Overview of the lattice Boltzmann method for nano- and microscale fluid dynamics in materials science and engineering

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING Modellng Smul. Mater. Sc. Eng. 12 (2004) R13 R46 PII: S (04) TOPICAL REVIEW Overvew of the lattce Boltzmann method for nano- and mcroscale flud dynamcs n materals scence and engneerng D Raabe Max-Planck-Insttut für Esenforschung, Max-Planck-Strasse 1, Düsseldorf, Germany E-mal: raabe@mpe.de Receved 15 March 2004, n fnal form 2 August 2004 Publshed 16 September 2004 Onlne at stacks.op.org/msmse/12/r13 do: / /12/6/r01 Abstract The artcle gves an overvew of the lattce Boltzmann method as a powerful technque for the smulaton of sngle and mult-phase flows n complex geometres. Owng to ts excellent numercal stablty and consttutve versatlty t can play an essental role as a smulaton tool for understandng advanced materals and processes. Unlke conventonal Naver Stokes solvers, lattce Boltzmann methods consder flows to be composed of a collecton of pseudo-partcles that are represented by a velocty dstrbuton functon. These flud portons resde and nteract on the nodes of a grd. System dynamcs and complexty emerge by the repeated applcaton of local rules for the moton, collson and redstrbuton of these coarse-graned droplets. The lattce Boltzmann method, therefore, s an deal approach for mesoscale and scale-brdgng smulatons. It s capable to tacklng partcularly those problems whch are ubqutous characterstcs of flows n the world of materals scence and engneerng, namely, flows under complcated geometrcal boundary condtons, mult-scale flow phenomena, phase transformaton n flows, complex sold lqud nterfaces, surface reactons n fluds, lqud sold flows of collodal suspensons and turbulence. Snce the basc structure of the method s that of a synchronous automaton t s also an deal platform for realzng combnatons wth related smulaton technques such as cellular automata or Potts models for crystal growth n a flud or gas envronment. Ths overvew conssts of two parts. The frst one revews the phlosophy and the formal concepts behnd the lattce Boltzmann approach and presents also related pseudo-partcle approaches. The second one gves concrete examples n the area of computatonal materals scence and process engneerng, such as the predcton of lubrcaton dynamcs n metal formng, dendrtc crystal growth under the nfluence of flud convecton, smulaton of metal foam processng, flow percolaton n confned geometres, lqud crystal hydrodynamcs and processng of polymer blends /04/ $ IOP Publshng Ltd Prnted n the UK R13

2 R14 Topcal Revew 1. Introducton to the lattce gas and lattce Boltzmann smulaton methods 1.1. Motvaton for the use of dscrete methods n computatonal flud mechancs The theoretcal pcture of flud dynamcs n the materals engneerng communty largely departs from the work of Naver and Stokes from the frst half of the 19th century. Ther dfferental formulaton of the mechancs of ncompressble flows, the so-called Naver Stokes equaton, accounts for the conservaton of mass, momentum and energy, and the requrement that these quanttes be conserved locally [1 3]. Tacklng hydrodynamcs and related problems wth ths equaton amounts to solvng coupled sets of nonlnear partal dfferental feld equatons by use of fnte dfference or fnte element methods. Although the Naver Stokes framework serves as a long-establshed bass for predctng flud behavour, t has stll not been possble to resolve some basc questons n the felds of modern materals scence and engneerng wth t. Ths s due to the fact that the Naver Stokes dfferental formulatons theoretcally do not apply and numercally also do not converge under condtons whch are characterzed by large Knudsen numbers (mean free molecule path dvded by characterstc system length). Such restrctons occur when the mean free path of the flud molecules s smlar to the geometrcal system constrants, such as, for nstance the obstacle spacng or the roughness wavelength whch may characterze mesoscopc system heterogenety. Promnent examples where such lmtatons occur are the smulaton of nano- and mcroflows n flters, foams, mcro-reactors or otherwse confned geometres; multcomponent flows n the area of polymer and metal processng; trbology and wear n the area of lubrcated contact mechancs and metal formng; lqud crystal processng; nanoscale process technology; lubrcaton n mnaturzed components, lqud phase separaton; jont flud gas flows; abrason and sedmentaton; flud percolaton n cellular structures; processng of metallc foams; as well as corroson and soldfcaton n non-quescent envronments to name but a few. These examples do not only challenge our basc understandng of flud mechancs but represent at the same tme key ssues n modern materals scence and engneerng of consderable practcal relevance. Lattce gas cellular automata [4, 5] and ther more mature (non-boolean) successors, the lattce Boltzmann automaton technques (see detals n the ensung sectons), seem to be predestned to tackle some of these challenges n the doman of materals-related computatonal flud dynamcs n a more effcent way than the conventonal Naver Stokes approach. The lattce Boltzmann technque belongs to a broader group of pseudo-partcle methods whch form a growng class of mult-scale smulaton approaches to computatonal flud mechancs, table 1, fgure 1. Other mportant partcle-based approaches n ths class are (besdes lattce gas cellular automata) the dsspatve partcle dynamcs method [6 11] and the drect smulaton Monte Carlo method [11 16] together wth ts hybrd mesh refnement varatons [17]. These pseudo-partcle approaches can be grouped nto lattce-based cellular automaton approaches (lattce gas method, lattce Boltzmann method) and off-lattce approaches (dsspatve partcle dynamcs method, drect smulaton Monte Carlo method). Whle the remnder of ths overvew deals exclusvely wth vectoral cellular automaton models of flud flow, the ensung secton provdes a concse summary of the off-lattce methods Off-lattce pseudo-partcle methods n computatonal flud mechancs The most mportant off-lattce pseudo-partcle approach to computatonal flud mechancs s the dsspatve partcle dynamcs method [6 11], table 1, fgure 1. Ths technque uses

3 Topcal Revew R15 Table 1. Overvew of models n computatonal flud mechancs. All approaches beyond the atomc-scale (molecular dynamcs) and below the conventonal contnuum scale (Naver Stokes solvers) use coarse-graned pseudo-partcles whch can ether move on a fxed lattce (lattce-based pseudo-partcle models) or contnuously n space (off-lattce pseudo-partcle models). Models n computatonal flud mechancs Molecular dynamcs Pseudo-partcle models Off-lattce models Lattce-based models Naver Stokes solvers Dsspatve partcle dynamcs Drect smulaton Monte Carlo methods Lattce gas automata Lattce Boltzmann automata Fgure 1. Varous approaches to computatonal flud dynamcs together wth ther preferred range of applcablty. Molecular dynamcs methods ntegrate Newton s equatons of moton for a set of molecules on the bass of an ntermolecular potental. Dsspatve partcle dynamcs and drect smulaton Monte Carlo are off-lattce pseudo-partcle methods n conjuncton wth Newtonan dynamcs. Lattce gas and lattce Boltzmann methods treat flows n terms of coarse-graned fctve partcles whch resde on a mesh and conduct translaton as well as collson steps entalng overall flud-lke behavour. Naver Stokes approaches solve contnuum-based partal dfferental equatons whch account for the local conservaton of mass, momentum and energy. These three methods have ther respectve strengths at dfferent Knudsen numbers, where the Knudsen number s the rato between the mean free molecule path and a characterstc length scale representng mesoscopc system heterogenety (e.g. the obstacle sze).

4 R16 Topcal Revew dscrete flud portons whch can freely move n contnuous space at dscrete tme ncrements. The method can be derved from molecular dynamcs by means of coarse-granng,.e. the pseudo-partcles do not represent sngle atoms or molecules but rather mesoscopc droplets or clusters of atoms whch carry the poston and momentum of coarse-graned flud elements. The phlosophy of usng such averaged partcles nstead of real molecules leads to a substantal gan n computatonal effcency compared wth conventonal molecular dynamcs methods, however, at the expense of a loss n mcroscopc detal. The pseudo-partcles nteract parwse accordng to a set of short-range nterpartcle central forces that nclude a repulsve conservatve force a dsspatve force and a random force actng symmetrcally between each par of pseudo-partcles. The dsspatve force acts to slow the partcles down and to remove energy from them. The random force acts between all pars of partcles and s uncorrelated between dfferent pars. It adds energy to the system on average. Together wth the dsspatve force t acts as a thermostat for the system. The conservatve force s derved from a pseudo potental energy smlar to that n molecular dynamcs. As n conventonal molecular dynamcs methods the dynamcal behavour s realzed by the ntegraton of the Newtonan equatons of moton. It dffers from molecular dynamcs n two respects. Frst, the conservatve parwse forces between the pseudo-partcles are softrepulsve, whch makes t possble to extend the smulatons to longer tmescales. Second, the system thermostat for the canoncal ensemble s mplemented by means of the dsspatve as well as the random parwse forces such that the momentum s locally conserved. The pseudo-partcle method s used to smulate hydrodynamcs at mesoscopc scales n whch both, hydrodynamc nteractons and Brownan moton are mportant. At large Mach numbers and large Knudsen numbers t s superor to the cellular automaton models. The drect smulaton Monte Carlo method s also an off-lattce pseudo-partcle smulaton method [11 16]. The state of the system s gven by the postons and veloctes of a set of pseudo-partcles. Frst, these flud or gas portons are moved as f they dd not nteract. Ths means that ther postons are updated wthout consderng nter-partcle collsons. After ths translaton step a fxed number of partcles are randomly selected for collsons. The collson step s typcally realzed by placng the partcles nto spatal collson cells, by calculatng the collson frequency n each cell, by randomly selectng collson partners wthn each of those cells and by the actual collsons. The probablty that a par colldes only depends on ther relatve velocty. The actual collsons,.e. the calculatons of the post-collson velocty vectors are determned for each colldng par by accountng for the conservaton of momentum as well as the conservaton of energy and by random selecton of the collson angle. Ths splttng of the evoluton between forward streamng and collsons s only accurate when the tme step elapsng durng one update step s a fracton of the mean collson tme for a pseudopartcle. The partcular strength of the drect smulaton Monte Carlo method les n the feld of dlute gases Basc phlosophy of lattce-based cellular automaton methods for flud mechancs The applcaton of automaton models to the feld of flud dynamcs represents a remarkable shft n modellng phlosophy when compared to the contnuum, molecular dynamcs and pseudo-partcle approaches. Lattce gas automata replace the macroscopc pcture underlyng the Naver Stokes framework by dscrete sets of fctve partcles whch carry some propertes of real flud portons, fgure 2 [18, 19]. These fctve partcles can be regarded as coarse-graned groups of flud (or gas) molecules the exact Newtonan dynamcs of whch are not explctly taken nto account as n molecular dynamcs approaches, or, to a certan extent, n the pseudopartcle methods. The flud portons n the lattce gas move at dfferent speeds n dfferent

5 Topcal Revew R17 Fgure 2. Pseudo-flud partcles n a lattce gas model wth a quadratc grd (HPP lattce gas model of Hardy, Pomeau, de Pazzs [18], see detals n the next secton). All partcles have the same unt mass and the same magntude of the velocty vector (Boolean partcles). Moton of the partcles conssts n translatng them from one lattce node to ther nearest neghbour n one dscrete unt of tme accordng to the drecton of ther unt momentum vector. The symmetry of the quadratc grd turned out not to be suffcent for the reproducton of the Naver Stokes equaton. drectons on a fxed lattce and nteract by smple local rules. Durng each tme step they move accordng to ther current momentum vector. If two partcles happen to end up on the same lattce ste, they collde and change ther veloctes accordng to a set of dscrete collson rules. The only restrcton s that collsons have to conserve the partcle number, the momentum and the energy. Usng ths small set of rules offers the frst and coarsest way of approxmatng flud dynamcs n terms of lattce gas automata. An mportant computatonal advantage of ths method s that any lattce node can be marked as sold, allowng for the ntegraton of arbtrarly complex geometres that would be dffcult to model wth conventonal contnuum methods owng to convergence problems. The basc dea of lattce gas models, lke generally of cellular automata, s to mmc complex dynamcal system behavour by the repeated applcaton of smple local translaton and reacton rules. These rules smulate, n a smplfed and coarse-graned mesoscopc fashon, some of the mcroscopcal effects occurrng n a real flud. Ths means that lattce-gas automata take a mcroscopc, though not truly molecular, vew of flud mechancs by conductng fctve mcrodynamcs on a lattce. Solutons of the Naver Stokes dfferental contnuum equaton can be regarded as a top-down approach to flud mechancs for small Knudsen number regmes, whle the pseudopartcle and lattce-based automaton methods pursue a bottom-up strategy vald also at larger Knudsen numbers. In the macroscopc world of the Naver Stokes equaton one drectly analyses the pressure, densty, vscosty and velocty of the flow. In the mcroscopc vew taken by the pseudo-partcle and lattce gas automata, such macroscopc quanttes can be computed by averagng the nteracton and densty of the pseudo-partcles locally. It must be noted though that lattce-gas automata themselves are coarse-graned methods,.e. the fctve flud droplets whch they use as elementary consttuents are averaged pseudo-partcles, whch do not perform ndvdual Newtonan dynamcs as n a molecular dynamcs smulaton, fgures 1 and 3.

6 R18 Topcal Revew Fgure 3. Valdty regmes of a gas or flud smulaton method as a functon of densty relatve to ar and length scale. The fgure shows that the contnuum descrpton becomes naccurate when the characterstc length scale s wthn an order of magntude of the mean free path (fgure adopted from the works of Brd [20, 21] and Garca [22] Some mportant measures for flow mechancs In the feld of flud mechancs one typcally uses some elementary mesoscopc and contnuum measures for the consttutve, geometrcal and dynamcal quantfcaton of flows. Some of them are relevant n the context of ths artcle, table Boolean lattce gas cellular automata (HPP and FHP models) Lattce gas cellular automata wth Boolean partcle states resdng on fxed nodes were orgnally suggested by Frsch, Hasslacher and Pomeau n 1986 (FHP lattce gas model) [19] for the reproducton of Naver Stokes dynamcs. A prevous formulaton for vector automata was already n 1973 suggested by Hardy, Pomeau and de Pazzs (HPP lattce gas model) [18]. However, ths earler verson of a lattce gas method was based on a square grd and could, therefore, not fulfll the requrement of rotatonal nvarance. The FHP lattce gas model publshed later [19] used a hexagonal two-dmensonal lattce whch fulflls both, conservaton of partcle number and rotatonal nvarance. All partcles n a Boolean lattce gas have the same unt mass and the same magntude of the velocty vector. The model mposes, as an excluson prncple, that no two partcles may st smultaneously on the same node f ther drecton s dentcal. For the square lattce orgnally suggested by the HPP model, ths mples that there can be at most four partcles per node. Ths occupaton prncple, orgnally meant to permt smple computer codes, has the consequence that the equlbrum dstrbuton of the partcles follows a Ferm Drac dstrbuton. Moton of

7 Topcal Revew R19 Dynamc vscosty (absolute or Newtonan vscosty) Table 2. Some mportant measures for flow analyss. Parameter Relevance Defnton Unts [ Ns m 2 Knematc vscosty Knudsen number Measure of the nternal molecular resstance of a flud to flow or shear under an appled force Vscosty dvded by the densty of the lqud; force free measure of the vscosty Rato between the mean free molecule path and a characterstc length scale representng mesoscopc system heterogenety (e.g. obstacle sze) τ = µ dyn γ l τ: shear stress, µ dyn : dynamc vscosty, γ : shear rate of one layer relatve to another, l: spacng of the layers µ kn = µ dyn ρ ρ: mass densty K = L 1 L 2 L 1 : mean free path of molecule, L 2 : characterstc system length ] [ ] kg = ms [ g ] [Pose] = ms [ m 2 ] s [ cm 2 ] [Stoke] = s [1] Mach number Reynolds number Rato of the speed of a partcle n a medum to the speed of sound n that medum; Mach number 1 corresponds to the speed of sound Measure of the relatve strength of advectve over dsspatve forces quantfyng the degree of turbulence naflow M = c c s c: speed of partcle n a medum, c s : speed of sound n the medum Re F nerta F vscous UL µ kn U: characterstc macroscopc flow speed, L: characterstc length scale of flow geometry [1] [1] the partcles conssts of movng them from one lattce node to ther nearest neghbour n one dscrete unt of tme accordng to ther gven unt momentum vector, fgure 4. The evoluton of system dynamcs of the lattce gas takes place n four successve steps. The frst one s the advecton or propagaton step. It conssts of movng all partcles from ther nodes to ther nearest neghbour nodes n the drectons of ther respectve velocty vectors. The second one s the collson step, fgure 5. It s conducted n such a way that nteractons between partcles arrvng at the same node comng from dfferent drectons take place n the form of local nstantaneous collsons. The elastc collson rules conserve both mass and momentum. Ths mples that partcles arrvng at the same node may exchange momentum f t s compatble wth the mposed nvarance rules. The thrd step s (usually) the bounceback step. It mposes no-slp boundary condtons for those partcles whch ht an obstacle. The fourth step updates all stes. Ths s done by synchronously mappng the new partcle coordnates and velocty vectors obtaned from the precedng steps onto ther new postons. Subsequently the tme counter s ncreased by one unt. Owng to the dscrete treatment of the pseudo-partcles and the dscreteness of the collson rules Boolean lattce gas automata reveal some ntrnsc flaws such as the volaton of Gallean nvarance 1 and the occurrence of large fluctuatons. The latter dsadvantage can to a certan extent be crcumvented by ntroducng localzed averagng procedures 1 A Gallean transformaton s a change to another nertal reference frame movng wth constant velocty. Ths should not affect the propertes of the flow.

8 R20 Topcal Revew Fgure 4. Postons of lattce gas pseudo-partcles at two successve tme steps (advecton only) on a hexagonal two-dmensonal lattce (FHP lattce gas model) [19]. Fgure 5. Collson rules of the lattce gas cellular automaton for the case of a hexagonal grd (FHP lattce gas model) [19]. where a group of neghbourng vectors s summarzed nto a coarse-graned net-vector, fgure 6. The man advantage of the lattce gas concept compared to classcal Naver Stokes solvers conssts of ts excellent numercal stablty under ntrcate geometrcal boundary

9 Topcal Revew R21 Fgure 6. Schematc sketch of averagng n a lattce gas smulaton. Such procedures are mportant n classcal Boolean lattce gas smulatons for reducng statstcal nose. condtons. Ths property qualfes them partcularly for the smulaton of mcroflow dynamcs n porous mcrostructures and related problems arsng n the feld of modern materals scence and engneerng (see examples n part 2 of ths artcle). Snce the basc structure of the lattce gas algorthm s that of a synchronous automaton t s also an deal platform for realzng combnatons wth related materals smulaton methods such as sold state cellular automata or Potts models Introducton to the phlosophy of the lattce Boltzmann approach The lattce Boltzmann approach has evolved from the lattce gas models n order to overcome the shortcomngs dscussed above. It corresponds to a space-, momentum- and tme-dscretzed verson of the Boltzmann transport equaton. The man ratonale behnd the ntroducton of the lattce Boltzmann automaton s to ncorporate the physcal nature of fluds from a more statstcal standpont nto hydrodynamcs solutons than n the classcal lattce gas method dscussed n the precedng secton, table 3. Accordng to the underlyng pcture of the Boltzmann transport equaton the dea of the lattce Boltzmann automaton s to use sets of partcle velocty dstrbuton functons nstead of sngle pseudo-partcles and to mplement the dynamcs drectly on those average values [23 32]. The partcle veloctes n the lattce Boltzmann scheme are not Boolean varables as n conventonal lattce gas automata [14, 19], but real-numbered quanttes as n the Boltzmann transport equaton, fgure 7. Ths means that Ferm-lke statstcs no longer apply. It s also mportant to note that n contrast to the conventonal lattce gas method the lattce Boltzmann approach may use pseudo-partcles wth zero velocty. These are requred for smulatng compressble hydrodynamcs by usng a tunable model sound speed. Another man dfference between the orgnal lattce gas and the lattce Boltzmann methods s the fact that the former approach quantfes the partcle nteractons n terms of dscrete local Boolean redstrbuton rules (collson rules) whle the latter approach conducts (non-boolean) redstrbutons of the partcle velocty dstrbuton (relaxaton rules, collson operator). The man advantage of the lattce Boltzmann method compared to the orgnal lattce gas s that small sets of neghbourng nodes n a Boltzmann lattce are capable of creatng smooth flow dynamcs as opposed to the lattce gas methods whch ental rather coarse dynamcal behavour. Ths means that the Boltzmann method requres less averagng and provdes ncreased performance Typcal mesh types for the lattce Boltzmann method Lattce Boltzmann models are spatally dscrete approaches to flud dynamcs. Ths means that the underlyng grds of such smulatons must fulfll certan symmetry condtons n order to

10 R22 Lattce gas automata Lattce Boltzmann automata Topcal Revew Table 3. Overvew of the lattce gas and the lattce Boltzmann model famly (see detaled explanatons of the lattce types DkQn n secton 1.7). The lattce gas and lattce Boltzmann automaton famly HPP-model (accordng to Hardy, Pomeau, and Pazzs). The orgnal form of the lattce gas automaton wth Boolean pseudo-flud partcles resdng on a dscrete two-dmensonal quadratc grd (Hardy et al [18]) FHP-model (accordng to Frsch, Hasslacher and Pomeau). Two-dmensonal lattce gas, hexagonal grd. FHP-I: 6 neghbour nodes; FHP-II: 6 neghbour nodes and one rest partcle FHP-III: 6 neghbour nodes, one rest partcle and complete collson rules (Frsch et al [19], d Humères and co-workers [35 37]) FCHC-model (face-centred-hypercubc). FHP-type three-dmensonal lattce gas model, four-dmensonal Bravas lattce wth 24 neghbour nodes projected on a three-dmensonal spatal lattce (d Humères and co-workers [35 37]) Two-colour FCHC-model (mult-phase model on the bass of FHP-III). FHP model wth a two-dmensonal hexagonal lattce, two types of (coloured) flud phases (red, blue), phase separaton occurs by the ntroducton of a local flux and a colour gradent vector (Rothman and Keller [33]) LB-model (sngle-phase lattce Boltzmann model). Real-numbered partcle velocty dstrbuton functons, typcal lattce types: D2Q9, D3Q15 and D3Q19, collson matrx (Frsch et al [19], McNamara and Zanett [26]) LBGK-model (lattce Boltzmann wth Bhatnagar Gross Krook relaxaton). The lattce Boltzmann model, assumpton of an equlbrum velocty dstrbuton, collson matrx replaced by a sngle-step collson relaxaton towards equlbrum (Bhatnagar et al [34], Hguera and Jmenez [27], Qan et al [38]) Mult-phase LBGK-model (mult-phase model on the bass of LBGK). LBGK models for mult-phase applcatons usng sngle-step relaxaton and gradent terms, pseudo-potentals, or free-energy functonals for phase separaton (Gunstensen et al [39, 40], Grunau et al [41], Shan and Chen [42, 43], Shan and Doolen [44, 45]) Fgure 7. Schematc example demonstratng the dea of a coarse-granng procedure whch renders a Boolean lattce gas nto a lattce Boltzmann gas. The arrows resdng at the nodes on the two grds represent dscrete lattce partcles (lattce gas, left-hand sde) and portons of a local vector dstrbuton functon, respectvely (lattce Boltzmann method, rght-hand sde). The fgure shows that the lattce gas method works wth dscrete velocty vectors and dscrete flud portons. The lattce Boltzmann gas works also wth dscrete velocty vector drectons but t uses real-numbered flud portons.

11 Topcal Revew R23 Table 4. Overvew of the weght factors w for the most mportant lattce types. Lattce Zero Smple cubc Dagonal Cubc model poston vectors [100] vectors [110] vectors [111] D2Q9 4/9 1/9 1/36 do not exst D3Q15 2/9 1/9 do not exst 1/72 D3Q19 1/3 1/18 1/36 do not exst Fgure 8. Velocty vectors for a D2Q9- (left) and a D3Q19-lattce geometry (rght). recover hydrodynamc behavour wth full rotatonal symmetry of space. Ths requres that the nvarance measures whch form from the respectve sets of underlyng lattce vectors up to fourth order are sotropc,.e. w = 1, w e α = 0, =0 =0 w e α e β = (2) δ αβ, =0 w e α e β e γ = 0, =0 w e α e β e γ e θ = (4) (δ αβ δ γθ + δ αγ δ βθ + δ αθ δ βγ ), =0 (1) where w represent weght factors whch must be properly chosen for each grd type n order to correct the lattce wth respect to sotropy, e are the lattce vectors wth the Greek ndces α, β, γ, θ for the spatal drectons and hyper-drectons, and (2) and (4) are lattce constants whch are related to the lattce sound speed c s. The ndex s the counter for the lattce vectors. The most frequent mesh types for lattce Boltzmann smulatons are the D1Q3-, the D2Q9-, the D3Q15- and the D3Q19-lattce, table 4, fgure 8. The termnology DkQn refers to the number k of dmensonal sublattces (equvalent to the number of ndependent speeds) and to the dscrete number n of spatal translaton vectors e consttutng the vector bass of the dstrbuton functon. In three dmensons, sotropy generally requres a mult-speed lattce. Lke for all lattce gas automata the unts of the correspondng set of velocty vectors, c, are calculated by the correspondng lattce vectors, e, dvded by the tme step (tme proceeds synchronously for all nodes n dscrete steps, t, as n all automata). The number of occurrng speeds, therefore, corresponds to the number of sublattce vector types. It s mportant to note that c 0 s a zero vector. The D1Q3-lattce has 1 sublattce and 3 dscrete velocty vectors (dentty, left and rght). The D2Q9-lattce has 2 sublattces and 9 dscrete velocty vectors (dentty, north, west, south,

12 R24 Topcal Revew Fgure 9. Schematc fgure showng some non-zero vectors of the partcle velocty dstrbuton functon at a node. The two-dmensonal lattce has 9 velocty vectors (8 neghbours and a zero velocty). Zero velocty vectors are requred for smulatng compressble flows. east, northwest, southwest, southeast and northeast). The bass vectors of the D2Q9-lattce are ( ) e = c t = ( e 1, e 2, e 3, e 4, e 5, e 6, e 7, e 8, e 9 ) = (2) The D3Q15-lattce has 3 sublattces and 15 dscrete velocty vectors (dentty, 6 towards face centres and 8 towards vertces of a cube). The bass vectors of the D3Q15-lattce are e = c t = ( e 1, e 2, e 3, e 4, e 5, e 6, e 7, e 8, e 9, e 10, e 11, e 12, e 13, e 14, e 15 ) = (3) The D3Q19-lattce has 3 sublattces and 19 dscrete velocty vectors (dentty, 6 veloctes to the face centres and 12 towards edge centres of a cube). e = c t = ( e 1, e 2, e 3, e 4, e 5, e 6, e 7, e 8, e 9, e 10, e 11, e 12, e 13, e 14, e 15, e 16, e 17, e 18, e 19 ) = (4) Whle the D3Q15-lattce model requres less computaton and less memory than the D3Q19- lattce model, t suffers more from fnte sze effects and s less accurate Formal descrpton of the lattce Boltzmann method for sngle-phase flow The lattce Boltzmann gas uses as a central quantty a partcle velocty dstrbuton functon, f ( x,t) [24, 26], whch quantfes the (real-numbered) probablty to observe a pseudo-flud partcle wth dscrete velocty c at lattce node x at tme t. The partcle velocty dstrbuton functon s defned for partcles movng synchronously along the nodes of a dscrete regular spatal lattce. The subscrpts = 0,...,m of the velocty vectors ndcate ther dscrete lattce drecton on the chosen grd, fgure 9. The occurrng velocty vectors depend on the number of sublattces and on the coordnaton sphere as outlned n the precedng secton. The flud partcles can collde wth each other as they move under appled forces.

13 Topcal Revew R25 In the lattce Boltzmann approach the temporal evoluton of the partcle velocty dstrbuton functon satsfes a lattce Boltzmann equaton of the type f new ( x + c t, t + t) f old ( x,t) = t ( = 0,...,n), (5) where t s the lattce tme step. The ndex stands for the n base vectors of the underlyng lattce type. The left hand term, f new ( x + c t, t + t) f old ( x,t), s the advecton term whch represents free propagaton of the partcle packets along the lattce lnks. The term f new ( x + c t, t + t) s the new dstrbuton functon after advecton and redstrbuton. When consderng an addtonal external source of momentum (e.g. body forces such as occurrng n pressure gradents or gravtatonal felds), F, one obtans f new ( x + c t, t + t) f old ( x,t) = t + tf ( = 0,...,n). (6) Normalzaton provdes f [0, 1]. (7) The symbol represents the collson operator. In the frst varant of the lattce Boltzmann model suggested by McNamara and Zanett [26], collsons where formulated as drect transcrptons of the lattce gas approach,.e. complete flud partcles were exchanged between the dfferent lattce vectors wthout alterng ther mass content. By usng ths method the partcles mean free path and thus the flud vscosty were stll fxed. Releasng that constrant and allowng the exchange of matter between the flud packets leads to varable vscosty. In ths case the collson operator s a vector. The collson term may be lnearzed by assumng that there s always a local equlbrum partcle dstrbuton, f eq ( x,t), whch depends only on the locally conserved mass and momentum densty. A frst-order approxmaton for the collson operator yelds =0 old (f ( x,t)) = (f eq ( x,t)) + (f old ( x,t) f eq ( x, t)). (8) In order to ensure the local conservaton of mass and momentum the equlbrum dstrbutons must at each node satsfy f eq ( x,t) = f old ( x,t) (9) and =0 f eq ( x,t) c = =0 =0 f old ( x,t) c, (10) where stands for the lattce vectors. Snce now only acts on the departure from equlbrum, the frst term n the frst-order approxmaton for the collson operator, namely eq (f ( x,t)), vanshes. A convenent formulaton for the remander, used by most current versons of the lattce Boltzmann automaton, has the form of a sngle-step relaxaton as suggested by the Bhatnagher Gross Krook approxmaton [32], namely, = 1 old (f ( x,t) f eq ( x, t)). (11) τ In ths expresson the relaxaton tme, τ, s a parameter whch quantfes the rate of change towards local equlbrum for ncompressble sothermal materals. The Bhatnagher Gross Krook relaxaton yelds maxmal local randomzaton. All partcle dstrbutons relax at the same rate, ω = 1/τ, towards ther correspondng equlbrum value. As was frst ponted out by Qan et al [38] the relaxaton rate must obey 0 <ω<2for the method to be stable and for the partcle densty and vscosty to be postve. The condton where 0 <ω<1s

14 R26 Topcal Revew called sub-relaxaton regme whle 1 <ω<2s referred to as over-relaxaton regme. The use of a Bhatnagher Gross Krook sngle-step relaxaton scheme has replaced the use of a dscrete collson matrx whch had to be formulated for collsons n earler lattce gas models. The lattce Boltzmann equaton n the sngle-step relaxaton approxmaton corresponds to the dscrete form of the classcal Chapman Enskog frst-order Taylor expanson of the Boltzmann equaton. For non-sothermal flows or fluds wth varable densty the relaxaton tme may devate from a constant accordng to τ = 1 ( τ 1 ), (12) ξ( x,t,t ) 2 where T s the temperature and ξ( x,t, T ) s the local partcle densty whch can be calculated as the local sum over the partcle velocty dstrbuton accordng to ξ( x,t,t ) = f ( x,t,t ). (13) =0 The relaxaton tme s a parameter whch characterzes the consttutve behavour of the fluent materal at a mcroscopc level. It s connected wth the macroscopc knematc vscosty of the smulated flud accordng to ν = cs 2 t ( τ 2) 1, (14) whch reduces to ν = 2τ 1 (15) 6 for ncompressble sothermal flows where c s = 1/ 3 s the lattce sound speed. In the lattce Boltzmann Bhatnagher Gross Krook method (LBGK) the partcle dstrbuton after propagaton s relaxed towards the local equlbrum partcle dstrbuton functon. The equlbrum dstrbuton, f eq ( x, t), depends, n the Bhatnagher Gross Krook approxmaton, only on locally conserved quanttes such as mass densty and momentum densty. It s carefully chosen so that Gallean nvarance and the sothermal Naver Stokes equaton n the ncompressble flud lmt are recovered. Macroscopc parameters are determned by the ntegraton of the dstrbuton functons. These ntegrands are referred to as moments. Important n that context are those parameters whch are relevant wth respect to local conservaton laws, namely, the local partcle densty, ξ( x,t), the local mass densty, ρ( x,t), and the local velocty vector, u( x,t), whch relates to the momentum densty vector, ρ( x,t) u( x,t), and the local knetc energy densty, ϑ( x,t). They can be calculated as moments of the partcle dstrbuton accordng to ξ( x,t) = f ( x,t), =0 ρ( x,t) = m f ( x,t) = mξ( x,t), =0 u( x,t) = 1 ξ( x,t) ρ( x,t) u( x,t) = m ϑ( x,t) = 1 2 f ( x,t) c = =0 f ( x,t) c, =0 n =0 f ( x,t) c n =0 f ( x,t), f ( x,t) c u( x,t) 2 = ξ( x,t) D 2 =0 ρ( x,t) D RT = m 2 RT, (16)

15 Topcal Revew R27 where m s the mass of a lattce partcle, D s the dmenson of the momentum space of the dscrete lattce veloctes and R s the gas constant. The last equaton defnes the local temperature. The moments must be conserved durng the collson phase,.e. the velocty moments of the collson term must vansh at each node accordng to = 0 and c = 0. (17) =0 =0 The momentum densty tensor, M eq αβ, can be calculated as the second moment of the dstrbuton functon accordng to the equaton M eq αβ = m =0 c α c β f eq ( x,t), (18) where α and β are lateral cartesans of the dfferent velocty vectors c. The stran rate tensor can be approxmated as the frst-order symmetrc part of the velocty gradent tensor accordng to the equaton ε αβ ( x,t) = 1 ( uα ( x,t) 2 x β ( x,t) + u ) β( x,t). (19) x α ( x,t) Rotaton rates can be approxmated as the frst-order skew-symmetrc part of the velocty gradent tensor αβ ( x,t) = 1 ( uα ( x,t) 2 x β ( x,t) u ) β( x,t). (20) x α ( x,t) The equlbrum dstrbuton for an ncompressble sothermal flud, f eq ( x,t), whch approxmates the Maxwell Boltzmann equlbrum dstrbuton up to a second-order Taylor seres, can be wrtten as f eq ( x,t) = w ξ( x,t)[c 1 + c 2 ( c u) + c 3 ( c u) 2 + c 4 ( u u)], (21) where c 1,c 2,c 3,c 4 are lattce constants whch depend on the lattce type and the lattce sound speed, c s,asc 1 = 1, c 2 = 1/c 2 s, c 3 = 1/(2c 4 s ), and c 4 = 1/c 2 s. The symbols w represent the weght factors. The strength of the lattce Boltzmann method s ts computatonal smplcty and ts localty. The latter aspect s an advantage whch predestnes the method for parallelzaton. The algorthm requres nformaton about the dstrbuton functon only at nearby ponts n space. It allows one to treat flows under complex boundary condtons n smple terms by accountng for reflectons and bounces at approprate spatal locatons flagged as sold obstacles. The use of an averaged quantty, f ( x,t), as a central state varable avods statstcal nose so that fnte sze effects practcally do not occur. As n all automata, the set of allowed veloctes n the lattce Boltzmann models s constraned by conservaton of mass and momentum, and by the requrement of rotatonal symmetry (sotropy). However, these restrctons turn out to be much less severe than n the lattce gas cellular automaton models. Ths means that lattce effects practcally do not occur n lattce Boltzmann models. By usng a small velocty Chapman Enskog expanson one can show that the lattce Boltzmann formulaton as outlned n ths chapter reproduces the Naver Stokes equaton for ncompressble flows n the lmt of small Knudsen and low Mach numbers (below 0.15), fgures 1 and 3.

16 R28 Topcal Revew 1.9. The lattce Boltzmann method for mult-phase flows Introducton. Mult-phase flow phenomena are characterzed by movable and deformable phase boundares at whch the propertes of the flow may dscontnuously change. One assumes further that the flows do not evaporate. An essental feature of mmscble mult-phase flows (as for mult-phase solds) s the occurrence of a Laplacan surface tenson for each of the phases whch gudes the system towards the reducton of nterface energy. Three types of lattce Boltzmann approaches have been suggested for the smulaton of complex reacton-free mult-phase flows, namely, the chromodynamc (colour), the pseudopotental and the free-energy models. The followng sectons wll gve a bref ntroducton to these dfferent approaches. An excellent dscusson of the dfferent approaches to lattce-based mult-phase flows s gven n [24] Chromodynamc or colour models of mult-phase lattce Boltzmann flows. The frst lattce-based model for mmscble two-phase flow was proposed by Rothman and Keller [33]. It was formulated as a lattce gas approach. The authors used as a startng pont the snglephase FHP model wth hexagonal lattce and ntroduced two types of (coloured) flud phases, termed red and blue (hence the term chromodynamc or colour models). Phase separaton was, n ther approach, ntroduced by a local flux and colour gradent term. The work of the colour flux aganst the feld mnmum was chosen to encourage the preferental groupng of dentcal phases. Owng to ts descend from the Boolean lattce gas method, the orgnal form of the Rothman Keller model suffered from the defcences assocated wth lattce artefacts and nose, dscussed above. A later verson of a lattce-based two-phase chromodynamc flow model was the two-phase lattce Boltzmann model of Gunstensen and co-workers [39, 40]. Ths method was nspred by the orgnal Rothman Keller lattce gas scheme, but t was based on the lattce Boltzmann method of McNamara and Zanett [26] n conjuncton wth the lnearzed collson operator proposed by Hguera and Jmenez [27]. Although the unphyscal propertes lke the lack of Gallean nvarance and statstcal nose nherent to the lattce gas were overcome, the pressure was stll velocty dependent n ths approach. In addton, the lnearzed collson operator was not computatonally effcent and the model could not handle two fluds wth dfferent denstes and vscostes. Grunau et al [41] developed the model further by ntroducng a sngle-tme relaxaton approxmaton wth a proper partcle equlbrum dstrbuton functon. These modfcatons elmnated the problems of the formulaton of Gunstensen and co-workers [39, 40]. The followng presentaton follows, therefore, essentally the approach of Grunau et al [41]. Mult-phase lattce models use at least two separate phases. Each phase s characterzed n terms of an ndvdual partcle dstrbuton functon and ndvdual equlbrum partcle dstrbuton functon. Ths means that the overall partcle occupaton state at each node s descrbed by a set of partcle velocty dstrbuton functons, each of whch follows n ts dynamc evoluton a lattce Boltzmann equaton, wth an ndvdual collson operator for each phase. It s mportant to note that these ndvdual collson operators do not descrbe the nteractons between dssmlar pseudo-partcles. In the smplest case of a two-component flow, the phases and the assocated partcle populatons are tradtonally labelled by colours [33, 39 41]. The two separate partcle velocty dstrbuton functons are f red ( x,t) and f blue ( x,t). Generalzaton to mult-phase flows leads to f φ ( x,t), where the ndex φ refers to the k dfferent flud phases (runnng from φ = 1tok). The ndex refers to the th velocty vector, x to the lattce node poston, and t to tme.

17 Topcal Revew R29 Assumng only a red and a blue phase these partcle dstrbutons are evolved by a set of modfed lattce Boltzmann equatons of the followng form f new,red f new,blue ( x + c t, t + t) f old,red ( x,t) = t red + S red, ( x + c t, t + t) f old,blue ( x,t) = t blue + S blue, where red and blue are the ndvdual sngle-phase collson operators for the two phases. They have the conventonal form of the sngle-step Bhatnagher Gross Krook relaxaton [32] accordng to red = 1 τ = 1 τ blue red [f red blue [f blue ( x,t) f red,eq ( x,t)], ( x,t) f blue,eq ( x,t)] wth the characterstc relaxaton tmes τ red and τ blue, and the equlbrum dstrbutons f red,eq ( x,t) and f blue,eq ( x,t). One should note that the vscosty of each flud phase can be ndvdually selected by choosng the desred relaxaton tmes for that phase, snce the correspondng operators account for collsons wth partcles of the same type only. The partcle velocty equlbrum dstrbuton for each ndvdual phase, f red,eq ( x,t), f blue,eq ( x,t), depends (n all mult-phase flow lattce models) on the local macroscopc varables pertanng to that phase,.e. ξ φ ( x,t), ρ φ ( x,t) and u φ ( x,t). The equlbrum dstrbutons can, hence, be wrtten as f φ,eq ( x,t) = w ξ φ ( x,t)[c 1 +c 2 ( c u φ ) + c 3 ( c u φ ) 2 + c 4 ( u φ u φ )], (24) where the ndex φ = 1,...,k refers to the k dfferent flud components and c 1,c 2,c 3,c 4 are the Taylor expanson coeffcents c 1 = 1, c 2 = 1/cs 2, c 3 = 1/2cs 4 and c 4 = 1/cs 2. The relevant moments of the ndvdual flows and of the total flow are ξ φ ( x,t) = f φ ( x,t) = f φ,eq ( x,t), =0 ρ φ ( x,t) = m φ ρ( x,t) = =0 k ρ φ ( x,t), φ=1 u φ 1 ( x,t) = ξ φ ( x,t) =0 k ρ( x, t) u( x, t) = m φ φ=1 =0 f φ ( x,t) = m φ =0 f φ,eq ( x,t) = m φ ξ φ ( x,t), n f φ =0 ( x,t) c = f φ ( x,t) c n =0 f φ ( x,t), k =0 f φ ( x,t) c = φ=1 m φ =0 f φ,eq ( x,t) c, where m φ s the mass of the consttuent partcles pertanng to the φth flow and ρ( x,t) u( x,t) s the total local momentum vector of the mult-phase flow. The source terms S red and S blue n the lattce Boltzmann equaton represent the nteracton between the two phases,.e. they must be desgned to capture the phase separaton and coarsenng dynamcs of the flow. In the chromodynamc model the source terms are defned n such a way that they nfluence the confguraton of neghbourng stes enablng the pressure tensor to become ansotropc near the flud nterface. (22) (23) (25)

18 R30 Topcal Revew The man ngredents to the formulaton of these nter-phase nteracton operators are, n the colour approach [33, 39 41], the chromodynamc or colour current vector K K( x,t) = [f blue ( x,t) f red ( x,t)] e = K blue ( x,t) K red ( x,t) (26) =0 and the chromodynamc or colour gradent vector G G( x,t) = [ρ blue ( x + e,t) ρ red ( x + e,t)] e. (27) =0 Accordng to the lattce Boltzmann models of Gunstensen and co-workers [39, 40] and Grunau et al [41] the two source terms, S red and S blue, can be wrtten as S red = A red G 2 ( x,t) ζ G 2 G, S blue = A blue G 2 ( x,t) ζ G 2 G. (28) In ths heurstc quadratc approach G s the magntude of the chromodynamc gradent vector, G = G e s the projecton of the gradent vector along the lattce node drecton e, ζ s a constant proportonal to the square of the lattce speed of sound, and A red and A blue are two adjustable parameters whch control the surface tenson for the two (or more) phases. Ths formulaton shows that the nteracton rules redrect the momentum of the components accordng to the gradent of a colour feld whch s defned by the spatal dstrbuton of the phases. One should also note that the colour gradent vanshes n each sngle-phase regon of the ncompressble flow. Therefore, the source terms only contrbute to nterfaces and mxng regons. The moments for the two flow phases amount to ξ red ( x,t) = =0 f red ( x,t), ξ blue ( x,t) = ξ( x,t) = ξ red ( x,t) + ξ blue ( x,t), =0 f blue ( x,t), ρ red ( x,t) = m red ξ red ( x,t), ρ blue ( x,t) = m blue ξ blue ( x,t), ρ( x,t) = ρ red ( x,t) + ρ blue ( x,t), where ξ red ( x,t) and ξ blue ( x,t) are the partcle denstes of the two flows, and ξ( x,t) s the total partcle densty at lattce pont x and tme t. The quanttes ρ red ( x,t), ρ blue ( x,t), and ρ( x,t) are the correspondng mass denstes. The local veloctes are u red ( x,t) = 1 ρ red ( x,t) u( x,t) = 1 ρ( x,t) =0 =0 f red ( x,t) c, u blue ( x,t) = (f red ( x,t) + f blue ( x,t)) c, 1 ρ blue ( x,t) =0 f blue ( x,t) c, where u red ( x,t), u blue ( x,t) and u( x,t) are the correspondng local veloctes at lattce pont x and tme t. (29) (30) Pseudo-potental models of mult-phase lattce Boltzmann flows. An alternatve to the chromodynamc approach of Rothman and Keller [33], Gunstensen and co-workers [39, 40] and Grunau et al [41] for the lattce-based smulaton of mult-phase flows was suggested

19 Topcal Revew R31 by Shan and Chen [42, 43] and Shan and Doolen [44, 45]. Ther formulaton uses a pseudopotental model and ntroduces a non-local nteracton force between dssmlar flow partcles. Ths potental s essental for the descrpton of non-deal fluds snce t controls the form of the resultng equaton of state of the flud as well as the knetcs of phase separaton. The model of Shan and Chen can be readly formulated for an arbtrary number of phases consstng of partcles wth dfferent molecular masses. The evoluton of the partcle populatons follow, for the k dfferent flud components at each lattce node, a form of the lattce Boltzmann equaton wth dfferent relaxaton and nterpartcle nteracton propertes, as outlned above for the chromodynamc model, accordng to f φ=1,new ( x + c t, t + t) f φ=1,old ( x,t) = t φ=1,old (f τ φ=1 ( x,t) f φ=1,eq ( x,t)) +S φ=1,...f φ=k,new ( x + c t, t + t) f φ=k,old ( x,t) = t φ=k,old (f τ φ=k ( x,t) f φ=k,eq ( x,t)) + S φ=k, (31) where τ 1,τ 2,...,τ k are the relaxaton parameters for the 1, 2,...,kndvdual flows (.e. they do not account for nteractons among dssmlar partcle types). The ndex stands for the n base vectors of the respectve lattce type. The ndex φ refers to the dfferent flud phases (runnng from φ = 1toφ = k). The equlbrum dstrbutons for the ndvdual phases are n the model of Shan and co-workers [42 45] formulated n the same way as outlned above for the chromodynamc model, equaton (24). The source terms S φ=1 S φ=k descrbe the nteracton between the phases usng a pseudo-potental formulaton. They are n the model of Shan and co-workers [42 45] usually formulated n the followng way S φ = F φ e, (32) where S φ s the nteracton source term for phase φ n the drecton of the lattce vector e and F φ s the total effectve nterpartcle force vector actng on the φth component assocated wth the pseudo-potental of the parwse nteracton between dfferent partcle types. Interactons between dentcal partcle types are consdered by the sngle-phase one-step relaxaton terms, as n all Bhatnagher Gross Krook versons of the lattce Boltzmann model. The nteracton force between partcles of component φ at ste x and of component φ at ste x s assumed to be proportonal to ther respectve effectve mass. The nteracton s approxmated n the form of an effectve free-energy potental, ψ φ (ρ φ ( x)), whch s n the Shan Chen model wrtten for phase φ at poston x as a functon of the local partcle mass densty. It takes the followng swtch-lke emprcal form ( ( )) ψ φ ( x) = ψ φ (ρ φ ( x)) = ρ φ 0 1 exp ρφ ( x) ρ φ, (33) 0 whch marks a sharp transton between the lght and the dense phase. In ths expresson ρ φ 0 s a tunable constant n the form of a reference densty whch defnes the transton between the lght and the dense phase. One should remark that the spacng between partcles of component φ at ste x and of component φ at ste x takes n the Shan Chen model only parwse nearest-neghbour nteractons nto account,.e. x x = e. The total nteracton force on component φ at ste x can be wrtten as k k F φ ( x) = V φφ ( x, x )( x x) = V φφ ( x, x + e ) e, φ =1 x φ =1 =0 (34) V φφ ( x, x ) = G φφ ( x, x )ψ φ ( x)ψ φ ( x ),

20 R32 Topcal Revew where V φφ ( x, x ) s an nteracton pseudo-potental between dfferent phases. The summaton symbol over x accounts for nearest neghbour nodes. The symbol G φφ ( x, x ) s the strength of the nteracton. It assumes the form of a Green s functon matrx whch satsfes the symmetry relatonshp G φφ ( x, x ) = G φφ ( x, x). The expresson for the parwse phase nteracton shows that the pseudo-potental force actng on the component phase φ at ste x s smply a neghbour sum of the forces between the flud partcles belongng to phase φ at ste x and the flud partcles belongng to phase φ at the neghbourng stes x. If only homogeneous sotropc nteractons between the nearest neghbours are consdered, the Green s functon G φφ ( x, x ) assumes the form of a smple symmetrc lattce matrx wth constant elements,.e. { 0 f x x G φφ ( x, x > e, ) = (35) g φφ f x x = e, where e s the magntude of the lattce parameter and g φφ s the ampltude factor of the strength of the nteracton potental between components φ and φ. It s an mportant feature of the Green s functon method that phase separaton starts spontaneously once the nteracton strength exceeds a crtcal threshold value. Ths means that, nversely, ths crtcal nteracton value acts lke a phase transformaton temperature [24, 30 32, 42 45]. Therefore, t can be used to calbrate the system wth respect to the thermodynamc propertes of the nteractng flows under consderaton of lattce type and ntal densty. The value of the ampltude factor g φφ can be related to the surface tenson. The moments of these k flows can be calculated, as outlned n the precedng subsecton. Alternatve formulatons of the pseudo-potental method for mult-phase lattce flows use a related approach, where the lattce Boltzmann equaton s not equpped wth a separate source term. In these approaches the actual par nteracton between the mmscble phases enters through a modfed form of the equlbrum dstrbuton functons rather than explctly through a separate source term. One can show, though, that these lattce formulatons are equvalent Free-energy models of mult-phase lattce Boltzmann flows. The mult-phase lattce Boltzmann models outlned n the precedng two subsectons are based on phenomenologcal sharp-nterface approaches to nterface energy and dynamcs. Although, partcularly, the model formulatons suggested by Shan and Chen [42, 43] and Shan and Doolen [44, 45] are well suted for descrbng spontaneous phase separaton ncludng Laplace-type capllary effects n sothermal mult-component flows, an mportant mprovement was suggested by Swft and co-workers [46 48] n terms of the free-energy approach. The basc approach of the free-energy model s that the equlbrum dstrbuton can be defned consstently, based on thermodynamcs, usng, for nstance, a classcal form of a dffuse nterface approach. Consequently, the conservaton of the total energy, ncludng surface, knetc and nternal energy terms, can be properly satsfed. The van der Waals or, respectvely, the Cahn Hllard formulaton of quaslocal thermodynamcs for a two-component flud n equlbrum at a fxed temperature has a free-energy functonal form whch s assumed to depend on densty and densty gradents accordng to ψ(ρ) = [ϕ(t,ρ) + W( ρ)]dv, (36) where the term ϕ(t,ρ) n the ntegral s the bulk free-energy densty and the second term, W( ρ), s the free-energy contrbuton from densty gradents and s related to the surface tenson. An mportant aspect of ths approach s that the free-energy functonal can be wrtten n the form of a Landau potental whch ncludes hgh-order gradent terms that act as a penalty

21 Topcal Revew R33 contrbuton wth respect to nterface curvature. When usng a quadratc nterface penalty approxmaton the non-local system pressure P s related to the free-energy densty functonal accordng to P = ρ dψ(t,ρ) ψ(t,ρ) = P 0 kρ 2 ρ 2 1 dρ 2 k ρ 2. (37) The full Gnzburg Landau-type pressure expresson, whch ncludes also off-dagonal terms (see dervatons n [24, 46]), enters fnally a modfed form of the equlbrum partcle velocty dstrbuton functon whch accounts also for some weak non-local terms [46 48] Thermal fluctuatons and movable nterfaces lattce Boltzmann smulatons of collodal partcle-flud suspensons Lattce Boltzmann automata are well-suted for the smulaton of collodal suspensons owng to ther conceptual potental to tackle ntrcate boundary condtons and to ncorporate fluctuaton forces [24, 25]. In order to smulate partcles suspended n fluds the lattce Boltzmann method must ncorporate dscretzed sold partcles that can move across the nodes of the statonery lattce as well as an approxmate treatment of the nteracton of those partcles wth the flud. The latter aspect can be treated n the framework of the fluctuaton dsspaton approach. Thermal fluctuatons on a mesoscopc scale can be ntroduced nto the lattce Boltzmann framework by means of stochastc Brownan hts. These thermal hts can be ncluded n the form of small random pulses each exertng an addtonal force term whch may shft the postons of the suspended partcles to any of the neghbourng nodes n a probablstc fashon. Whle each ndvdual force pulse may push the partcle n a arbtrary drecton, the overall drectonal and ampltude dstrbuton of the pulses must reproduce a Gaussan form. The varance of the dstrbuton s adjusted n such a way as to defne the temperature of the system by means of the fluctuaton dsspaton theorem [24, 25, 49, 51]. Smlar approaches are well known from solutons of Langevn-type contnuum-feld dfferental equatons whch requre the ncorporaton of stochastc terms that mmc small thermal hts on contnuum objects. The second open queston n ths context s the treatment of the sold obstacles suspended n the flud. Accordng to the work of Ladd and co-worker [25, 49] a sold boundary can be mapped onto the lattce and a correspondng set of boundary nodes, r b, can be defned n the mddle of lnks, whose nteror ponts represent a suspended partcle. A no-slp boundary condton on the movng partcle requres the flud velocty to have the same speed at the boundary nodes as the partcle velocty u b whch has a translatonal porton U and a rotatonal porton L. Assumng that the centre poston of the partcle s R, then u b = U + L ( r b R). (38) The dstrbuton functon f s then defned for grd ponts nsde and outsde the suspended partcle. To account for the momentum change when u b s not zero, Ladd proposed to add a term to the dstrbuton functon for both sdes of the boundary nodes: f ( x) = f ( x) ± B( e u b ), (39) where B s a coeffcent whch depends on the detaled lattce structure and whch s proportonal to the mass densty of the flud. The + sgn apples to boundary nodes at whch the partcle s movng toward the flud and the sgn for movng away from the flud The lattce Boltzmann method for reactve flows Reactve flows are ubqutous n materals scence and engneerng. Promnent examples occur n the felds of corroson and trbology. For descrbng such reactve flows consstng

22 R34 Topcal Revew Table 5. Overvew of some groups whch make smple tral versons of lattce Boltzmann source codes or executables avalable. Unversty of Braunschweg, Germany ( drttmttel/freudger/freudger.htm) Unversty of Erlangen, Germany ( Alfred-Wegener-Insttut, Germany ( Unversty of Tokyo, Japan ( nmura/nmura-qf) Unversty of Geneva, Swtzerland ( of a number of mscble speces n the framework of the lattce Boltzmann approach requres to ntroduce a set of dstrbuton functons, f φ, matchng the varous components φ. The correspondng lattce Boltzmann equatons amount to f φ,new ( x + c t, t + t) f φ,old ( x,t) = t φ,old (f τ φ ( x,t) f φ,eq ( x,t)) + R φ, (40) where R φ s the reactve term whch must have the property to reproduce the correct rates of the densty changes, ρ φ, and of the energy changes, q φ, for each of the reacton partners,.e. m φ R φ = ρ φ, m φ R φ e = 0, m φ R φ e 2 = qφ, (41) =0 =0 =0 where m φ s the mass of a flud partcle belongng to speces φ. Addtonal boundary condtons are due to the conservaton of mass,.e. k k m φ R φ = 0, m φ R φ e = q, (42) 2 φ=1 =0 φ=1 where q amounts to the total heat exchange n the reactve flow. The rates ρ φ and q φ are usually senstve (exponental) functons of the temperature. Whle the basc constrants for R φ are gven above n terms of the conservaton equatons, the detaled coeffcents of the reacton term approxmaton depend on the specfc problem addressed [24] Implementaton, boundary condtons and ntal value condtons for lattce Boltzmann smulatons Numercal aspects, mplementaton and parallelzaton. The man steps n a lattce Boltzmann algorthm are the defnton of the boundary condtons, the ntalzaton of start values for densty and momentum, the calculaton of the local equlbrum dstrbuton wth these gven values, the propagaton of the partcle portons to the next neghbour (except for the dstrbuton of the rest partcles), collson and the calculaton of the new densty and momentum dstrbuton. After ths step the tme ncrement s ncreased by one unt and the algorthm starts agan wth the calculaton of the equlbrum dstrbuton, table 5. The lattce Boltzmann method s bascally resource ntensve when t comes to larger threedmensonal arrays. Ths means that runnng smulatons on systems n excess of nodes s not practcal because of the lack of memory resources and long processng tmes. However, one should underlne that the method has very low memory use and hgh processng speed when counted per lattce ste, partcularly when t comes to complcated boundary condtons. Ths makes t an deal and effcent method for materals-related applcatons whch are often characterzed by rough nterfaces and flow percolaton problems. Because of these lmtatons set by conventonal sngle processor archtectures and owng to the fact that the lattce Boltzmann method generally requres only near-feld neghbour =0

23 Topcal Revew R35 nformaton (lke most cellular automata), the algorthm s a good canddate for parallel mplementatons. Parallelzaton s typcally realzed by multple nstructon multple data (MIMD) systems whch run n sngle program multple data (SPMD) mode. Ths means that the data set s dvded nto spatally contguous blocks along one axs. Multple copes of the same program are then executed smultaneously on dfferent processors belongng to the parallel computer, each operatng on ts own block of data (SPMD concept). Each copy of the program runs as an ndependent process, and typcally each process runs on ts own processor. At the end of each teraton, data for the lnes (two-dmensonal) or planes (threedmensonal) that le on the boundares between blocks are passed between the approprate processes. Ths means that almost all parts of an algorthm must be carred out fully parallel n order to obtan maxmum acceleraton upon parallelzaton. The exchange of data between the processors must be provded wthn the code by usng communcaton lbrary routnes such as the message passng nterface (MPI) lbrary or the parallel vrtual machne (PVM) lbrary Boundary condtons. The frst step n the set-up of the boundary condtons for a lattce Boltzmann smulaton conssts n the defnton of the character of each lattce node. Flud nodes are those grd ponts on whch the flow collson operator s fully appled. All other grd ponts are referred to as sold nodes. The relevant ones among them are the boundary nodes. These are the ones where flows mpnge on at least one sold node whch may belong to a movable partcle or to the system wall. The node type can be dentfed by a Boolean marker. Collsons of flud partcles wth sold objects at the boundary nodes can be grouped nto three types of obstacle stuatons [4, 24, 32]. These are collsons wth statc sold objects, e.g. statc wall elements, collsons wth movng walls, e.g. to shear the system, and collsons wth movng partcles, e.g. such as occurrng for freely suspended collods (see separate secton). In each of these cases the addtonal possblty of wall reactvty can be taken nto account by separate rules. Such contact stuatons are n lattce Boltzmann smulatons usually mplemented by applyng so-called no-slp, or stck, boundary condtons n the case when a sold obstacle mposes frcton, fgure 10. Ths s acheved by mplementng a bounce-back algorthm on the lnks [4, 24, 32]: durng propagaton, the component of the dstrbuton functon that would propagate nto the sold node s bounced back and ends up back at the flud node, but pontng n the opposte drecton. Ths means that ncomng partcle portons are reflected back towards the nodes they came from. Ths rule produces stck boundary condtons at roughly one-half the dstance along the lnk vector jonng the sold and flud nodes, ensurng that the velocty of the flud n contact wth the sold equals the velocty of the latter. In the case when the zerovelocty plane must be located exactly nsde the boundary layer,.e. on the correspondng boundary layer nodes rather than beng shfted from the locaton of the boundary nodes halfway nto the flud, one can use suted nterpolaton algorthms [24, 30 32]. An alternatve to the ntroducton of a nodal bounce-back nterpolaton rule s to place the boundary nodes mdway between sold and flud nodes. Frctonal slp or the lmtng case of free-slp boundary condtons may be approprate for smooth boundares wth small or neglgble frcton exerted on the flow. The surface forces resultng from partcle bounce-back are calculated from the momentum transfer at each boundary node and summed to gve the force and torque on each obstacle object. In contrast to fnte-dfference and fnte-element methods, where local surface normals are requred to ntegrate the stresses over the obstacle surface, the bounce-back rule elmnates these complcatons by drectly summng the surface forces.

24 R36 Topcal Revew Fgure 10. Schematc fgure showng the mplementaton of the bounce-back algorthm both, for no-slp (rgd wall) as well as for slp boundary condtons (movable or deformable wall) Intal condtons. Intal condtons can be defned by startng from an equlbrum dstrbuton. Ths means that the flow densty s equal to a constant everywhere on the grd, snce ρ( x,t) = m n =0 f eq ( x,t), and the speed s equal to 0 at each node n the system before the frst translaton and collson operatons. The ntaton of flows can than be nduced by mposng constant velocty boundary condtons at the flud nlet for nstance n conjuncton wth perodc boundary condtons. Such settngs can typcally approxmate the expermental practce of constant flow rates. Perodc boundary condtons are partcularly useful for modellng bulk systems because they tend to mnmze fnte-sze edge effects. Another mportant ntal standard condton s the assumpton of constant pressure Conventonal cellular automata and the lattce Boltzmann method The basc structure of the lattce Boltzmann method resembles that of a conventonal cellular automaton algorthm, whch has been successfully used partcularly for the smulaton of growth, recrystallzaton and coarsenng phenomena n metals [50, 51]. The classcal cellular automata typcally used n materals scence and engneerng dffer from both lattce gas and lattce Boltzmann methods, snce they do not use flud flow vectors or momentum vector dstrbuton functons but scalar (e.g. energy) or tensoral (e.g. orentaton) parameters as nternal state varables. Otherwse they follow a scheme common to all automata [52 55], that s, they are dscrete n tme and space and use Boolean or real-valued state varables to descrbe the consttutve behavour of the materals at a mcroscopc level. They may be defned on dfferent regular or non-regular two-dmensonal or three-dmensonal lattces consderng the frst, second or thrd neghbour shells for the calculaton of the state change of a node. The system complexty

25 Topcal Revew R37 emerges from the repeated and synchronous applcaton of certan cellular automaton rules equally to all nodes of the lattce. These local rules can for many cellular automaton and lattce Boltzmann models n materals scence be derved through fnte-dfference formulatons of the underlyng dfferental equatons that govern the system dynamcs at a mcroscopc level. Important felds where materals-orented cellular automata have been successfully used for mcrostructure predctons are prmary statc recrystallzaton and recovery [56 65] and soldfcaton [66 70]. The automaton propertes of the lattce Boltzmann method makes t an deal platform for combnatons wth related materals smulaton methods such as cellular automata for nstance for the case of crystal growth [50, 51]. An overvew on the relatonshp between the lattce Boltzmann method and conventonal cellular automaton s gven n [71]. 2. Some applcatons of the lattce Boltzmann method n materals scence and engneerng 2.1. Introducton The second part of the artcle provdes an ntroducton to applcatons of the lattce Boltzmann method n the felds of materals scence and engneerng. Although the lattce Boltzmann method s consderably ganng momentum n the felds of general computatonal flud mechancs, knetc theory, chemcal process engneerng and sol mechancs, the materals engneerng communty has not yet fully exploted ths approach. Important topcs whch are of nterest n the context of materals scence and engneerng are flow dynamc ssues assocated wth trbology and frcton durng metal formng, flud dynamcs durng meltng, castng, sem-sold processng of metals and polymers ncludng mult-component flows, hydrodynamc effects durng lqud lqud and lqud sold phase transformatons, flows n mcroporous mcrostructures such as those occurrng durng processng and nfltraton of metallc foams or related composte pre-forms, collodal flows, lqud crystal flows, lubrcated contact mechancs, mcrodevce engneerng, abrason and crystal growth knetcs n conjuncton wth flud flow. All these examples have three ponts n common. Frst, they mark challengng topcs n current materals scence, engneerng, and processng. Second, t s dffcult to yeld numercal convergence when smulatng such stuatons wth the ad of classcal Naver Stokes solvers, owng to the ntrcate boundary condtons and consttutve behavour nherent to such flows. Thrd, these problems are typcally too large n terms of ther respectve spatal dmensons and characterstc tmescales, so that off-lattce pseudo-partcle or molecular dynamcs approaches cannot be used. Ths means that most of the materals-related problems mentoned above are excellent canddates for the applcaton of the lattce Boltzmann method. An mportant aspect that must be consdered, though, before the use of a lattce-based flud dynamcs smulaton method to an engneerng problem s ts valdty regme wth respect to the stuaton encountered, as already dscussed n greater detal n the frst part of ths artcle. The two mportant crtera n ths context are the rato of the mean free partcle path relatve to the characterstc system length (e.g. obstacle spacng) as expressed by the Knudsen number and the occurrng characterstc macroscopc flow speed regmes as quantfed by the Mach number. As a rule of thumb the lattce Boltzmann scheme s partcularly well suted for small Mach numbers (below 0.15) and small Knudsen numbers (below 0.2), fgures 1 and 3. Some of the engneerng topcs mentoned above wll be dscussed n the followng. It must be noted, though, that the ntenton of ths part of the work does not to present n-depth treatment of the varous topcs, but rather, to present some representatve examples whch document the huge potental of the lattce Boltzmann smulaton technque n the feld of advanced materals

26 R38 Topcal Revew Fgure 11. Applcaton of the lattce Boltzmann method for the predcton of turbulence as a functon of surface roughness at deformable metallc surfaces [50, 71]. The whte area ndcates the rough sheet surface. The greyscale pattern marks the flud pressure where lght tones corresponds to hgh and dark tones to low pressure. The arrows n the upper fgures track some flud portons vsualzng turbulence. scence and engneerng. The ntenton, therefore, s to stmulate the reader s nterest n ths method wth respect to current and new problems n materals research. Further detals whch are beyond the lmts of ths artcle must, therefore, be obtaned from the orgnal references provded n each subsecton Lubrcaton dynamcs n metal formng The precson whch s nowadays requred n the area of metal formng and tool desgn requres detaled knowledge of the underlyng contact mechancs between workpece, lubrcant and tool. An essental example s the doman of large-scale automotve sheet formng where the overall shape accuracy after formng, ncludng elastc sprngback, must be of the order of some hundred mcrons. Another example s the feld of mcrodeformaton processng such as used when formng metallc parts n the mllmetre and centmetre range. Predctng the processng of such products s even more ntrcate when t comes to the treatment of contact mcromechancs at a quanttatve level. Related ssues occur n the felds of sheet and fol rollng or for flows and corroson n narrow tubes wth rough surfaces. A man aspect n the context of sheet formng at least as far as flud mechancs s concerned s the mportance of the surface roughness and the resultng (Prandtl-type) boundary layer flow dynamcs n the vcnty of such a rough nterface. Of partcular nterest are scalng effects n boundary layers whch arse from changes n the surface topography of metals such as occurrng durng plastc formng. Scalng s mportant because metallc surfaces become rougher durng deformaton whle the flud propertes may reman unchanged at least wthn certan bounds (temperature changes due to dsspated heat as well as abrason are neglected at ths pont). An mportant observaton s the transton from lamnar to turbulent flow as a functon of the roughness of the deformed metal surface. Flud dynamcs for such a stuaton can be smulated by the use of a lattce Boltzmann automaton. For the example gven n fgure 11 the smulaton strategy was desgned to study the transton from lamnar to turbulent flow as a functon of the ncreasng roughness of a

27 Topcal Revew R39 Fgure 12. Smulaton of Mller and Succ [74] of dendrtc growth n a flud envronment. Fgures (a), (b) and (c) show the evoluton of crystal shapes for dfferent seeds and tltng angles n the case of dffusve transport only. Fgures (d), (e) and ( f ) show the evoluton of the same crystal shapes for dfferent seeds and tltng angles f buoyancy convecton s present. surface and of the vscosty of the flud. The transton s characterzed by the formaton of turbulences n the vcnty of the tps of the roughness peaks. The rough metallc surface s modelled as a snusodal wall. One mportant parameter n the study s the varaton of the perod and ampltude of the snusodal surface. The flow s modelled by usng a standard lattce Boltzmann automaton wth sngle-step relaxaton, fgure Dendrtc crystal growth under the nfluence of flud convecton The group of Mller and co-workers [72 75] has recently desgned a three-dmensonal parallel lattce Boltzmann code for the smulaton of lqud sold phase transformatons, n partcular, for predctng dendrtc growth n a flow envronment, fgure 12. The basc challenge of such an approach conssts of provdng a numercal tool for calculatng growth knetcs together wth flud flow on a mesoscopc scale n one ntegrated smulaton approach. Related poneerng work about the conjuncton of forced flows wth crystal growth was publshed by Tönhardt and Amberg [76] as well as by Beckermann et al [77]. The am of such smulaton studes s to study the nfluence of flud convecton on the crystal growth knetcs and on the resultng mcrostructures of the crystals. The engneerng perspectve of such approaches s at hand. For nstance, the growth of large sngle crystals wth hgh qualty for electronc and optcal purposes s of huge ndustral mportance. Sngle crystals wth certan lattce defects such as twns, small angle boundares or related dslocaton arrays resultng from growth do no longer have the same functonal propertes as a perfect sngle crystal. It s lkely that a strong relatonshp exsts between the flow dynamcs on the one hand and the elementary atomc-scale and mesoscale soldfcaton mechansms on the other. It s, hence, of substantal mportance to combne these two aspects n one theoretcal framework,.e. the development of jont smulaton approaches may help to predct the optmal condtons for crystal growth experments wth respect to achevng crystals wth good propertes. The approach of Mller and co-workers [72 75] s based on a jont phase-feld lattce Boltzmann automaton concept. The two phases, lqud and sold, are, n ths approach,

28 R40 Topcal Revew dstngushed by a phase-feld structure varable, smlar as n the Gnzburg Landau or Allen Cahn models. In contrast to conventonal contnuum sold-state phase feld models, where the tme evoluton of the phase feld s computed by the ntegraton of a Gnzburg Landau-type dfferental equaton, Mller et al descrbe the phase transton by a reacton model as orgnally suggested by de Fabrts et al [78]. Ths model descrbes the phase transformaton n terms of transton rates across the nterface from one phase nto the other and vce versa. The transton rates are calculated by usng frequency factors from the nverse tmescale for soldfcaton and meltng, respectvely, together wth swtch functons whch control the onset of meltng and soldfcaton around the crtcal temperature Smulaton of metal foam processng Metal foams are a novel class of energy absorbng structural materals wth a consderable perspectve for applcatons n the feld of lght-weght materals engneerng. Ther wdespread commercal use, though, s stll mpeded by the unsatsfactory reproducton of materal homogenety. The mcrostructure and pore dstrbuton of foams are to a large extent determned by the process parameters and by the detals of the producton strategy. Therefore, t s sensble to accompany further ntatves for mcrostructure and property optmzaton of metallc foams wth systematc process smulatons. These should be desgned to dentfy process wndows for optmum foam homogenety and reproducblty of the cellular mcrostructure. The smulaton of the evoluton and decay of metallc foams produced by powder metallurgcal routes s a very demandng target for applcatons of the lattce Boltzmann method, snce such processes are characterzed by ntrcate boundary condtons, phase transformaton, melt dynamcs, gas dynamcs and gas melt nteracton. Also, gravty occurs n such processes as a relevant body force. The group of Snger [79] has recently publshed such an nvestgaton on the formaton of metal foams by usng a lattce Boltzmann method. They had chosen a formulaton wth free surface boundary condtons whch allowed them to ncorporate the gas lqud nterface that s typcal of the cellular foam mcrostructure evolvng durng processng. The study amed n partcular at the clarfcaton of the relatonshp between the processng parameters and the resultng mcrostructures placng attenton on pore nucleaton, pore growth, pore coalescence and soldfcaton. The smulatons were used to better understand correspondng experments whch were conducted wth an alumnum alloy whch was mxed and subsequently processed wth the TH 2 as an agent provdng the gas by a powder metallurgcal processng route. The study provded basc nsght nto the nfluence of vscosty, surface tenson, body forces and mould form on the knetc and structural evoluton of the foam mcrostructure. Related felds where the lattce Boltzmann method has reached the necessary maturty as a smulaton tool for optmzng producton processes s the modellng of flows n complex and tme-dependent geometres, as they are encountered n the context of composte materals that are manufactured by nfltratng fbre or powder pre-forms, fgure Hydrodynamcs of lqud crystallne polymers Lqud crystallne polymers are n a state of matter n whch lqud-lke order exsts at least n one drecton of space and n whch some degree of structural ansotropy s present. Typcally such materals consst of rod- or plate-lke molecular consttuents whch can algn to a certan extent. One dfferentates between two types of lqud crystallne polymers. Nematc ones

29 Topcal Revew R41 Fgure 13. Smulaton of flow nfltraton through hghly dealzed porous mcrostructures. The upper fgure shows the pressure dstrbuton and the two lower fgures show detals of the flow vectors. are composed of molecules wth ther long axes algned along a specfc drecton whle ther centres of mass are dstrbuted randomly n space. The smectc lqud crystal state has a hgher degree of order than the nematc one due to exstence of quas-long range order n the postons of the centres of gravty of the molecules n one or two dmensons. Ths means that the nematc state s characterzed by orentatonal order whle the smectc one reveals both translatonal and orentatonal order. Snce the state of structural order n these lqud crystals s between the tradtonal sold and lqud phases they are sometmes synonymously referred to as mesogenc materals. To quantfy just how much order s present n a materal, an order parameter can be defned whch quantfes the angular devaton between the drector and the long axs of each molecule. For a perfect crystal, the order parameter s one. Typcal values for the order parameter of a lqud crystal range between 0.3 and 0.9, wth the exact value a functon of temperature, as a result of knetc molecular moton. The algnment of the lqud crystal molecules entals tensoral ansotropy of the propertes. Studyng the relatonshp between the structure of lqud crystallne materals and the underlyng hydrodynamcs s essental for understandng the propertes of these materals. Other than conventonal Newtonan flows, lqud crystals reveal a strong couplng between ther mcroscopc structure and the velocty felds mposed by the flow. For nstance, shear flow can nduce non-equlbrum phase transton from the sotropc (flud) to the nematc state or lead to phenomena such as shear-thnnng and thckenng.

30 R42 Topcal Revew Fgure 14. Lattce Boltzmann smulatons of lqud crystal hydrodynamcs. The couplng between the tensor order parameter and the flow s treated consstently, allowng nvestgaton of a wde range of non-newtonan flow behavour [82]. The fgure shows two dfferent states n Poseulle flow, where the lnes represent the drector orentaton of the nematc phase projected down onto the x y plane, and the shadng represents the ampltude of the order parameter. Flow s from top to bottom, and the walls are at the left and rght. At the walls, the drector s algned perpendcular to the boundary. (a) A stable confguraton at low flow. (b) Snapshots of an oscllatng confguraton where the central regon s n the log-rollng state (drector perpendcular to the plane) and the boundary regon conssts of a transton from a confguraton n the shear plane to a tumblng and kayakng regon (drector rotatng n and out of the plane) nterfacng to the central log-rollng state. In order to better understand the mechancs of such flows t s pertnent to use smulaton methods whch can take nto account the dfferent length and tmescales that are relevant for the consttutve behavour of such materals. Edwards and co-workers [80, 81] and the group of Dennston and Yeomans [82 84] have suggested to use a mult-phase lattce Boltzmann free-energy method n conjuncton wth the Bhatnagher Gross Krook sngle-step approxmaton for solvng the hydrodynamc equatons of moton for nematc lqud crystals, fgure 14. The man modfcaton of ther formalsm when compared to conventonal mult-phase lattce Boltzmann schemes s the ntroducton of

31 Topcal Revew R43 an addtonal symmetrc traceless tensor-valued structural dstrbuton functon S. f new S new ( x + c t, t + t) f old ( x,t) = t τ ( x + c t, t + t) S old ( x,t) = t old (f f τ S (Sold ( x,t) f eq ( x, t)), ( x,t) S eq ( x, t)), where S eq s the correspondng equlbrum dstrbuton. The values of ths tensor densty varable are related to a tensoral order parameter whch can descrbe the crystallne ansotropy of those volume portons that assume the nematc or smectc state. Backflow, the hydrodynamcs of topologcal defects, and the possblty of transtons between the lqud crystallne and sotropc phases appear naturally wthn the formalsm. The method can also be used to study the velocty dependence of the crtcal temperature n the nematc-sotropc transton Flow percolaton n confned geometres Fgure 15 shows a two-dmensonal example (takng a perspectve nto the plane n whch the flud flows) where a Boltzmann-based lattce smulaton has been appled to the stuaton of a confned lubrcaton flow. The roughness data were expermentally obtaned from a plastcally deformed steel surface. The expermental roughness analyss conducted on the surface of the sample allowed us to separate the free volume portons from the regons where the sample was n closed contact wth the tool. The percolatve flow n the remanng confned nterface represents a typcal example of flow n a porous envronment. The jont experments and flud dynamcs smulatons am at a better understandng of trbology and contact mechancs durng steel sheet rollng. Parameters to be vared n the experments and smulatons are the vscosty of the lubrcants, the surface roughness of the tool and sheet materals, and the deformaton rate. The upper left-hand sde of fgure 15 shows the pressure dstrbuton n the lubrcant when compressed and redstrbuted n the expermentally determned obstacle (contact) feld. The data show an n-plane vew nto the contact layer. The other two fgures show magnfcatons of flow detals Processng of polymer blends breakup and coalescence of flud droplets Another mportant area for applyng the lattce Boltzmann method to materals engneerng s the feld of polymer processng, partcularly the mxng of mmscble polymers. Because most chemcally dfferent polymers are relatvely mmscble, flud blendng of such materals s an ubqutous challenge n the feld of ndustral polymer processng. The two most mportant classes of engneerng polymers produced by blendng are rubbertoughened plastcs and stffened elastomers. Such compostes are characterzed by synergetc mechancal propertes whch arse from the two mmscble polymer compounds n them. Blendng processes for polymers are typcally desgned to produce fne sphercal nclusons for an ncrease n mpact resstance or fbre-type dspersods for enhanced undrectonal strength. Typcal products made of such materals are structural macroscopc parts. Examples are blends of nylon and rubber and rubber-toughenng of brttle glassy polymers where the rubber nclusons can stop propagatng cracks through the brttle materal and dsspate energy. For these reasons assocated wth mechancal propertes and mcrostructure homogenety the am of the blendng process s to produce very fne dspersons of the order of submcronsze droplets. Optmzaton of the process wndow for achevng these goals requres an mproved understandng of the dynamcs and mechansms assocated wth polymer droplet breakup durng mxng of such mmscble polymers. Important subtasks n ths context are (43)

32 R44 Topcal Revew Fgure 15. A two-dmensonal example of the applcaton of a Boltzmann lattce gas smulaton to measured the roughness data of plastcally deformed steel. The upper left fgure shows the pressure dstrbuton n the lubrcant when compressed and redstrbuted n the expermentally determned obstacle (contact) feld. The data show an n-plane vew nto the contact layer. The other two fgures show magnfcatons of flow detals [71]. the smulaton of the mcroscopc breakup processes of droplets n shear flow and added block copolymers, the predcton of the droplet sze dstrbuton n heavly sheared multphase flows, the coalescence of polymer droplets n blends durng processng, as well as the detaled analyss of such mcroscopc processes under realstc ndustral boundary condtons n terms of geometry, processng rates, pressure and shear rates. The lattce Boltzmann method represents an excellent approach to the numercal analyss of such mult-phase polymer processng operatons, partcularly when amng at the smulaton of the breakup of droplets under shear. The approach s very effectve n systems whch nvolve low Reynolds numbers, dfferent phases and complex geometrcal boundary condtons [85, 86]. 3. Conclusons The study gave an overvew of the lattce Boltzmann smulaton method as an advanced tool for predctons n the feld of materals scence and engneerng. The frst part presented the basc