Numerical Simulations of Particulate Suspensions via a Discretized Boltzmann Equation Part I. Theoretical Foundation

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1 Under consderaton for publcaton n J. Flud Mech. 1 Numercal Smulatons of Partculate Suspensons va a Dscretzed Boltzmann Equaton Part I. Theoretcal Foundaton By ANTHONY J. C. LADD Lawrence Lvermore Natonal Laboratory, Lvermore, Calforna (Receved 4 September 2007) A new and very general technque for smulatng sold-flud suspensons s descrbed; ts most mportant feature s that the computatonal cost scales lnearly wth the number of partcles. The method combnes Newtonan dynamcs of the sold partcles wth a dscretzed Boltzmanuaton for the flud phase; the many-body hydrodynamc nteractons are fully accounted for, both n the creepng-flow regme and at hgher Reynolds numbers. Brownan moton of the sold partcles arses spontaneously from stochastc fluctuatons n the flud stress tensor, rather than from random forces or dsplacements appled drectly to the partcles. In ths paper, the theoretcal foundatons of the technque are lad out, llustrated by smple analytcal and numercal examples; n the companon paper, extensve numercal tests of the method, for statonary flows, tme-dependent flows, and fnte Reynolds number flows, are reported. 1. Introducton Numercal smulatons, whch take explct account of the hydrodynamc forces between the suspended partcles, are becomng useful tools for studyng the dynamcal and rheologcal propertes of suspensons. There are at least three mportant flow regmes whch can be addressed by numercal smulaton: collodal suspensons of sub-mcron szed partcles, where Brownan forces and vscous forces balance; suspensons of macroscopc partcles (.e. larger than 10µm), where the vscous forces alone are mportant; and flows at small but non-zero Reynolds number (1 < R e < 100). At present, computatonal cost s the lmtng factor; even wth supercomputers, t s not feasble to smulate more than about 100 partcles wth current methods. Thus, development of relable and more effcent smulaton technques, able to cope wth thousands of suspended partcles, would have a sgnfcant mpact on our understandng of partculate suspensons, complementng present expermental and theoretcal knowledge. In these two papers a new smulaton technque for partculate suspensons s descrbed; t combnes Newtonan dynamcs of the sold partcles wth a dscretzed Boltzmann model (McNamara & Zanett (1988); Hguera et al. (1989)) for the flud. The basc dea s llustrated n Fg. 1, whch shows 5 sold partcles suspended n a background flud. The flud can be modeled as a contnuum (Fg. 1a), as a molecular lqud (Fg. 1b), or as a dscrete velocty (lattce) gas (Fg. 1c). Because of the large scale separatons, the dynamcs of the sold partcles are largely ndependent of the detaled mechancs of the suspendng flud. Dscrete velocty models of the flud combne most of the features of a fully molecular smulaton of sold and lqud phases, but are orders of magntude faster computatonally. They have many advantages over conventonal methods of smulatng partculate suspensons, whch are usually based

2 2 A.J.C. Ladd Fgure 1. Mcroscopc models of a collodal suspenson. The dynamcal propertes of the large partcles are nsenstve to the detaled motons of the background flud, so that the contnuum flud n Fg. 1a can be equally well replaced by ether a molecular solvent (Fg. 1b) or a lattce gas (Fg. 1c). Lattce-gas/lattce-Boltzmann smulatons are many orders of magntude faster computatonally. on complcated, computatonally ntensve solutons of the Stokes equatons. By contrast, lattce-gas/lattce-boltzmann smulatons are fast, flexble, and smple. The new method s closely related to earler suspenson modelng usng lattce-gas cellular automata (Ladd et al. (1988); Ladd & Frenkel (1989, 1990); Ladd (1991); van der Hoef et al. (1991)), but the large and uncontrollable statstcal fluctuatons present n lattce-gas models are suppressed, reducng the need for computatonally expensve ensemble averagng. In the classcal theory of suspensons (Happel & Brenner (1986)), the hydrodynamc nteractons are assumed to be fully developed; n other words, there s a complete separaton of tme scales between the dynamcs of the flud and the moton of the sold partcles. Most smulaton methods (for nstance Ermak & McCammon (1978); Brady & Bosss (1988); Ladd (1988); Tran-Cong & Phan-Then (1989); Karrla et al. (1989)) utlze ths approxmaton, even though t mposes a crpplng numercal burden, assocated wth the global nature of the nteractons; n such cases one must ether make drastc smplfcatons (as n Ermak & McCammon (1978)), or pay the steep computatonal cost of an algorthm that scales as at least the square and often as the cube of the number of partcles (Brady & Bosss (1988); Ladd (1988); Tran-Cong & Phan-Then (1989); Karrla

3 Numercal Smulatons of Partculate Suspensons...I 3 et al. (1989))., In realty, hydrodynamc nteractons develop n tme and space from purely local forces generated at the sold-flud surfaces, whch then dffuse throughout the flud. The Reynolds number ntroduces an effectve range for the hydrodynamc nteractons, proportonal to Re 1/2. By contrast, truncatng the creepng-flow hydrodynamc nteractons beyond some crtcal dstance s unlkely to be feasble; n analogous smulatons of charged partcles, sphercal truncaton of the Coulomb nteracton between ons leads to large errors (Allen & Tldesley (1987)). Recently a new smulaton technque for partculate suspensons has been developed (Ladd (1993)) whch explots the spatal localty of tme-dependent hydrodynamc nteractons; as a result the computatonal cost scales lnearly wth the number of partcles. The method s also very flexble; the partcle sze and shape, the electrostatc nteractons, the flow geometry, the Péclet number (the rato of vscous forces to Brownan forces), and the Reynolds number (the rato of nertal forces to vscous forces), can all be vared ndependently. The popularty of boundary-ntegral methods, despte ther obvous computatonal drawbacks, s a reflecton of the dffcultes nvolved n solvng the Naver-Stokes equatons n complex geometres. The drawbacks of an elementary fnte-dfference method, usng a regular mesh, are dscussed n secton IIIE of Part II. More complex fnte-dfference and fnte-element technques must deal wth the challenge of fndng a sutable computatonal mesh on whch to calculate the flud flow. In these two papers t wll be shown that an algorthm based on a dscretzed Boltzmanuaton can successfully solve complex flud flow problems usng only a smple cubc array of nodes. Our computatonal method s based on the well establshed connecton between the dynamcs of a dlute gas and the Naver-Stokes equatons (Chapman & Cowlng (1960)). Thus, the problem s to determne the tme evoluton of the one-partcle velocty dstrbuton functon n(r, v, t), whch defnes the densty of partcles wth velocty v around the space-tme pont(r, t). By ntroducng the assumpton of molecular chaos,.e. that successve bnary collsons n a dlute gas are uncorrelated, Boltzmann was able to derve the ntegro-dfferental equaton for n named after hm (see Chapman & Cowlng (1960)); t n + v n = ( ) dn. (1.1) dt coll The Boltzmanuaton has the form of a contnuty equaton for the velocty dstrbuton functon, wth the addton of sources and snks due to ntermolecular collsons. The hydrodynamc felds, mass densty ρ and momentum densty j = ρu, together wth The exact scalng depends on the problem to be solved. In some nstances, partcle veloctes are computed for a gven set of forces, n whch case the computaton can scale as N 2. However n many cases, for nstance to smulate Brownan moton (Bosss & Brady (1987)), the full 6N 6N dffuson coeffcent matrx s needed; here the computatonal cost s of order N 3. Moreover, determnng lubrcaton forces (Durlofsky et al. (1987)) also nvolves an order N 3 calculaton of the frcton coeffcent matrx. The Stokes equatons can be solved very effcently by spectral methods (Fogelson & Peskn (1988); Sulsky & Brackbll (1991)). However, there are then the usual dffcultes assocated wth ncorporatng sold boundary condtons nto spectral codes; n the referenced work the partcle surfaces are represented by pont forces. Numercal studes of the Naver-Stokes equatons ndcate that, even for pure flud flows, the lattce-boltzmanuaton s qute compettve wth the best spectral methods (Chen et al. (1992)).

4 4 A.J.C. Ladd the momentum flux Π, can be descrbed by approprate moments of n; ρ(r, t) = mn(r, v, t)dv, j(r, t) = (mv)n(r, v, t)dv, Π(r, t) = (mvv)n(r, v, t)dv, (1.2) where m s the molecular mass. In what follows, only the transport of mass and momentum wll be consdered; smlar results can be derved for energy transport although the analyss s consderably more nvolved. Snce ntermolecular collsons are relatvely rare n a dlute gas, the collsonal contrbutons to the fluxes can be gnored; thus takng moments of Eq. 1.1 we fnd the expected conservatouatons for mass and momentum, t ρ + j = 0, t j + Π = 0. (1.3) The rght-hand sdes of Eqs. 1.3 vansh because mass and momentum are conserved n the collson process, but fndng a soluble equaton for the momentum flux requres a more detaled consderaton of the collson operator. The dffcultes assocated wth solvng the Boltzmanuaton arse from the complexty of the collson operator, wrtten smply as (dn/dt) coll n Eq The earler knetc theory of Maxwell leads to a smplfed form for the collson operator ( dn dt ) coll = n neq, (1.4) τ where the relaxaton tme τ s an emprcal parameter, and s the equlbrum Maxwell-Boltzmann dstrbuton. Maxwell was able to show that the dstrbuton, = ρ ( ) 3/2 ( ) m m(v u) (v u) exp, (1.5) m 2πk B T 2k B T s a collsonal nvarant,.e. (d /dt) coll = 0. Thus, f the system should once reach such a unform state, t would reman there ndefntely. Boltzmann s H-theorem showed that any ntal dstrbuton proceeds monotoncally to, thus establshng t as the true equlbrum dstrbuton for an deal gas. The Maxwell form for the collson operator s nadequate n some respects; t predcts a fxed rato between the vscosty and thermal conductvty, and a vanshng coeffcent of thermal dffuson. Boltzmann showed that these dscrepances could be removed by a detaled consderaton of the effects of ntermolecular forces on the collson process. Nevertheless, Eq. 1.4 s suffcent to gve a smple and more or less correct understandng of the connecton between the Boltzmann knetc equaton and Naver-Stokes flud dynamcs. The dstrbuton functon s wrtten as a sum of equlbrum and non-equlbrum terms n = + n neq, (1.6) Ignorng energy conservaton leads to the sothermal expresson for the sound speed c s = p k BT/m, rather than the adabatc result c s = p 5k BT/3m. Ths s of no consequence to the later parts of ths paper where the sound speed s treated as a varable parameter. Actually τ can be explctly related to the rather unusual case of an r 5 ntermolecular force law.

5 Numercal Smulatons of Partculate Suspensons...I 5 where the non-equlbrum dstrbuton, n neq, has the mportant property that ts zeroth and frst velocty moments vansh (c.f. Eq. 1.2), along wth the trace of the second moment (mv 2 )n neq (r,v, t)dv. The momentum flux can be smlarly decomposed; ts equlbrum contrbuton can be evaluated drectly from Eq. 1.5, Π eq = ρk BT m 1 + ρuu, (1.7) and s equvalent to the Euleran form for the momentum flux, wth an deal gas equaton of state p = ρk B T/m. The non-equlbrum momentum flux Π neq (the overbar ndcates the traceless projecton) can be obtaned from a knetc equaton for Π analogous to Eq. 1.3 t Π eq + (mvvv) (r,v, t)dv = ( τ 1 )Π neq. (1.8) In dervng Eq. 1.8 t has been assumed that the system s close to equlbrum, so that n can be replaced by, and that the hydrodynamc tme scales are large compared wth τ. The left-hand sde of Eq. 1.8 can be evaluated explctly, usng Eqs. 1.5 and 1.7; the result s Π neq = ρk ( BTτ u + u t), (1.9) m whch s the Naver-Stokes form for the vscous momentum flux wth vscosty η = ρk B Tτ/m. In the precedng dscusson t was shown, n a rather approxmate fashon, that the Naver-Stokes equatons follow drectly from the Boltzmanuaton n the lmt that the dmensons of the macroscopc flow felds are much larger than the mean free path between molecular collsons; a rgorous dervaton of ths result can be found n Chapman & Cowlng (1960). Because of ts complexty, there are few drect numercal solutons of the Boltzmanuaton (an excepton s Yen (1984)), but stochastc, partcle-based smulatons are qute commonly used n molecular gas dynamcs; the merts of ths approach are dscussed n Brd (1976) and Brd (1990). A varant of ths approach has recently been ntroduced to smulate partcle suspensons (Hoogerbrugge & Koelman (1992); Koelman & Hoogerbrugge (1993)). However, a key realzaton s that the onepartcle dstrbuton functon n(r, v, t) contans much more nformaton than s strctly necessary to solve flud-dynamcs problems. Hence n recent years, there has been a resurgence of nterest n dscrete-velocty or lattce-gas models (Frsch et al. (1986, 1987)), n whch the contnuous dstrbuton of molecular veloctes s replaced by a few dscrete values, carefully chosen to ensure that the moment equatons 1.3, 1.7, and 1.9 are reproduced correctly. Numercal studes have shown that lattce-boltzmann smulatons are comparable n accuracy and computatonal cost to state-of-the-art Naver-Stokes solvers, ether fnte dfference (McNamara & Alder (1992)) or spectral (Chen et al. (1992)). In the lattce-boltzmann approxmaton, the fundamental quantty s the dscretzed one-partcle velocty dstrbuton functon n (r, t), whch descrbes the number of partcles at a partcular node of the lattce r, at a tme t, wth a velocty c ; r, t, and c are dscrete, whereas n s contnuous. As before, the hydrodynamc felds, ρ, j, and Π, are moments of ths dscrete velocty dstrbuton (c.f. Eqs. 1.2): ρ = n, j = n c, Π = n c c. (1.10) The lattce-boltzmanuaton has two mportant propertes whch are valuable for smulatons of partculate suspensons. Frst, the connecton to molecular mechancs makes t possble to derve smple local rules for the nteractons between the flud and the

6 6 A.J.C. Ladd suspended sold partcles (Ladd (1993)); ths was demonstrated n our earler lattce-gas smulatons (Ladd & Frenkel (1990); Ladd (1991); van der Hoef et al. (1991)). Second, the dscrete one-partcle dstrbuton functon n contans addtonal nformaton about the dynamcs of the flud beyond that contaned n the Naver-Stokes equatons; n partcular, the flud stress tensor, although dynamcally coupled to the velocty gradent (Frsch et al. (1987)), has an ndependent sgnfcance at short tmes. Ths addtonal flexblty allows us to smulate molecular fluctuatons, leadng to Brownan moton of the suspended partcles. To do ths, a random fluctuaton, uncorrelated n space and tme, s added to the flud stress tensor (Ladd (1993)); the varance of the fluctuatons defnes the effectve temperature of the fluctuatng flud (Landau & Lfshtz (1959)). Ths approach s qute dfferent from Brownan dynamcs (Ermak & McCammon (1978)) or Stokesan dynamcs (Bosss & Brady (1987)), where random fluctuatons are appled drectly to the partcles; to nclude hydrodynamc nteractons n these methods requres samplng from the 6N 6N dffuson matrx, whch s extremely tme consumng. The layout of the remander of ths paper s as follows. In secton II the lattce- Boltzmann model s descrbed and ts connecton to Naver-Stokes flud dynamcs establshed. Secton III descrbes the mplementaton of the sold-flud boundary condtons at the mcroscopc level, together wth analytc and numercal results for shear flows and channel flows. Fluctuatons are ntroduced n secton IV; t s verfed that the smulatons satsfy the fluctuaton-dsspaton theorem and that the correct shear vscosty can be obtaned from an approprate Green-Kubo formulae. In the companon paper (Ladd (1994)), referred to hereafter as Part II, results of extensve tests of creepng-flow hydrodynamcs are reported, for both perodc arrays and random dstrbutons of spheres; tme-dependent and fnte Reynolds number flows are also dscussed. Three-dmensonal smulatons of up to 1024 collodal partcles, movng under the acton of Brownan forces, are also reported. 2. Dscrete Boltzmann Approxmaton The computatonal utlty of the lattce-boltzmanuaton s related to the realzaton that only a small set of dscrete veloctes are necessary to smulate the Naver-Stokes equatons (Frsch et al. (1986)). It may be helpful n what follows to magne an underlyng mechancal model n whch dentcal partcles move wth dscrete veloctes from node to node of a regular lattce; much of the knetc theory dlute gases, outlned n secton I, can then be carred over drectly to the dscretzed verson. The specfc model used n ths work has 18 dfferent veloctes correspondng to the near-neghbor and second-neghbor drectons of a smple cubc lattce. Thus there are sx veloctes of speed 1, correspondng to (100) drectons n the lattce and 12 veloctes of speed 2, correspondng to the (110) drectons, for a total of 18. All quanttes n ths paper are expressed n lattce unts, for whch the dstance between nearest-neghbor nodes and the tme for the partcles to travel from node to node are both unty. Note that the veloctes are such that all partcles move from node to node smultaneously. The tme evoluton of the dstrbuton functons n s descrbed by a dscrete analogue of the Boltzmanuaton (Frsch et al. (1987)), n (r + c, t + 1) = n (r, t) + (r, t), (2.1) where s the change n n due to nstantaneous molecular collsons at the lattce nodes. The post-collson dstrbuton n + s propagated for one tme step, n the drecton c. The collson operator (n) depends on all the n s at the node, denoted collectvely by n(r; t); t can take any form, subject to the constrants of mass and mo-

7 Numercal Smulatons of Partculate Suspensons...I 7 mentum conservaton. An exact expresson for the Boltzmann collson operator has been derved for several dfferent lattce-gas models (Frsch et al. (1987); McNamara & Zanett (1988)), under the usual assumpton that the dstrbuton functons n(r; t) are uncorrelated from those at prevous tmes. However, such collson operators are complex and ll-suted to numercal smulaton. A computatonally useful form for the collson operator, smlar to Eq. 1.4, can be constructed by lnearzng the collson operator about the local equlbrum (Hguera et al. (1989)),.e. (n) = ( ) + j L j (n j j ), (2.2) where L s the lnearzed collson operator, and ( ) = 0 by defnton. It s not necessary to construct a partcular collson operator and from ths calculate L; rather t s suffcent to consder the general prncples of conservaton and symmetry and then to construct the egenvalues and egenvectors of L. However, before dong ths, the proper form for the equlbrum dstrbuton functon wll be determned. To establsh the connecton between molecular mechancs and flud dynamcs, t s agan necessary to splt the dstrbuton functon nto aulbrum part and a nonequlbrum part n = + n neq. (2.3) The equlbrum dstrbuton s a collsonal nvarant (.e. ( ) = 0), and depends only on the local hydrodynamc varables (mass densty, stream velocty and, n some cases, energy densty); for a molecular gas s the Maxwell-Boltzmann dstrbuton (Eq. 1.5). It s well known that a Maxwell-Boltzmann local equlbrum leads to the Euler equatons of hydrodynamcs (Eqs. 1.3 and 1.7); however the most-probable (equlbrum) dstrbuton functons of dscrete velocty lattce gases gve rse to densty dependent advecton veloctes and velocty dependent pressures (Frsch et al. (1987)). Therefore we seek a constraned equlbrum dstrbuton for our dscrete velocty model that wll lead to the correct macroscopc flud dynamcs at the Euler level; the requred form for the moments of the velocty dstrbuton functon are ρ =, j = c, Π eq = c c = p1 + ρuu, (2.4) where p s the pressure and Π eq s the non-dsspatve part of the momentum flux. As n the usual knetc theory of gases, the vscous fluxes come from the non-equlbrum part of the dstrbuton functon. The equlbrum dstrbuton can be expressed as a seres expanson n powers of the flow velocty u, = ρ [ a c 0 + ac 1 u c + a c 2 uu : c c + a c 3 u2], (2.5) where uu = uu (1/3)u 2 1 s the traceless part of uu; Eq. 2.5 has the same functonal form as a small u expanson of the Maxwell-Boltzmann dstrbuton functon. The moments of the dstrbuton functon (Eq. 2.4) can be expressed n terms of the coeffcents a c,

8 8 A.J.C. Ladd whch are functons only of the speed, c ; ρ 1 = 6a a [ 6a a 2 3 ] u 2, (2.6) ρ 1 ρ 1 ρ 1 ρ 1 c α = [2a a 2 1 ]u α, (2.7) c 2 = 6a a [ 6a a c α c β = { 2a 1 2 [δ αβγδ (1/3)δ αβ δ γδ ] 2 3 ] u 2, (2.8) + 4a } 2 2 [ δ αβγδ (1/3)δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ] u γ u δ, (2.9) c α c β c γ = { 2a 1 1δ αβγδ } + 4a 2 1 [ δ αβγδ + δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ] u δ. (2.10) The tensor δ αβγδ s unty when all the subscrpts are the same (.e. δ xxxx = δ yyyy = δ zzzz = 1) and zero otherwse; δ αβ s the Kronecker delta. The thrd-order moments (Eq. 2.10) do not contrbute to the Euler equatons, but they do contrbute to the vscous stresses because of the second-order Chapman-Enskog expanson (see Eq. 2.28). Thus they must be proportonal to the fourth-rank dentty tensor (δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ); otherwse the vscous part of the momentum flux wll not be sotropc. A comparson of Eqs wth Eq. 2.4, together wth the sotropy condton n Eq. 2.10, s suffcent to determne all the coeffcents, a 1 0 = (2 3c 2 s )/6, a1 1 = 1/6, a 1 2 = 1/4, a 1 3 = 1/6, a 2 0 = (3c 2 s 1)/12, a 2 1 = 1/12, a 2 2 = 1/8, a 2 3 = 1/12. (2.11) The defnton of the pressure (Eq. 2.4) ndcates that t s proportonal to the densty,.e. p = (1/3)( neq c2 ρu2 ) = ρc 2 s; later t wll be shown that c s s the speed of sound (see Eq. 2.26), as n a normal gas. Snce the dstrbuton functon must always be postve, the speed of sound s bounded by the lmts 1/3 c 2 s 2/3. However, as the sound speed approaches ether of the two lmts, the smulaton becomes unstable wth respect to varatons n flud velocty. In our smulatons, the ntermedate value c s = 1/2 s used, to maxmze the stablty wth respect to varatons n flow velocty; n ths case a 1 0 = 1/12, a 2 0 = 1/24. (2.12) Havng constructed aulbrum dstrbuton approprate for the nvscd (Euler) equatons, let us next consder how to obtan the correct form for the vscous terms n the flud equatons. We requre that the lnearzed collson operator satsfy the followng egenvalue equatons; L j = 0, c L j = 0, c c L j = λc j c j, c 2 L j = λ B c 2 j. (2.13) The frst two equatons follow from conservaton of mass and momentum and the last two equatons descrbes the sotropc relaxaton of the stress tensor; the egenvalues λ and λ B are related to the shear and bulk vscostes (Eq. 2.34). Equatons 2.13 account for 10 of the 18 egenvectors of L. The remanng 8 modes, comprsng some hgher-order moments of L, are not relevant to smulatons of the Naver-Stokes equatons and wll be gnored.

9 Numercal Smulatons of Partculate Suspensons...I 9 Ther egenvalues are set to -1 so that these modes are then projected out entrely from the post-collson dstrbuton; ths both smplfes the smulaton and ensures the fastest possble relaxaton of the non-hydrodynamc modes. The computatonal procedure to update the lattce-boltzmanuaton s therefore qute straghtforward. At each ste the moments ρ, j, and Π (Eq. 1.10), and the equlbrum momentum flux Π eq (Eq. 2.4) are calculated; the momentum flux s then updated accordng to Eq Π αβ = Π eq αβ + (1 + λ)(π αβ Π eq αβ) + (1 + λ B )(Π γγ (1/3)Π eq γγ)δ αβ. (2.14) The post-collson dstrbuton, n + (n), s determned by the requrement that the new populatons are consstent wth Eq. 1.10, so that n + (n) = a c 0 ρ + ac 1 j αc α + a c 2 Π αβc α c β + a c 3 (Π αα 3ρc 2 s); (2.15) the term 3ρc 2 s ac 3 keeps Π orthogonal to ρ. Next we examne the macrodynamcal behavor arsng from the lattce-boltzmann equaton; our method of soluton s the usual mult-tme-scale analyss (Frsch et al. (1987)). We begn wth conservatouatons for the moments of the dstrbuton functon n (r + c, t + 1) = n (r, t), (2.16) n (r + c, t + 1) c α = n (r, t)c α, (2.17) n (r + c, t + 1) c α c β = n (r, t)c α c β + λ n neq (r, t)c α c β. (2.18) n (r + c, t + 1) c 2 = n (r, t)c 2 + λ B n neq (r, t)c 2. (2.19) To fnd the long-tme, long-wavelength dynamcs, a scalng parameter ǫ s ntroduced, defned as the rato of the lattce spacng to a characterstc macroscopc length; the hydrodynamc lmt corresponds to ǫ 1. In a molecular gas the approprate scalng parameter s the Knudsen number, the rato of the mean-free path between collsons to the macroscopc length scale. The parameter ǫ plays a smlar role to the Knudsen number n the Chapman-Enskog method (Chapman & Cowlng (1960)); t s used, frst of all, to separate the relaxaton of the equlbrum and non-equlbrum dstrbutons, (c.f. Eq. 2.3) n = + ǫn neq. (2.20) However, because the lattce spacng and the mean-free path are comparable, there are addtonal contrbutons to the vscous momentum flux, whch do not appear n the ordnary knetc theory of gases (see Eq. 2.32). In order to remove dscrete lattce artfacts from the macroscopc equatons, t s convenent to defne a macroscopc space scale r 1 = ǫr, and two macroscopc tme scales t 1 = ǫt and t 2 = ǫ 2 t; ths enables a separaton to be made between the propagaton of sound (t 1 ) and the dffuson of vortcty (t 2 ) (Frsch et al. (1987)). Expandng the fnte dfferences, n (r + c, t + 1) n (r, t) (Eqs ), to second order about r and t, and collectng terms that are frst order n ǫ we obtan the relaxaton on the t 1 tme scale; t1 + α c α = 0, (2.21)

10 10 A.J.C. Ladd t1 c α + β c αc β = 0, (2.22) t1 c αc β + γ c αc β c γ = λ n neq c α c β λ B n neq c 2 δ αβ. (2.23) The gradent operator refers to dervatves on the macroscopc r 1 space scale,.e. r1. The equatons for mass and momentum conservaton (Eqs and 2.22) can be rewrtten usng Eq. 2.4; t1 ρ + (ρu) = 0, (2.24) t1 (ρu) + (p1 + ρuu) = 0, (2.25) whch are the Euler equatons of hydrodynamcs. Substtutng the equaton of state p = ρc 2 s and lnearzng the Euler equatons wth respect to δρ and u, t s found that, for small densty fluctuatons, 2 t 1 ρ = c 2 s 2 ρ. (2.26) Equaton 2.26 shows that densty fluctuatons relax va the propagaton of sound waves, on a tme scale t 1, and therefore decouple from the t 2 tme scale evoluton of the vscous stresses. The tme dervatve that appears n Eq can be evaluated by usng Eqs and 2.25 to express the tme dervatves of ρ and ρu n terms of spatal dervatves; t1 c αc β = t1 (ρc 2 sδ αβ + ρu α u β ) = γ (ρu α u β u γ ) c 2 s [u α β ρ + u β α ρ + γ (ρu γ )δ αβ ]. (2.27) The spatal dervatve of the thrd-order moment can be evaluated drectly from Eqs and 2.11, γ c α c β c γ = (1/3)[ α (ρu β ) + β (ρu α ) + γ (ρu γ )δ αβ ]. (2.28) In the ncompressble lmt, varatons n densty can be gnored, so that t1 c αc β + γ c αc β c γ = (ρ/3)[ α u β + β u α (2/3) γ u γ ], (2.29) wth errors of order u 3. Then, from Eq. 2.23, the Naver-Stokes form for the vscous stresses can be obtaned, σ αβ = n neq c α c β = (ρ/3λ)( α u β + β u α (2/3) γ u γ δ αβ ). (2.30) Hereafter the non-equlbrum (vscous) contrbutons to the pressure wll be gnored. Snce the Mach number n our smulatons s typcally 10 2 or less, the effects of compressblty can be safely gnored. It s nterestng to note that f the speed of sound c s were set to 1/3 nstead of 1/2, then nspecton of Eqs and 2.28 ndcates that the correct form for the vscous stresses, ncludng the non-equlbrum pressure, would be obtaned (wth correctons of order u 3 ), even for non-zero Mach numbers. For such smulatons, an addtonal densty of zero-velocty partcles s requred to mantan stablty (McNamara & Alder (1993)). The t 2 relaxaton of the mass and momentum denstes can be found from the order ǫ 2 terms n the expanson of Eqs and 2.17; t2 = t2 ρ = 0, (2.31)

11 t2 Numercal Smulatons of Partculate Suspensons...I 11 c α +(1/2) β ( t1 c αc β + γ c αc β c γ )+ β n neq c α c β = 0. (2.32) Equaton 2.31 shows that the flud s ncompressble on the t 2 tme scale; all relaxaton of densty fluctuatons takes place on the t 1 tme scale. Usng Eqs and 2.30 the long-tme varaton of the vscous stresses can be expressed n a form dentcal to the ncompressble Naver-Stokes equatons, where t2 (ρu) = η 2 u (2.33) η = ρ(2/λ + 1)/6 (2.34) s the shear vscosty; once agan terms proportonal to u are neglected. The shear vscosty (Eq. 2.34) contans two dstnct contrbutons: the frst, proportonal to λ 1, arses from molecular-lke collsons (Eq. 2.13); the second term comes from the dffuson of momentum caused by the fnte lattce (Eq. 2.29). In most stuatons of practcal nterest the collsonal and lattce contrbutons to the vscosty are of comparable magntude. Fortunately they both have exactly the same dependence on velocty gradent, so that they may be combned nto a sngle transport coeffcent. A lnear stablty analyss shows that λ must be bounded n the range 2 < λ < 0 (Hguera et al. (1989); McNamara & Alder (1993)), otherwse the shear stress grows exponentally n tme and the smulaton s unstable. The bounds on λ correspond to the smple physcal requrement that the vscosty s postve. At non-zero Reynolds numbers the vscosty s bounded by a more strngent non-lnear stablty crteron, whch has not yet been worked out n detal, but whose general character s known. Essentally the quantty ρv 2 /η (v s a characterstc flow velocty) must be smaller than some postve constant of order 1; thus there s an upper bound to the flow velocty, proportonal to the square root of the vscosty. Combnng the relaxaton of the momentum densty on the t 1 and t 2 tme scales leads to the ncompressble Naver-Stokes equaton t (ρu) + (ρuu) = p + η 2 u, (2.35) wth equaton of state p = ρc 2 s. Once agan we agan pont out that for the 18 velocty model used n ths work, the smulatons are only vald at low Mach numbers; slghtly more complex models are needed to capture compressblty effects correctly. In the remander of ths paper, t wll be assumed that the smulatons wll be run under condtons of low Mach number, wth partcle veloctes U much less than the sound speed c s ; thus u = 0 to a good approxmaton. Many flows nvolvng partculate suspensons occur at low Reynolds number, and can be modeled by the creepng-flow or Stokes equatons u = 0, p = η 2 u; (2.36) or, n terms of the momentum densty j = ρu and the knematc vscosty ν = η/ρ, j = 0, p = ν 2 j. (2.37) In our smulatons the Stokes equatons are not modeled drectly, but rather as a longtme lmt of the lnearzed Naver-Stokes equatons t ρ = j, t j = p + ν 2 j; (2.38) Eq can be smulated drectly by a change n the equlbrum dstrbuton (c.f.

12 12 A.J.C. Ladd Fgure 2. Locaton of the boundary nodes for a crcular object of radus 2.5 lattce spacngs. The veloctes along lnks cuttng the boundary surface are ndcated by arrows. The locaton of the boundary nodes are shown by sold squares and the lattce nodes by sold crcles. Eq. 2.5), = a c 0 ρ + ac 1 j c. (2.39) Fnally, a sgnfcant smplfcaton of the code occurs when λ = 1, correspondng to a vscosty η = ρ/6. Although such a large vscosty s not sutable for hgh Reynolds number flows, n the creepng flow lmt t allows for a consderable smplfcaton of the collson operator n + (n) = a c 0 ρ + ac 1 j c, (2.40) whch requres less than half the number of floatng-pont operatons as Eq. 2.15; most of our R e = 0 smulatons use ths vscosty. 3. Sold-Flud Boundary Condtons To smulate the hydrodynamc nteractons between sold partcles n suspenson, the lattce-boltzmann model must be modfed to ncorporate the boundary condtons mposed on the flud by the sold partcles. The basc methodology s llustrated n Fg. 2. The sold partcles are defned by a boundary surface, whch can be of any sze or shape; n Fg. 2 t s a crcle. When placed on the lattce, the boundary surface cuts some of the lnks between lattce nodes. The flud partcles movng along these lnks nteract wth the sold surface at boundary nodes placed halfway along the lnks. Thus a dscrete representaton of the partcle surface s obtaned, whch becomes more and more precse as the partcle gets larger.

13 Numercal Smulatons of Partculate Suspensons...I 13 Lattce nodes on ether sde of the boundary surface are treated n an dentcal fashon, so that flud flls the whole volume of space, both nsde and outsde the sold partcles. The boundary node update rules descrbed later n the secton (see Eqs. 3.2 and 3.3) decouple the nteror and exteror flud regons, so fluctuatons n the nteror flud have no effect on the exteror flow. The nteror flud s kept for computatonal convenence only, snce t avods the necessty of creatng and destroyng flud as the partcle moves. In the creepng flow regme the nteror flud has relaxed to a rgd body moton, characterzed by the partcle velocty and angular velocty, and exerts no force or torque on the partcle. However the nteror flud does exert a tme-dependent force, whch, to leadng order n a 2 /νt, s equvalent to ts nertal mass. Further devatons at short tmes are dscussed n secton V of Part II; n general they are neglgble. In comparson wth our prevous work (Ladd et al. (1988); Ladd & Frenkel (1989, 1990); Ladd (1991); van der Hoef et al. (1991)), here we have chosen to place the boundary nodes on the lnks connectng the nteror and exteror regons, whereas n our lattce-gas smulatons they were located on the nodes closest to the boundary surface. There s lttle to choose between the two methods; the lnk method has the advantage that t provdes a somewhat hgher resoluton of the sold boundary surface, as can be seen (Fg. 2) from the much larger number of boundary nodes compared wth the number of lattce nodes just nsde the surface. On the other hand the node method s faster, although ths s of less sgnfcance n the computatonally more ntensve lattce-boltzmann (as opposed to lattce-gas) smulatons. Although at ths pont n tme our lattce-boltzmann smulatons have been lmted to smple symmetrcal objects; spheres, dsks and plane walls, ths restrcton s not fundamental: n fact a lmted number of lattce-gas smulatons contanng elongated objects have already been reported (van der Hoef (1992)). At each boundary node there are two ncomng dstrbutons n (r, t + ) and n (r + c, t + ), correspondng to veloctes c and c (c = c ) parallel to the lnk connectng r and r + c ; the notaton n (r, t + ) = n (r, t) + (r, t) s used to ndcate the post-collson dstrbuton (Eq. 2.15). In some cases boundary nodes for two dfferent lnk drectons, perpendcular to one another, may be concdent (see Fg. 2); these are treated ndependently. The velocty of the boundary node u b s determned by the sold partcle velocty U, angular velocty Ω, and centre of mass R, u b = U + Ω (r c R). (3.1) By exchangng populaton densty between n and n the local momentum densty of the flud can be modfed to match the velocty of the sold partcle surface at the boundary node, wthout affectng ether the mass densty or the stress, whch depend only on the sum n +n. Because the stress tensor s unaffected by the boundary-node nteractons, t then follows that the hydrodynamc stck boundary condton apples rght up to the sold surface, wthout any ntervenng boundary layer (Ladd & Frenkel (1990)). Ths pont wll be dscussed n more detal later. The mechansm for the boundary-node nteractons s llustrated n Fg. 3. In Fg. 3a we see the two ncomng populatons, n (r, t + ) and n (r + c, t + ), nteractng wth a statonary boundary node. In ths case, the populatons are smply reflected back n the drecton they came from (Frsch et al. (1987); Cornubert et al. (1991)), so that n (r + c, t + 1) = n (r + c, t + ), and n (r, t + 1) = n (r, t + ). (3.2) In Fgs 3b and 3c the effects of a movng object can be seen. In addton to reflecton, populaton densty s now transferred across the boundary node, n proporton to the

14 14 A.J.C. Ladd Fgure 3. Populaton denstes before and after a collson wth a boundary node. The effects of statonary (Fg. 3a) and movng (Fgs. 3b and 3c) boundary nodes on the ncomng populatons are shown. The arrows ndcate the velocty drecton and the lengths of the sold lnes are proportonal to the populaton denstes. The dfferences between populaton denstes are hghly exaggerated for clarty. Note that the effects of the movng boundary are the same n Fgs 3b and 3c, because the velocty component parallel to the lnk drecton s the same. velocty of the node u b, n (r + c, t + 1) = n (r + c, t + ) + 2a c 1 ρu b c, n (r, t + 1) = n (r, t + ) 2a c 1 ρu b c ; (3.3) these results are ensemble averages of our earler boundary-node collson rules for lattce gases (Ladd & Frenkel (1989); Ladd (1991)). Only the velocty component of the boundary node along the lnk drecton (c ) s ncluded n the calculaton of populaton transfer; thus the outcome n Fgs. 3b and 3c s the same. The general form for the boundary node nteractons n Eq. 3.3 s determned by the requrement that the local mass densty and stress tensor are conserved; thus rearrangements of populaton can only be made among pars of opposte veloctes. Furthermore, for statonary nodes the usual bounce-back condton (Eq. 3.2) must be recovered. The exact amount of populaton densty transferred (.e the magntude of the u b c term) s determned by the requrement that any dstrbuton consstent wth the boundary-node velocty u b s statonary wth respect to nteractons wth the boundary nodes. It s not obvous that Eq. 3.3 satsfes ths condton, but t wll be verfed n the next paragraph that ths s ndeed so. Let us now examne boundary-node nteractons n more detal. From secton II we know that the dstrbuton functon at a node can be wrtten as the sum of equlbrum (Eq. 2.5) and non-equlbrum contrbutons. The non-equlbrum part s proportonal to c c ; the proportonalty constant can be determned from Eq. 2.30, usng the relaton ac 2 c αc β c γ c δ = 1 2 (δ αγδ βδ + δ αδ δ βγ 2 3 δ αβδ γδ ), n neq = a c 2 σ : c c. (3.4) Ignorng terms proportonal to u, the collsonal stress tensor n Eq. 3.4 can be

15 Numercal Smulatons of Partculate Suspensons...I 15 expressed n terms of velocty gradents (Eq. 2.30), λn neq here we have substtuted a c 1 for 2 3 ac (r, t) + (1 + λ)nneq 2 (Eq. 2.11). The post-collson dstrbuton n (r, t + ) = (r, t) (c.f. Eqs and 2.15) s then gven, to the same approxmaton, by = a c 1 ρc c : u; (3.5) n (r, t + ) = n (r, t)+a c 1 ρc c : u(r) = n (r, t)+2a c 1 ρu(r) c +a c 1 ρc c : u(r), (3.6) where the symmetres n the dstrbuton functons for velocty drectons and have been exploted. If there s a boundary node located at r c, then the populaton n (r, t + 1) s modfed accordng to Eq. 3.3,.e. n (r, t + 1) = n (r, t) + 2a c 1 ρ [ u(r) + (1/2)c u(r) u b (r c ) ] c ; (3.7) thus the dstrbuton s statonary when the flud velocty u(r c ) = u(r) + (1/2)c u(r) s equal to the boundary-node velocty u b. To llustrate the acton of the boundary nodes more clearly, we consder, as an explct example, planar Couette flow. Fgure 4 shows a two-dmensonal projecton of the lattce- Boltzmann model onto the xy-plane; the system s assumed to be tme ndependent, and translatonally nvarant n the y- and z-drectons. As an dealzed model of a sold partcle surface, two nfnte planes of boundary nodes are set up, at x = 0 and x = L. In the flud between the boundary surfaces (0 < x < L) there s a unform velocty gradent x u y (x) = γ; outsde the boundary planes, the flud moves wth unform velocty equal to the wall velocty. Note that n ths example, the lattce nodes are more convenently set to half-nteger values of x. The problem s to fnd the dstrbuton functon for ths flow geometry that s statonary under the acton of the boundary-node mcrorules, wth veloctes u b (x = 0) = 0 and u b (x = L) = γl. A related problem, nvolvng mxed stckslp boundary condtons at a statonary wall, has been addressed by Cornubert et al. (1991). The expected velocty dstrbuton n a unform velocty gradent can be constructed from the equlbrum dstrbuton for Stokes flow (Eq. 2.39) and the non-equlbrum dstrbuton Eq. 3.5, n neq = a c 1 ρc xc y γ. (3.8) Usng the notaton of Fg. 4, the velocty dstrbuton functon n the flud (0 < x < L) can be wrtten explctly as n 0 (x) = 4, n 1 (x) = 4, n 2 (x) = 4, n 3 (x) = 4(1 + 2γx), n 4 (x) = 4(1 2γx), n 5 (x) = (1 + 2γx + 2γ/λ), n 6 (x) = (1 2γx + 2γ/λ), n 7 (x) = (1 2γx 2γ/λ), n 8 (x) = (1 + 2γx 2γ/λ); (3.9) the denstes n 0 through n 4 have been multpled by 4 to account for the number of projected veloctes, and the mass densty has, for convenence, been set equal to 24. The velocty dstrbuton away from the boundares s updated accordng to the usual tme evoluton of the lattce-boltzmanuaton. The post-collson dstrbuton s computed from Eqs. 1.10, 2.14, and 2.15, then propagated to the neghborng nodes usng Eq. 2.1.

16 16 A.J.C. Ladd Fgure 4. A two-dmensonal projecton of the lattce-boltzmann flud, bounded by plane walls. The boundary nodes are shown as squares and the boundary planes (x = 0 and x = L) by dashed lnes. The crcles are the lattce nodes; the set of nodes explctly consdered n the text are shown flled. The flud between the walls s subjected to a velocty gradent by the relatve moton of the walls; the flud outsde the walls moves wth unform velocty equal to the wall velocty. The labelng of velocty drectons used n the text s also shown; velocty components n the z-drecton have been projected onto the xy-plane. There s an addtonal densty of statonary partcles (not shown), labeled 0, correspondng to veloctes [0, 0, ±1]. The lower porton of the fgure s a plot of the velocty profle, wth the crosses showng the flud velocty u y at the nodes and the dotted lne the nterpolaton between nodes. The new velocty dstrbuton, denoted by n, s n 0 (x) = 4, n 1(x) = 4, n 2(x) = 4, (x) = 4(1 + 2γx), n (x) = 4(1 2γx), n 3 n 5 (x) = [1 + 2γ(x 1) + 2γ(1 + λ)/λ], n n 7(x) = [1 2γ(x 1) 2γ(1 + λ)/λ], 4 6 (x) = [1 2γ(x + 1) + 2γ(1 + λ)/λ], n 8(x) = [1 + 2γ(x + 1) 2γ(1 + λ)/λ]; (3.10) whch s dentcal to the ntal dstrbuton (Eq. 3.9), as requred. For lattce-nodes adjacent to the sold-flud boundares, the update of some of the populatons denstes s affected by the boundary nodes (Eq. 3.3): explctly n 5( 1 2 ) = [1 2γ(1/2) + 2γ(1 + λ)/λ], n 7( 1 2 ) = [1 + 2γ(1/2) 2γ(1 + λ)/λ], n 6(L 1 2 ) = [1 + 2γ(L 1 (3.11) 2 ) + 2γ(1 + λ)/λ] 4γL, n 8(L 1 2 ) = [1 2γ(L 1 2 ) 2γ(1 + λ)/λ] + 4γL; whch, once agan, s dentcal to the ntal dstrbuton (Eq. 3.9). Thus the boundarynode collson rules generate an exact lnear shear flow; ths s because they mantan the second order accuracy of the pure flud model. Furthermore, the velocty dstrbutons outsde and nsde the partcle are solated from one another; thus a sharp change n

17 Numercal Smulatons of Partculate Suspensons...I 17 velocty gradent from the nsde to the outsde the partcle surface can be supported, as llustrated n Fg. 4. As a result of the boundary-node nteractons (Eq. 3.3), forces are exerted on the sold partcles at the boundary nodes,.e. f(r c, t ) = [n (r + c, t + 1) n (r, t + 1) n (r, t + ) + n (r + c, t + )]c = 2[n (r, t + ) n (r + c, t + ) 2a c 1 ρu b c ]c ; (3.12) thus momentum s exchanged locally between the flud and the sold partcle, but the combned momentum of sold and flud s conserved. The forces and torques on the sold partcle are obtaned by summng f(r c ) and (r c ) f(r c ) over all the boundary nodes assocated wth a partcular partcle. As an example, Eq can be used to compute the drag force per unt area on a planar wall adjacent to a steadly shearng flud. We compute the force on one face of each sold boundary surface, assumng that the flud on the other sde s movng unformly wth the velocty of the boundary (as shown n Fg. 4) and therefore exerts no force on the wall; ths corresponds to replacng one of the dstrbutons n Eq by ts equlbrum form (Eq. 2.39) wth a velocty equal to the wall velocty. Usng the dstrbutons at t + just after the molecular collson process (Eq. 3.9 wth 1/λ replaced by (1 + λ)/λ) the wall forces are found to be f y (0) = 2[ n 6 ( 1 2, t +) + n 8 ( 1 2, t +)] = 4(2/λ + 1)γ = ηγ, f y (L) = 2[n 5 (L 1 2, t +) n 7 (L 1 2, t +) 4γL] = 4(2/λ + 1)γ = ηγ; (3.13) the last equalty follows from summng the collsonal and lattce contrbutons to the vscosty (Eq. 2.34), usng ρ = 24. Thus the wall force s computed exactly for lnear shear flows. As a prelmnary applcaton of the method to tme-dependent flows, we consder the evoluton of the flow feld from an mpulsvely started flat plate. The geometry s smlar to Fg. 4, wth the plates beng suffcently far apart that they do not nteract over the duraton of the smulaton. We focus on a sngle plate (x = 0). Intally the system s at rest; at t = 0, an mpulsve force gves the plate a constant velocty [0, U, 0]. In ths problem t s agan convenent to defne the lattce nodes at half-nteger values of x, and for the flud to resde at the lattce nodes at half-nteger values of the tme; then the boundary condtons at the plate are appled at x = 0 and t = 0 precsely. We compute numercally the evoluton of the flow feld [0, u(x, t), 0], created by the dffuson of vortcty nto the flud, and compare wth the analytc solutons for the velocty feld (Batchelor (1967)) and the force per unt area u(x, t) = U [1 Φ(x/ 4νt)], (3.14) f(t) = η x u(0, t) = ηu(πνt) 1/2 ; (3.15) ν = η/ρ s the knematc vscosty of the flud and Φ s the error functon. The results are shown n Fg. 5 (for a vscosty ν = 1/6) at several dfferent tmes. It can be seen that there s complete agreement, except at very short tmes and dstances. Ths smple test mples that both statonary and tme-dependent flows can be smulated accurately, as wll be confrmed by further results n Part II. Next, consder flow perpendcular to the wall. The ncompressblty condton means that there can be no velocty gradents n steady flow; thus the flud velocty and the wall veloctes are [u, 0, 0]. For ths flow the model shown n Fg. 4 can be further smplfed to just three velocty drectons; statonary partcles, and partcles movng n the postve

18 18 A.J.C. Ladd Fgure 5. Flow nduced by an mpulsvely loaded flat plate, U(t) = 0 for t < 0 and U(t) = U for t 0. The plots show the velocty feld and force per unt area on the wall at varous tmes; the sold crcles are the numercal smulatons and the sold lnes are analytcal results (Batchelor (1967)). and negatve x drectons. The projected dstrbutons are n 0 (x) = 12, n 1 (x) = 6(1 + 2u), n 2 (x) = 6(1 2u), (3.16) Clearly ths dstrbuton s statonary wth the respect to the evoluton of the lattce- Boltzmanuaton; t s also statonary wth respect to the boundary-node update rules (Eq. 3.3). However ths s not the only statonary dstrbuton that satsfes the boundary condtons. A more general form for the dstrbuton functon, whch exhbts a two tme-step repeat cycle s n 0 (x) = 12, n 1 (x) = 6(1 + 2u + ( 1) t+x 2w), n 2 (x) = 6(1 2u ( 1) t+x 2w); (3.17) t s straghtforward to verfy that ths dstrbuton also satsfes both the tme evoluton equaton and the boundary condtons at the walls. Thus the flud has a unform momentum ρu and a staggered momentum ( 1) t+x ρw. Staggered momenta are an artfact of all lattce models (Zanett (1989)); the precse value of the staggered momentum depends on the channel wdth (n ths example) and the ntal condtons. Although staggered momenta cannot arse spontaneously n the flud, they can be generated at sold surfaces, as seen n ths example. However, t can be shown by technques smlar to those used above, that staggered momentum parallel to the walls s damped by the boundary condtons, so that for plane Couette flow the steady solutons have no staggered momentum component. In the more complex geometres that are of nterest n partculate suspensons, t s mpossble to analyze the staggered momenta analytcally. Numercal results show that large oscllatons n partcle torques can be bult up by a feedback mechansm n whch the staggered momenta are fed by ever ncreasng angular veloctes of the partcles. To overcome these nstabltes we average the force and flud velocty over two successve tme steps whch effectvely cancels out the staggered momentum contrbuton. In the above example, ths gves a unform flow feld wth velocty [u, 0, 0] regardless of the magntude of the staggered momenta. Snce the forces at the boundary nodes are generated at the half-nteger tme steps, the smoothly varyng force f at the ntermedate nteger

19 Numercal Smulatons of Partculate Suspensons...I 19 tme s f(r c, t) = (1/2)[f(r c, t 1 2 ) + f(r c, t )]. (3.18) We can calculate the smooth part of the flud velocty feld at half-nteger tme steps n a smlar way ū(r, t ) = (1/2)[u(r, t) + u(r, t + 1)], (3.19) or at nteger tme steps usng the 3-pont formula ū(r, t) = (1/4)[u(r, t 1) + 2u(r, t) + u(r, t + 1)], (3.20) The veloctes of fnte mass partcles (as opposed to nfntely massve fxed objects) are updated every two tme steps, U(t + 1) = U(t 1) + 2M 1 F(t), Ω(t + 1) = Ω(t 1) + 2I 1 T(t). (3.21) The partcle mass M and moment of nerta I are preassgned parameters whch control the rate at whch partcles respond to the flud flow; usually M and I are on the order of several thousands (n lattce unts). Snce the partcle veloctes vary slowly on the tme scale of a lattce-boltzmann cycle, the precse form for the update s usually not too mportant; however, t s mportant to use tme-smoothed forces and torques, as descrbed n Eq So far only lnear shear flows have been consdered. In the next and last example a statonary, two-dmensonal channel flow s examned. The geometry s agan shown n Fg. 4, ths tme wth statonary walls (u b = 0). The flud s drven by a pressure gradent, whch s represented n the smulaton by a unform force densty n the flud. Thus a constant ncrement j y s appled to the y momentum at each node, so that the pressure gradent down the channel s y p = j y. The flud velocty at each node s measured after half the force has been appled; t was found emprcally that ths prescrpton gves the fastest convergence as a functon of system sze. The steady flow profles are compared wth the analytc soluton, u y (x) = ( j y /2η)x(L x), (3.22) n Fg. 6. It can be seen that the agreement s very good for channels more than about 9 lattce spacngs wde. Furthermore, the force on the walls s exact, no matter what the channel wdth. Ths result follows from the balance between forces on the walls and the total force from the pressure gradent. Snce the pressure forces are dstrbuted equally on the walls t follows that the wall force per unt area f = j y L/2, whch s the correct result for Poselle flow. In ths secton t has been shown that movng sold boundares can be ncorporated nto a lattce-boltzmann smulaton, and t has been ndcated how they functon for a few smple examples. In Part II numercal results for sphercal partcles are descrbed, both n perodc arrays and random assembles. Results are compared wth known analytc and numercal solutons of the creepng-flow and Naver-Stokes equatons. 4. Fluctuatons In recent years, t has become ncreasngly apparent that the lattce-boltzmanuaton s a much better smulaton tool for hydrodynamcs than lattce gases. However, n ts normal state the lattce-boltzmanuaton cannot model the molecular fluctuatons n the solvent that gve rse to Brownan moton. Of course n many stuatons Brownan moton s unmportant, but, for suspensons of sub-mcron szed partcles, t s a fundamental component of the dynamcs. It has been shown recently (Ladd (1993)) that