaton repreentaton for the Stoke flow ntead of the multpole repreentaton. The lubrcaton effect mentoned earler lkely to play le mportant a role for thee cae, and conequently the traghtforward ntegral equaton coupled wth the fat ummaton method decrbed here expected to be adequate for the tudy of uch upenon. The bac method outlned n Sec. II where we conder a mple cae of Laplace nteracton. We have choen to treat thee nteracton frt nce the method much eaer to undertand for th cae and becaue of t applcaton to the mulaton of bubbly lqud at large Reynold and mall Weber number ee Sangan and Ddwana 19. Although the general prncple are the ame a n the method outlned by Greengard and Rokhln, the detal are qute dfferent. In Sec. III we decrbe the method for computng Stoke nteracton. In Sec. IV we ae the effcency of the algorthm by applyng t to a number of problem. Frt we conder two Laplace nteracton problem: determnaton of the effectve reacton rate contant n a dffuon-lmted reactng medum; and determnaton of the added ma coeffcent for partcle n nvcd upenon. Next, we conder three Stoke flow nteracton problem: a unform flow through fxed bed of partcle; effectve vcoty of upenon; and edmentaton velocty and hydrodynamc fluctuaton n upenon. II. THE METHOD FOR LAPLACE INTERACTIONS A mentoned n Sec. I, we hall frt conder a mpler problem of determnng Laplace nteracton of phercal partcle. We hall explan the method n reference to a problem of dffuon-controlled reacton. Th wll be applcable wth mnor modfcaton to the other problem of Laplace nteracton. When the ze of one of the reactant pece much greater than the other, the larger pece may eentally be regarded a mmoble and the rate of reacton then depend on the rate at whch the maller pece dffue through the medum and arrve at the urface of the larger, mmoble pece. To model th tuaton, we conder a upenon contng of N phercal partcle each of radu a placed wthn a unt cell of a perodc array. The upendng flud contan a pece wth a lnear dmenon much maller than a whch dffue through the flud wth a contant dffuvty D. The pece react very rapdly wth the phere uch that t concentraton at the urface of the phere may be taken to be vanhngly mall. We hall aume that the pece contnuouly produced n the flud at a contant rate throughout the flud medum. At teady tate the average concentraton C of the pece n the upenon determned by the balance between the rate at whch t produced n the bulk and the rate at whch t conumed by the reacton. The problem then to determne the non-dmenonal reacton rate contant R defned by Q4aDR C. 1 Here, Q the average quantty of the pece reactng per unt tme on a ngle phere. When, the volume fracton of the phere, mall, the nteracton among phere can be neglected, and R 1 a reult frt gven by Smoluchowk. 20 An etmate of the frt correcton for mall but fnte wa gven by Felderhof and Deutch, 21 and, more recently, numercal mulaton have been ued to compute R a a functon of for dene upenon ee, for example, Felderhof 22. Our goal wll be to calculate R for a few elected confguraton of N phere. The flud aumed to be at ret o that the pece concentraton C atfe the Poon equaton 2 CS0 wth the boundary condton C0 on the urface of the phere. Here, DS the net rate at whch the pece produced per unt volume of the flud and related to Q by DS(1)nQ, n beng the number denty of the phere. It may be noted that the preence of S n Eq. 2 render t a Poon equaton ntead of the Laplace equaton but we hall contnue to refer to the nteracton a Laplacan nce Eq. 2 a rather trval pecal cae of the more general Poon equaton n whch the nk term a functon of the poton. In Sec. IV, where we preent the reult of computaton for R, we hall alo conder the problem of Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo

6 added ma whoe governng dfferental equaton ndeed the Laplace equaton, and the oluton for that cae wll be obtaned mply by ettng S0. A. A revew of an O(N 2 ) algorthm Before decrbng the O(N) algorthm n detal, t ueful to preent a more conventonal method of multpole expanon n whch the computaton grow a N 2 a the ytem ze ncreaed. The method ha a cloe connecton to the boundary ntegral method but enoy an advantage of a fater convergence for mple partcle hape uch a phere condered n the preent tudy. Th method wa outlned n reference to the problem of determnng the effectve thermal conductvty and the added ma coeffcent for a gven confguraton of phere n our earler tude. 23,24 The concentraton C of the dffung pece can be expreed n term of the Green functon or the fundamental ngular oluton S 1 of the Poon equaton a N CxC 1 G S 1 xx, 3 where C to be choen uch that the average concentraton equal C, G a dfferental operator that wll be defned more precely later n the ecton, x the center of the partcle, and S 1 the patally perodc Green functon atfyng 2 S 1 x4 1 xl xx L. 4 Here, x L repreent the lattce pont of the perodc array, the volume of the unt cell of the perodc array, and the Drac delta functon. The contant nk term 1 n the above expreon needed to balance the ource term at the lattce pont. An Ewald technque for evaluatng S 1 decrbed n detal by Hamoto. 25 More detal ncludng expreon for the dervatve of S 1 are gven n Sangan et al. 24 and Cchok and Felderhof. 26 A hown by Hamoto, S 1 (x) ha a ngular, ource-lke, behavor near lattce pont where t behave a 1/xx L. The ue of patally perodc Green functon enure that the feld nduced by each partcle,.e., G S 1 (xx ), patally perodc, and hence content wth the mpoed perodc boundary condton. Thu, we only need to atfy the boundary condton at the urface of the partcle. For the cae of phercal partcle t convenent to expre C near each partcle n term of phercal harmonc n a polar coordnate ytem wth t orgn at the center of that partcle. Thu, near partcle, we expre C a 1 CSr 2 /6 0 n0 where rxx, and wth n m0 E, nm A, nm r 2n1 r, 5 are the old phercal harmonc Y 0 nm r n P m n com, Y 1 nm r n P m n nm. 6 Here, co and the phercal polar angle and are defned by r 1 rco, r 2 rnco, and r 3 rnn. Now the boundary condton of vanhng C at ra yeld E, nm a 2n1 A, nm 1 6 Sa2 n0 m0 0 0, where n0 a Kronecker delta functon whoe value unty for n0 and zero otherwe. In order that Eq. 3 can be recat nto Eq. 5, we defne the dfferental operator G uch that the ngular term at x n Eq. 3 are exactly the ame a thoe n Eq. 5. Snce the ngular part of S 1 equal 1/r, we requre that G r 1,n,m r 2n1 A nm, 7, 8 where the ummaton over,n,m the ame a that n Eq. 5. In Appendx A, we have compled a number of ueful reult on the dfferentaton of 1/r and the other phercal harmonc. Ung Eq. A1, we ee at once that G,n,m 1 nm A, nm D nm, 9 where nm gven by Eq. A2 and D nm the dfferental operator defned by Eq. A3. The contant A, nm wll be referred to a the nduced multpole. Now the coeffcent E, nm of the term that are regular at r0 n Eq. 5 are related to the nth order dervatve of the regular part of C at r0 bycf. Eq. A6-A8 E, nm nm D nm C reg Sr 2 /6 r0, 10 where nm gven by Eq. A8 and C reg equal C mnu the ngular part at r0,.e. C reg CG r 1. Subttutng for G from Eq. 9 nto Eq. 3 and combnng t wth Eq. 10 yeld E, nm nmd nm k 1 N l0 k0 0 1 C Sr 2 /6 r0 1 kl A kl, D nm D kl S 1 x x, 11 where the ngular part 1/r mut be removed from S 1 before dfferentatng t for. For later reference, we note that S related to the um of monopole by mean of a mple relaton N S 4 1 0, A obtaned by combnng Eq. 4 and 9. Here, we made ue of the fact that all ngularte are tuated nde the partcle o that, for a pont lyng n the flud, Eq. 4 mplfe to 2 S 1 4/. Now the O(N 2 ) algorthm cont of truncatng the nfnte et of equaton repreented by Eq. 7 and 11 by conderng only the equaton and multpole A, nm wth nn. Th reult n a total of N t N(N 1) 2 number of equaton n an equal number of unknown multpole A, kl. Thee equaton are cat nto a form A xb where x an Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo 1993

7 N t -vector of unknown multpole trength, A an N t N t matrx whoe coeffcent are the dervatve of S 1 (x x ), and b an N t -vector that related to C, or, equvalently, C. The computatonal cot typcally governed by the calculaton of N t 2 element of the matrx A. Th computatonally ntenve nce S 1 telf to be computed ung ere n real and recprocal pace lattce vector. 25 When hgh accuracy n numercal mulaton not crtcally requred, t poble to avod the repeated calculaton of S 1 for all par of partcle by ung a grd nterpolaton cheme n whch the unt cell frt dvded nto a number of maller cube wth the help of a grd and all the dervatve of S 1 needed n the calculaton are evaluated at the grd pont and tored for the nterpolaton purpoe n the ubequent calculaton. Although th reduce the computatonal effort conderably, the computaton tll grow quadratcally wth N t. The et of lnear algebrac equaton ubequently olved ung an approprate teratve olver and th requre computaton of O(N t 2 ) tme the number of teraton requred for the convergence to wthn a dered accuracy. Thu, the overall computatonal effort and the memory torage for the matrx A) cale a N t 2. In earler calculaton, 8,24 we olved the ytem of equaton ung a Gauan elmnaton algorthm whch requred an O(N t 3 ) effort, but for mall N, the computatonal tme wa motly governed by the tme for computng the matrx element and thu th tep wa not crucal. B. Far- and near-feld repreentaton of the dturbance nduced by a group of partcle In order that the overall computaton for determnng the multpole cale lnearly wth N t, we mut be able to determne E, nm wth O(N t ) computaton. The method decrbed n Sec. II A neffcent for large N t nce t compute the dturbance created by each partcle eparately at the center of each partcle. Clearly, the feld created by partcle that are eparated by a large dtance from partcle can be grouped together for the purpoe of evaluatng ther effect on partcle. Smlarly, all the partcle near feel mlar regular feld (C reg ) from the group of partcle far away from them and therefore the calculaton of the regular feld for the partcle could alo be grouped together. If we mply create all the group of partcle wth each group contanng nearly an equal number P of partcle, then we would requre O((N/P) 2 ) group group nteracton computaton. In addton, we mut eparately account for the nteracton among partcle that are neghbor and th would requre O(NP) computaton reultng n a total computatonal effort that cale roughly a N 2 /P 2 NP. Th ha a mnmum for PO(N 1/3 ), and the total computatonal tme for th optmum P cale a N 4/3. In order to further reduce the order of computaton we mut create a herarchy among group of partcle and adopt a trategy n whch the regular feld near partcle evaluated by combnng greater number of partcle that are further away from t and fewer partcle that are cloer to t. Th can be accomplhed ung the algorthm of Greengard and Rokhln 12 whch we hall preent n more detal n Sec. II C. Here, we hall derve the expreon that are needed for combnng the feld nduced by a group of partcle and the regular feld felt by a group of partcle. In partcular, we need to know how to tranlate a feld nduced due to a ngularty at x c to a feld wth ngularty at another pont x p uch that both feld are dentcal at a pont x uffcently far away from both x c and x p ; and how to tranlate a feld whch regular and expreed n old phercal harmonc at one pont to a regular feld expanded around another pont n t vcnty. The frt one wll be ueful, for example, n combnng the feld nduced by a group of partcle whle the econd one wll be ueful n determnng C reg around a number of partcle near. Greengard and Rokhln accomplhed thee two tak through the ue of addton theorem for Legendre functon. We hall ue a dfferent procedure here, one that we have found more utable to treat the cae of Stoke flow to be condered n Sec. III. Alo, nce the method preented here ncorporate the perodc boundary condton mpoed by the preence of the unt cell at the outet, t ha the advantage of dealng more ealy wth varou knd of non-abolutely convergent um that otherwe are n calculaton nvolvng the Green functon for nfnte doman. The cae of nteracton among fnte number of partcle n an nfnte medum can of coure be trvally recovered by ubttutng 1/r n place of S 1 (r). 1. Tranlaton of ngularte We wh to tranlate a feld C c G c S 1 (xx c ) wth t ngularte at x c to an equvalent feld C p wth t ngularte at x p uch that both C c and C p gve the ame value of C or t dervatve at a pont x far from both x c and x p.we tart wth a Green dentty f 2 CC 2 f dv r V fccf nda r, 13 V where V any volume enclong pont x c and x p, V t urface, n the unt outward normal on V, and rxx p. Now we chooe f to equal (r)(0,1) and ubttute n turn for C both C c and C p. Snce C c C p and C c C p on V, the urface ntegral n both mut be equal and therefore we obtan r 2 C c dv r V r 2 C p dv r, 14 V where we have made ue of the fact that 2 f 2 (r)0. Note that th doe not aume that C c and C p are equal at all pont wthn V, only ther equvalence on V.) Care mut be taken n evaluatng the above ntegral nce the Laplacan of C c or C p a ere n generalzed functon 2 C c G c 2 S 1 rr cp 4A 0,c kl A kl,k,l,c D kl rr cp, Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo

8 where r cp x c x p. Here, we have ued Eq. 9 to repreent C c n term of multpole A,c kl at x c and Eq. 4 for the Laplacan of S 1, the pont x c and x p beng aumed to le nde the bac unt cell wth x L 0. Now nce C p mut be patally perodc, the mot general form for t wth ngularte at x p C p e p 1 kl A kl,k,l,p D kl S 1 r, 16 where e p a contant that may are n tranlatng the ngularte from x c to x p, and A,p kl are the multpole at x p. Subttutng Eq. 15 and 16 nto Eq. 14 we obtan 1 n A,p nm nm nm 1 k 1 kl A,c kl D kl,k,l Here, we have ued the reult that D kl r cp. 17 (r) at r0 nonzero only for, nk, and ml, and that t value for th pecal cae 1/ nm. Alo, n dervng the above reult we have aumed that the monopole at x c and x p, are equal,.e., A 0,c 00 A 0,p 00, a reult that verfed a poteror from Eq. 17. Thu, the term contanng 1 n Eq. 4 made no contrbuton to Eq. 17. Fnally, we alo made ue of the followng reult for the ntegraton of generalzed functon: rd kl rr cp dv1 k D kl V r cp. 18 Expreon 17 allow one to compute the multpole at x p gven ther value at x c. A more convenent form that ueful for computng thee multpole can be obtaned by ung the reult gven n Appendx A where we have preented more detaled formulae for evaluatng the dervatve of phercal harmonc. It may be noted that the frt few multpole at x p could alo be obtaned by a traghtforward Taylor ere expanon of C c around x p. Thu, ung G S 1 rr cp GS 1 rr cp GS 1 r..., 19 the relaton among frt few multpole can be readly obtaned A 0,p 00 A 0,c 00, A 0,p 10 A 0,c 10 r cp 1 A 0,c 00, A 0,p 11 A 0,c 11 r pc 2 A 0,c 11, It eay to verfy that thee are n agreement wth the more general reult gven by Eq. 17. Calculaton of hgher-order multpole ung the Taylor ere expanon, however, become cumberome and the method preented here baed on generalzed functon prove more convenent. To complete the tranlaton, we need to determne the contant e p. For th purpoe we tart wth the dentty V C 1 6 r2 2 CdV 3 1 n rc 1 V 2 r2 CdA 21 and once agan ubttute for C n turn both C p and C c. The volume V choen to be the bac unt cell n whch both x c and x p le and V the urface of the unt cell. Snce both C p and C c are requred to be equvalent at all pont on the urface of the unt cell, the urface ntegral n both cae mut be dentcal leadng thereby to the equalty of the volume ntegral cdv C c 1 6 r2 2 C C p 1 6 r2 2 C pdv. 22 Subttutng for C c and C p, notng that the ntegral of G S 1 over the unt cell vanhe, and ung the generalzed functon repreentaton of Laplacan of C c and C p, we obtan e p 2 3 A 20 0,p 2 3 A 00 0,c r pc r pc 2A 0,c 10 r pc 1 2A 0,c pc 11 r 2 2A 1,c 11 r pc 3 A 0,c 20, 23 whch can be further mplfed by ubttutng for A 0,p 20 from Eq. 17 to obtan e p 2 A 00 0,c r pc 2 Y 0 20 r pc 3A 0,c 11 r pc 2 A 1,c 11 r pc 3 24 Equaton 17 and 24 allow u to hft the multpole ngularte at pont x c to that at x p. Thee wll be ueful n combnng the dturbance created a group of partcle nto an equvalent dturbance created at a ngle pont x p. 2. Tranlaton of regular oluton We now conder the problem of tranlatng a feld C reg,p whch regular at both x p and x c thee are not to be confued wth the ngular pont we ued n the prevou dervaton and for whch a phercal harmonc expanon around x p known to the correpondng feld wth t expanon around x c. Let C reg,p 1 6 fr2,n,m E nm,p r 25 be the regular expanon around rxx p 0. We then wh to determne the coeffcent that appear n the expanon around x c C reg,c 1 6 frrcp 2,k,l E kl,c Y kl rr cp. 26 For th purpoe we ue the fact that E kl related to a kth order dervatve of C reg,c evaluated at rr cp E,c kl kl D kl C reg 1 6 frrcp. 27 2rr cp Subttutng for C reg from Eq. 25 we obtan the dered reult E,c kl 1 6 f kld kl r cp 2 2r r cp rr cp kl,n,m E,p nm D kl r cp. 28,c Once agan, expreon for the frt few coeffcent E kl could alo be obtaned ung the Taylor ere expanon, and the reult obtaned that way can be hown to be n agreement wth the above more general reult. Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo 1995

9 C. An O(N) algorthm We now decrbe the O(N) algorthm for computng the Laplace nteracton. Th cont of the followng tep: 1 Create a herarchy tree. The frt tep to create a herarchy among group of partcle. For mplcty, we hall aume that our bac unt cell cubc. We dvde th nto 8 equal-zed cube each wth t lnear dmenon half that of the bac cell. Thee are referred to a the level 0 boxe. Next, each box at level 0 further ubdvded nto 8 maller level 1 boxe leadng to a total of 64 boxe at level 1. The proce contnued to the fnet level m lev at whch the box ze uch that on average there are P partcle per fnet level box, P beng a contant of O(1) whoe prece value mut be determned by optmzng the total computatonal tme. Note that there are a total of m lev 1log 8 (N/P) level. Fnally, each partcle agned the fnet level parent box n whch t center le. 2 Upward pa. The econd tep to determne the multpole repreentaton of the feld nduced by a group of partcle that vald at large dtance from the group. It aumed that we hall determne the multpole of the partcle by a utable teratve procedure cf. Step 5. Thu, at the begnnng of each teraton we tart wth the aumed value of the multpole A, nm for each partcle and compute the contrbuton from each partcle multpole to t parent box multpole and the contant e p at m lev level ung Eq. 17 and 24 wth x p n that expreon beng the poton vector of the center of the parent box and x c and A,c kl, repectvely, the center and the multpole of partcle. Next, wth the multpole and the contant e for all the fnet level boxe computed, we determne the multpole and e for the next coarer m lev 1 level boxe wth each parent box multpole now determned from the multpole of t eght chldren at level m lev. Th procedure repeated to larger ze boxe to compute the contant e and the multpole of all the boxe at all the level. 3 Downward pa. The multpole and the contant e determned n Step 2 gve the far-feld repreentaton of the effect of partcle whoe center located n a gven box. We next want to compute f and E kl,.e., the coeffcent that appear n decrbng the regular feld, for all the boxe at all the level. Th acheved by tartng wth the boxe at level 1 or level 0 f the bac unt cell not cubc but oblong ntead, for example and determnng the contrbuton to the regular feld expanon about the center of the boxe from the dturbance due to partcle n the other boxe at the ame level but the one that are not t nearet neghbor. Here, and n the ubequent dcuon, we hall refer to all the 26 nearet neghbor of a gven box at a gven level and the box telf a the nearet neghbor of the box for the ake of brevty. Thu, a gven box ha 27 nearet neghbor. At level 1, there are boxe that are further away from a gven box and contrbuton to f and E kl of a gven box from the partcle n thee 37 boxe can be determned ung Eq. 11 wth the ummaton over n that expreon replaced by the ummaton over thee 37 equal generaton boxe. Of coure, x mut be replaced by the poton vector of the center of the box whoe regular coeffcent are beng computed and x by the center of the equal generaton box from whch the contrbuton beng computed. Alo, S to be ued equal the net nk S eq due to all the partcle repreented by the equal generaton boxe. Th can be determned from Eq. 12 wth the ummaton over once agan replaced by the ummaton over the equal generaton boxe. Now comparng wth the regular expanon gven by Eq. 25, we ee that at th level f for a gven box the um of S eq over t 37 equal generaton dtant neghbor. Next, we compute f and E kl of the boxe at the next fner level,.e., level 2. Unlke level 1, n addton to the contrbuton from t equal generaton level 2 boxe there are equal generaton boxe for each box at th level, we mut alo determne the contrbuton from the regular expanon of t parent box at level 1. Denotng the box at level 2 under conderaton by a upercrpt c, the parent by p, and the equal generaton box by eq, we wrte f c f p S eq, E,c kl E,p c kl eq eq E,eq c kl, 29 and ue Eq. 28 to determne the contrbuton from the parent (p c); the equal generaton contrbuton to f and E kl determned, a before, wth the ue of Eq. 11 and 12. It hould be noted that the parent of a box account for the feld nduced by all the partcle lyng n the dtant boxe of level 1. Thu, for each level 2 box, we have now accounted for all the partcle that are outde t nearet 27 level 2 boxe. The partcle n thee 27 boxe are too cloe to an arbtrarly elected partcle n the box under conderaton and therefore we mut wat for the calculaton of the coeffcent for the fner level boxe to account for ther effect. The above procedure of combnng contrbuton from the equal generaton boxe and the parent box contnued to level 3,4,...,m lev. At all thee level, the total number of equal generaton boxe from whch the contrbuton are computed equal 189, except for the fnet m lev level, for whch we um over all the 216 boxe. Th nclude addtonal 27 nearet neghbor boxe wth one mall dfference: the ngular part 1/r removed from S 1 before computng the contrbuton from thee nearet 27 boxe. Phycally, th account for all the partcle that are lyng n the perodc mage of the nearet neghbor boxe at the fnet m lev but not the partcle n the nearet boxe themelve whch are too cloe to permt the ue of far-feld repreentaton n determnng the regular feld expanon. We hall account for thee partcle eparately va Step 4. Fnally, we compute the contrbuton to f and E kl of each partcle from the fnet level parent box. There, of coure, no contrbuton from the equal generaton boxe at the partcle level. 4 Partcle to partcle contrbuton. The contrbuton from the partcle n the nearet 27 boxe are evaluated n the ame way a for the contrbuton from the equal generaton boxe n the prevou tep except that the functon S 1 (r) now replaced by 1/r becaue the regular part of S 1 ha already been accounted for n Step 3. 5 Determne new gue for the multpole. The Step 2 4 conttute one teraton n olvng for the multpole 1996 Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo

10 of the partcle. A utable teratve procedure, uch a the generalzed moment redual GMRES method, ued to obtan the new gue for the multpole. Step 2 5 mut be repeated untl the multpole converge to wthn a pecfed accuracy. We now make everal remark regardng the procedure outlned above. Remark 1. For problem n upenon mechanc, we typcally ue the perodc boundary condton. For th pecal cae, creatng the herarchy tree a trval matter. Once the bac unt cell dvded nto a pecfed number of level, th tree reman unchanged throughout the dynamc mulaton. In order that th reman computatonally effcent, the number of partcle n any of the fnet level boxe mut not become much greater than t average value P. Th wll be true provded that no olated cluter wth a large number denty develop a the mulaton proceed. Th an mportant conderaton n tellar dynamc where the overall number denty of partcle tar/planet very mall and the cluter galaxy formaton an mportant phenomenon to be nvetgated through mulaton. In uch a cae, m lev may have to be changed durng the mulaton and may not reman unform throughout the bac cell. The computatonal effort for the determnaton of the tree for uch hghly nonunform ytem cale a N(logN) 4 a hown by Aluru and co-worker. 27,28 The number denty of partcle n mot upenon problem typcally large and the probablty of developng a hghly nonunform upenon generally mall. In few exceptonal cae, uch a ga old fludzed bed where large vod devod of any partcle may form, creatng tree wth nonunform m lev may prove ueful. Remark 2. If the multpole moment repreentng the effect of group of partcle are computed up to nn p, the computatonal effort for the upward pa cale a (N p 1) 4 N: there are a total of (N p 1) 2 multpole coeffcent to be evaluated and each depend lnearly on the ame number of multpole of t chldren. The computatonal cot for computng the parent to chld contrbuton to the coeffcent E kl n the regular expanon alo O((N p 1) 4 N), aumng that thee coeffcent are alo computed up to kn p. The cot of computng the contrbuton from the equal generaton boxe roughly 216/P tme that for the parent to chld calculaton, P beng the average number of partcle per box. Fnally, the partcle to partcle contrbuton requre an O(27P(N 1) 4 N) effort. Here, N the order of multpole retaned n decrbng the feld nduced by the partcle. Thu, a a frt approxmaton, the total computatonal cot per one teraton controlled by the equal generaton contrbuton and the partcle to partcle contrbuton. A rough etmate of the total operaton count therefore 216(N p 1) 4 /P27P(N 1) 4 N and th ha a mnmum for P3(N p 1)/(N 1) 2. Of coure, th to be ued only a a rough gude to etmate how optmum P mght depend on N and N p. More accurate etmate can be obtaned through numercal expermentaton. The total operaton count and the etmate of optmum P obtaned here are dfferent from that of Greengard and Rokhln 12 who ued a lghtly more complex algorthm 3 whch cale a N p ntead of the fourth power dependence obtaned n the preent algorthm. Smlar reducton n the exponent of N p obtaned n a related calculaton by Znchenko. 29 Thee nvetgator condered very hgh value of N p for whch the reducton gnfcant. A wll be hown n Sec. IV, a very good accuracy obtaned even wth N p a mall a 3 and therefore we have not mplemented ther method here. Remark 3. If the dmenon of the unt cell doe not change n dynamc mulaton, then t poble to ave conderable computatonal tme by torng varou matrce that are needed n computng the parent to chld or chld to parent contrbuton, and the contrbuton from the equal generaton boxe. In partcular, the only place where one need to ue Ewald technque for determnng S 1 and t dervatve n the equal generaton computaton and thee calculaton need to be done only once, at the begnnng of the mulaton. Alo the total number of dervatve to be evaluated O(4N 2 p logn), whch amount to a neglgble cot compared wth a total dervatve of O(4N 2 N 2 ) that one mut evaluate at every tme tep n the O(N 2 ) algorthm decrbed n the prevou ecton. III. THE METHOD FOR STOKES INTERACTIONS Havng decrbed n detal the method for Laplace nteracton, we now conder the method for Stoke nteracton. The bac dea ame a before and we need to addre only two mportant ue: how to nclude the lubrcaton effect uch that reaonably accurate partcle traectore are obtaned wth very few unknown per partcle; and how to tranlate the ngular and regular oluton of Stoke equaton. Of coure, the lubrcaton effect could alo be mportant n ome problem nvolvng Laplace nteracton, e.g., the problem of determnng the effectve thermal conductvty of dene upenon contng of hghly conductng ncluon, but we choe to defer the dcuon of the ue to the preent ecton to explan the mportant apect of the algorthm through a relatvely mple problem for whch the lubrcaton effect are abent. We hall follow the method of Sangan and Mo 8 to account for the lubrcaton force n Stoke flow. Th method eparate the force denty on the urface of the partcle nto a ngular dtrbuton of the force denty near the narrow gap between the partcle and a regular dtrbuton of force denty over the entre urface of the partcle. The ngular force denty gve aymptotcally correct force on the partcle n term of ther velocte and the gap wdth whle the regular dtrbuton expanded n the cae of phercal partcle n a ere of multpole at the center of the partcle, and ther value are determned by atfyng the boundary condton on the urface of the partcle. In addton to gvng correct lubrcaton force and torque on the partcle n cloe proxmty, the method alo account for the effect of the velocty nduced by the lubrcaton force on the other partcle n the upenon. The velocty of the flud gven by N u xu x 1 M v xx u lub x, 30 Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo 1997

11 where u the average velocty of the upenon, v a patally perodc Green functon for the Stoke equaton, M a dfferental operator, and u lub the velocty nduced by the lubrcaton force denty. Detaled expreon for each of thee quantte may be found n Mo and Sangan 7 and Sangan and Mo. 8 In partcular, (F /4)v (r) the velocty at r due to pont force F actng at the lattce pont of the perodc array. A hown by Hamoto, 25 v S 1 2 S 2 r r, 31 where S 1 the ame functon a ntroduced earler n the Laplace nteracton calculaton, and S 2 atfe 2 S 2 S 1. v (r) ha a ngular behavor near r0 a gven by v v 1 r r r r, 32 the well-known Oeen tenor for the flow nduced due to a pont force at orgn n a flud at ret at nfnty. The actual expreon for the dfferental operator M omewhat nvolved but, fortunately, wll not be needed for our dcuon. The only thng that we need to note that t defned uch that, when operated on v, t produce term that concde wth the ngular term n the Lamb general oluton n term of phercal harmonc. More pecfcally, let the velocty of the flud near the urface of partcle be expanded n the Lamb oluton a u u, u r, 33 wth u, and u r, beng, repectvely, the ngular and regular part of u at xx. Thee are defned by u, r n1 c n r 2 p, n b n rp, n xr, n, n, 34 where rxx, c n 2n 2n2n1, b n n1 n2n1, 35 and p n, n, and n are phercal harmonc of degree n1. For th ecton we temporarly uppre our prevou notaton accordng to whch the volume fracton of the partcle. We defne the above phercal harmonc n term of multpole coeffcent P nm, etc., by mean of p n, m, P nm n, m, T nm n, m, nm,,, r 2n1, r 2n1, r 2n1, 36 where the ummaton over m from 0 to n and for from 0 to 1. Lkewe, the regular part wrtten a u r, r n1 wth c r n c n1 p n r, m, P nm n r, m, T nm n r, m, nm c n r r 2 p n r, b n r rp n r, xr n r, n r,,, b r n b n1, and r, r,,, r, In Mo and Sangan, 7 we have defned the dfferental operator M n term of the coeffcent P, nm, etc., that appear n Eq. 36 uch that u, M v, 39 where v the Oeen tenor cf. Eq. 32. We alo gave expreon for evaluatng the coeffcent that appear n the regular part of the velocty at x n term of the ngular coeffcent P, kl, etc., of all the partcle n the upenon. Th analogou to the expreon we cted for the Laplace nteracton cf. Eq. 11 except that the correpondng expreon for the Stoke nteracton are conderably more nvolved. The drect evaluaton of thee regular coeffcent requre an O(N 2 ) computatonal effort. In the preent ecton we hall derve the reult for the tranlaton of regular and ngular oluton that wll allow u to determne the regular coeffcent wth an O(N) effort. A. Tranlaton of Stoke ngularte We wh to tranlate u c M c v (xx c ) wth t ngularte at x c to a velocty feld wth t ngularte at x p uch that both are equvalent at a pont x uffcently far away from both x c and x p. Snce the feld wth ngularte at x p mut alo be patally perodc, the mot general form for t gven by u p e M p v xx p, 40 where e a contant. Let p c and p p be the correpondng preure feld. Subttutng p for C n Eq. 14 we obtan r 2 p c dv r V r 2 p p dv r, 41 V where rxx p. Now nce the preure atfe the Laplace equaton except at t ngularte, the ntegral n the above expreon can be evaluated mply from the ngular behavor of p whch can be wrtten a p P kl k,l, Y kl r 2k1 kl k,l, 1 P kl D kl r 1, 42 where we have made ue of Eq. A1 n wrtng the lat equalty. Notng that 2 r 1 4(r), t relatvely eay to carry out ntegraton n Eq. 41 to obtan a relaton mlar to Eq Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo

12 P,p nm nm nm 1 kl P,c kl D kl,k,l r pc, 43 where r pc x p x c. Now we determne T,p nm. Let u be the vortcty. Ung Eq. 34 and 35, t can be hown that the ngular part of the vortcty gven by,a n 1 n ra p n,a n n,a r a 2 n,a, 44 where the upercrpt a tand for c a well a p, and r a xx a. Now we note that r a atfe the Laplace equaton at all pont except at t ngular pont x a. Th can be een by multplyng Eq. 44 wth r and ung r a rr ap to yeld r,a n 1 n rap rp n,a nn1nr ap r 2 2 r ap r 2 n. 45 Takng Laplacan of the above equaton and ung reult uch a r a n,a (n1) n,a nce n,a a homogeneou polynomal of degree n1 nr a ) and rr a r ap we obtan 2 r,a n 1 n rap r 2 p n,a n 2 5n6 n2r ap r r ap r 2 2,a n. 46 Now ubttutng r for C n Eq. 14, ung the generalzed functon repreentaton of Laplacan of p n and n, and mplfyng the reultng ntegral we obtan nn1 T,p 1 nm nm nm k,l, kl klkn1t,c D kl D kl r rrr pc. 1 k P kl,c r 47 Th can be further mplfed ung the general reult for dfferentaton of phercal harmonc gven n Appendx A. A convenent et of formula for computng all the multpole at x p from thoe at x c gven n Appendx B. To compute the coeffcent,p nm we tart wth the dentty x u u u p u, 48 x x where p (u /x u /x ) the tre tenor correpondng to a feld (u,p) and the tre correpondng to a regular feld (u,p). (u,p) on the other hand, allowed to be ngular at ome pont n the pace. We now chooe the regular feld to be gven by p r, uc r n r 2 b r n r, 49 ubttute for (u,p) both (u c,p c ) and (u p,p p ) n turn, ntegrate the dentty 48 over a volume V large enough to contan both x c and x p, apply the dvergence theorem, and ue the equvalence of the two feld at all pont on the boundary V to obtan c r n r 2 rb r n r r c r u c dv r V c r n r 2 rb r n r r p V r u p dv r. 50 Snce the dvergence of tre and velocty are zero except at the ngular pont, only the ngular part of the velocty and tre wll contrbute to the above ntegral. Subttutng the ngular part of the velocty for u a, where a tand for c or p, the ntegrand n the above expreon reduce to k 1 k c r nr 2 c r n r 2 b r n r b r n r r a c k r a 2 2 p k r a 2 k 2 k. 51 Ung the generalzed functon repreentaton of Laplacan of p k, etc., and carryng out the ntegraton n Eq. 50 we obtan,p nm nm nm kl,k,l D kl D kl D kl r r 2,c D kl 1 n1 T kl,c r pc P kl,cc r n c k kn1 1 1 kn1 k2k1r 2k r 2k 2k2k1 r2 D kl. 52 rr pc A convenent formula for evaluatng nm baed on the above expreon gven n Appendx B. Fnally, to complete the tranformaton of the ngular oluton at x c to that at x p, we need to determne the contant e n Eq. 40. For th purpoe we ue the dentty u c r u c dv u p r u p dv ru n da, 53 where the unt cell enclong both x c and x p and n a unt outward normal on t urface. A before, we have ued the equvalence of u c and u p on. Snce v and t dervatve are olenodal, and nce ther ntegral over the unt cell vanh, ubttutng for u c and u p cf. Eq. 40 yeld e0. Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo 1999

13 B. Tranlaton of regular oluton of Stoke equaton We now conder a oluton of Stoke equaton whch regular both at x p and x c and for whch the coeffcent (P r,p nm,t r,p nm, r,p nm ) n the regular Lamb oluton around x p are known. Our goal to derve expreon for t expanon around x c,.e., to determne the coeffcent P r,c r,c nm,t nm and r,c nm. Snce the preure atfe the Laplace equaton, the coeffcent n t expanon are related by the ame expreon a for E kl n Sec. II cf. Eq. 28 wth f 0 P r,c kl kl,n,m P nm r,p D kl r cp. 54 Smlarly, we ue the fact that r r wth rxx c atfe the Laplace equaton and obtan kk1t r,c kl kl r r r0 kl,n,m kn1t r,p nm D kl P r,c nm r D kl r rr cp. r cp 1 n1 55 Fnally, we ue the fact that r u r bharmonc, and therefore r,c kl can be evaluated from 7 r,c kl kl k D kl klkl1 4k2 2 D k2,l r u r. rr cp 56 Once agan, the detaled expreon for determnng varou coeffcent of the regular part of the velocty are gven n Appendx B. C. The O(N) algorthm for Stoke nteracton The O(N) algorthm for Stoke nteracton cont of the ame tep a outlned n Sec. II. In addton to computng the contrbuton from the ngularty at the center of partcle to the regular feld near partcle, we alo calculate the flow nduced by the lubrcaton force between each par of partcle n cloe proxmty. In Sangan and Mo, 8 we gave the expreon for the flow due to lubrcaton force n term of a force dpole ngularty tuated at the center of the gap between the partcle. The upward pa now determne the equvalent force multpole of the fnet level boxe from both the force multpole of the partcle and the lubrcaton ngularte. The remander of the upward pa calculaton n whch the multpole are evaluated for the coarer level boxe reman unaffected by the lubrcaton ngularte. In the downward pa calculaton, the contrbuton from the equal generaton boxe evaluated by the expreon gven n Mo and Sangan 7 wth the center of partcle n that tudy now replaced by the center of the equal generaton box, and the center of partcle replaced by the center of the box whoe regular coeffcent are beng evaluated. Fnally, n the partcle to partcle tep, we evaluate the contrbuton from the partcle and the lubrcaton ngularte lyng n the 27 nearet neghbor boxe. For th we need addtonal expreon for computng the contrbuton to coeffcent n the regular part of the velocty near each partcle from the ngularte tuated at the center of the partcle and the lubrcaton gap. Thee expreon are gven n Appendx B. IV. APPLICATION TO FEW SPECIFIC SUSPENSION PROBLEMS In th ecton we apply the method decrbed n the prevou two ecton to few pecfc problem wth the am of aeng the utlty of the method n tudyng ytem wth large N. Snce the computatonal effort ncreae a N 4 p, we hall be partcularly ntereted n determnng the accuracy of the method for maller N p. To valdate the analytcal reult for computng the tranlaton of ngular and regular oluton of Laplace and Stoke equaton, and to tet the accuracy of the computer program, we found t very ueful to compare the reult of the program agant O(N 2 ) program whch were extenvely teted prevouly for ther accuracy. 7,8,18,24 Snce the computatonal tme requred by thee O(N 2 ) algorthm very large, the accuracy for large N wa teted by arrangng the N partcle wthn the bac unt cell n a perodc array wth each ub-unt cell contanng N 0 partcle. Typcally, the calculaton were checked wth N 0 1, whch correpond to a truly perodc array, and wth N 0 16, the partcle wthn the ub-unt cell arranged n the latter cae n a random array. Snce n the O(N 2 ) method we compute each element of the matrx A eparately and then evaluate the product A x, the mot mportant tet of the O(N) algorthm requre the drect evaluaton of th product to match wth the correpondng product evaluated by the O(N 2 ) method for any gven x. Here, for example, for the cae of Laplace nteracton, x the vector of A, nm whle the product the vector of E kl cf. Eq. 11 plu a contant tme the vector of A, kl, wth the contant dependng on the boundary condton at the urface of the partcle. The element of the matrx A beng related to the dervatve of S 1 evaluated at xx x. Takng A, nm 1 for all n,m,, and, we evaluate the mean value of E, kl over all the partcle and t varance from the mean. For the pecal cae of a perodc array wth N 0 1, the varance mut, of coure, be zero. However, the O(N) algorthm wth fnte N p ntroduce ome fluctuaton even n the cae of a perodc array. Thee fluctuaton were found to decreae rapdly wth the ncreae n N p. The mean value for each E kl wa alo found to agree well wth that obtaned from the O(N 2 ) algorthm a we hall how n more detal below. Smlar tet were alo made for the Stoke nteracton code. A. Laplace nteracton A few typcal reult for the relatve error for the pecal cae of dffuon-controlled reacton are gven n Table I. The boundary condton on the urface of partcle for th problem yeld Eq. 7. Denotng the left-hand de lh of th equaton by r, nm we calculate two meaure of the relatve error. The frt defned by E 1 1 n eq n,m, r nm, r nm, r nm,, Phy. Flud, Vol. 8, No. 8, Augut 1996 A. S. Sangan and G. Mo