Cahn Hilliard modeling of particles suspended in two-phase flows

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS In. J. Numer. Meh. Fluds (211) Publshed onlne n Wley Onlne Lbrary (wleyonlnelbrary.com) Cahn Hllard modelng of parcles suspended n wo-phase flows Young Joon Cho and Parck D. Anderson *, Deparmen of Mechancal Engneerng, Endhoven Unversy of Technology, Endhoven, The Neherlands SUMMARY In hs paper, we presen a model for he dynamcs of parcles suspended n wo-phase flows by couplng he Cahn Hllard heory wh he exended fne elemen mehod (XFEM). In he Cahn Hllard model he nerface s consdered o have a small bu fne hckness, whch crcumvens explc rackng of he nerface. For he drec numercal smulaon of parcle-suspended flows, we ncorporae an XFEM, n whch he parcle doman s decoupled from he flud doman. To cope wh he movemen of he parcles, a emporary ALE scheme s used for he mappng of feld varables a he prevous me levels ono he compuaonal mesh a he curren me level. By combnng he Cahn Hllard model wh he XFEM, he parcle moon a an nerface can be smulaed on a fxed Euleran mesh whou any need of re-meshng. The model s general, bu o demonsrae and valdae he echnque, here he dynamcs of a sngle parcle a a flud flud nerface s suded. Frs, we apply a small dsurbance on a parcle resng a an nerface beween wo fluds, and nvesgae he parcle movemen owards s equlbrum poson. In parcular, we are neresed n he effec of nerfacal hckness, surface enson, parcle sze and vscosy rao of wo fluds on he parcle movemen owards s equlbrum poson. Fnally, we show he movemen of a parcle passng hrough mulple layers of fluds. Copyrgh 211 John Wley & Sons, Ld. Receved 25 February 211; Revsed 3 May 211; Acceped 8 May 211 KEY WORDS: Cahn Hllard heory; dffuse-nerface model; exended fne elemen mehod (XFEM); emporary ALE scheme; fne elemen; hydrodynamcs; ncompressble flow; lamnar flow; wo-phase flows 1. INTRODUCTION Small sold parcles adsorbed a lqud nerfaces arse n many ndusral producs and processes, such as anfoam formulaons, crude ol emulsons, fludzed suspensons, slurry ranspor, maerals separaon, rae of mxng enhancemen, ec. In parcular, f parcles are suspended n emulsons, hs emulson can be sablzed by sold parcles ha adsorb ono he nerface beween he wo fluds, whch s usually called a Pckerng emulson [1]. They ac n many ways lke radonal surfacan molecules, bu offer dsnc advanages. Unforunaely, he undersandng of how hese parcles operae n such sysems s lmed. In hs paper, we presen a numercal mehod for he dynamcs of parcles n wo-phase flows based on he Cahn Hllard heory. Dffuse-nerface modelng s based on he van der Waals approach of he nerface problem [2] and developed by Cahn and Hllard [3]. The man assumpon s ha he nerface s no sharp, bu has a hckness ha s no explcly prescrbed, bu follows from he governng equaons ha couple hermodynamc and hydrodynamc forces. The man elemens of he heory, and he couplng of hermodynamcs and hydrodynamcs are summarzed n he revew paper by Anderson e al. [4] and references heren. *Correspondence o: Parck D. Anderson, Deparmen of Mechancal Engneerng, Endhoven Unversy of Technology, PO Box 513, 56 MB Endhoven, The Neherlands. E-mal: p.d.anderson@ue.nl Copyrgh 211 John Wley & Sons, Ld.

2 Y. J. CHOI AND P. D. ANDERSON Dffuse-nerface mehods have been appled o a varey of mulphase flow problems rangng from phase separang polymer blends o smulang sold umor growh usng mxure models. For example, Prusy e al. used he Cahn Hllard echnque o he coarsenng dynamcs for PMMA/SAN28 blends and a quanave comparson beween he expermenal and numercally predced phase separaon knecs was presened [5]. Wse e al. presened smulaons of umor growh n wo and hree dmensons ha demonsrae he capables of he dffuse-nerface model n accuraely and effcenly smulang he progresson of umors wh complex morphologes [6]. Khaavkar e al. used he dffuse-nerface mehod o model mcron-szed drop spreadng and mpac on smooh and srucured subsraes [7 9]. Recenly, Tufano e al. appled he Cahn Hllard heory coupled wh hydrodynamc neracons o descrbe a hree-phase sysems where he effecs of muual dffuson on nerfacal enson, drop drop and drop wall neracons n quescen condons are nvesgaed and compared wh expermenal observaons [1]. Mlle and Wang nroduced a dffuse-nerface feld descrpon of each flud phase n addon o he se of sold parcles. Ther model can nclude parcles of arbrary shapes and orenaons, and he ably o ncorporae elecrosac parcle neracons [11]. The mos nuve mehod o smulae parcle movemen n wo-phase flows s usng a boundaryfed mesh, whch means ha he parcle surface s algned wh elemen boundares of he flud [12 14]. In hs mehod, he governng equaons are solved only n he flud doman, akng no accoun he nerface condons on he boundares of parcles. To handle movng parcles, hs approach ncorporaes he ALE echnque ha reles on a movng mesh scheme. The generaon of a new mesh s needed f he old mesh becomes oo dsored, and he soluon mus be projeced ono he new mesh. The generaon of boundary-fed meshes s, however, sll a challengng ask n vew of algorhms needed and compuaonal coss f complex geomeres are nvolved, especally n hree-dmensonal smulaons usng second-order hexahedron elemens. An alernave approach s he fcous doman mehod developed by Glownsk e al. [15 17]. The basc dea of hs mehod s o fll each doman of every parcle wh he surroundng flud, assumng and subsequenly prescrbng ha he flud nsde he parcle doman moves lke a sold objec. Hence, he problem s ransferred from a geomercally complex flud doman o a smpler doman ncludng boh flud and parcles, whch elmnaes he need of re-meshng. In hs mehod, parcles move n a Lagrangan sense on a fxed Euleran mesh. For sngle-componen problems hs approach has been que successful and he dynamcs of parcles n complex fluds has been suded n a varey of flow condons [18, 19]. However, f a parcle s suspended n a wo-componen sysem, a fcous doman approach would requre addonal consrans for he moon of fcous fluds nsde he parcle, whch s no rval and n hs work an alernave approach s followed. The exended fne elemen mehod (XFEM) has been recenly developed o smulae parcle suspended sngle-componen flud flows. In hs mehod, he fne elemen shape funcons are locally exended, or enrched, o decouple he flud doman from he parcle doman whle sll usng a mesh ha s no boundary fed. Orgnally, XFEM was developed for he smulaons of cracks n srucures whou he need of re-meshng [2,21]; laer, was appled o flow problems as well [22 24]. A recen revew on XFEMs appled o maeral modelng s presened n [25]. Cho e al. proposed a emporary arbrary Lagrangan Euleran (ALE) scheme o handle movng parcles whou any need of re-meshng hroughou he whole compuaons n he exended fne elemen conex [26]. In he presen paper, we presen a numercal mehod for he dynamcs of parcles n wophase flows by couplng he Cahn Hllard heory wh he XFEM whle usng a fxed Euleran mesh whou any need of re-meshng. Because he flud doman s decoupled from he parcle doman n XFEM, we do no need exra condons nsde he parcles, conrary o he fcous doman mehod. The conen of hs paper s as follows. In Secon 2 we gve a bref revew of he Cahn Hllard heory. In Secon 3 he numercal algorhm of he XFEM combned wh a ALE scheme s dscussed. The nroduced model s appled o sudy he dynamcs of a sngle parcle a a flud flud nerface n Secon 4, and he movemen of a parcle passng hrough mulple layers of fluds s demonsraed n Secon 5. Fnally, a dscusson follows n Secon 6.

3 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS 2. MATHEMATICAL FORMULATION The classcal expresson for he specfc Helmholz free energy used n dffuse nerface modelng s based on he work of Cahn and Hllard [3] f.c, rc/ D 1 2 c2 C 1 4ˇc4 C 1 2 " jrcj2, (1) where and ˇ are posve consans and " s he graden-energy parameer and c s he mass fracon of one of he wo componens [27]. The chemcal poenal s obaned from he varaonal dervave wh respec o concenraon D ıf ıc D c C ˇc3 "r 2 c. (2) Ths generalzed chemcal poenal allows for he descrpon of he nerface beween he wo maerals by a connuously varyng concenraon profle. For example, for a planar nerface, wh x beng he drecon normal o he nerface, he analycal soluon of Equaon (2) reads r c.x/ D ˇ anh x p 2, (3) wh p =ˇ he equlbrum bulk soluons (n he approach oulned here 1) and (D p "= ), whch s a measure for he nerfacal hckness. In order o comply wh mass conservaon for boh componens, he balance equaon should be C u rc Dr.M r/, wh M he mobly, whch n general s a funcon of he composon, bu s here aken consan for smplcy. The dffuson flux s assumed o be proporonal o he graden of he chemcal poenal, whch s more general han he common Fckan dffuson, based on he concenraon gradens (rc), whch does no hold for mulphase sysems, even a equlbrum. The more general expresson used n Equaon (4) reflecs Gbbs fndngs ha he chemcal poenal becomes unform n a nondeal mxure a equlbrum, and s known as he Cahn Hllard equaon [28]. To oban momenum conservaon, a generalzed Naver Sokes equaon can be derved for he velocy feld C.u r/ u D rgcr.2d/ C rc, ru D, (6) where D D.ruCru T /=2 s he rae-of-deformaon ensor; g s he Gbbs free energy gdf Cp=, wh p he local pressure and he densy. The vscosy generally depends on c because he wo fluds, n general, have dfferen vscoses. The vscosy, s assumed o have he followng lnear relaonshp wh he concenraon c, D 1 c C 1 2 c 1 2, (7) 2 where 1 and 2 are he vscoses of he wo fluds, respecvely. Compared wh he Naver Sokes equaons, n Equaon (5) only one exra capllary erm rc appears reflecng he nerfacal enson. Ths modfcaon accouns for hydrodynamc neracons, ha s, he nfluence of he concenraon c or he morphology on he velocy feld and, hence, descrbes he spaal varaons of he velocy feld because of he presence of nerfaces. To focus on he couplng of he parcles wh he mulphase sysem, whou any loss of generaly, we now furher assume quas-saonary flow and neglec nera n he momenum balance. Then, he momenum balance Equaon (5) reduces o r.2d/ Crg D rc. (8)

4 Y. J. CHOI AND P. D. ANDERSON 2.1. Scalng of he equaons Usng c D c=c B, u D u=u, D 2 =."c B /, D U=L, wh c B D p =ˇ he bulk concenraon, U a characersc velocy, and L a characersc doman sze, and omng he asersk noaon, Equaons (2), (4), (6), and (8) read n dmensonless form dc d D 1 Pe r2, (9) D c 3 c C 2 r 2 c, (1) r.2d/ Crg D 1 rc, (11) CaC ru D. (12) Three dmensonless groups appear n he governng equaons: Pécle number Pe, he capllary number Ca and he Cahn number C,defnedas PeD 2 LU M" U I Ca D "cb 2 I C D L. (13) 2.2. Rgd-body moon of parcles We suppose ha N parcles are suspended n an ncompressble flud. Le be he enre doman ncludng he flud and parcles, and le P./. D 1, :::, N/ be he embedded doman of he -h parcle a me wh he number of parcles N. The collecve parcle regon a a ceran me s denoed by P./ D S N D1 P./. Boundares are denoed by For he unknown rgd body moons (U,! ) of he parcles, we need balance equaons for forces and orques on parcle surfaces. In he absence of nera, exernal forces F ex, and orques T ex, acng on he parcle P./ are balanced by he ne hydrodynamc force F and orque T on he parcle, respecvely Z F D Z n ds D F ex,, (14).x X /. n/ ds D T ex,, (15) where n s he ouwardly-dreced un normal vecor on he parcle The parcle posons X and angular orenaons are obaned from he followng knemac equaons: dx d D U, X. D / D X,, (16) d D!,. D / D,. (17) d A he flud parcle nerface, we use he no-slp boundary condon u D U C!.x X / D 1, :::, N/. (18) 2.3. Paral weng boundary condons The parcle may be neural or preferably weed by one of he componens of he bnary flud. Ths effec s accouned for by followng he approach gven by Cahn [3], where sold flud neracons

5 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS are assumed o be shor ranged. Because of hs assumpon, he oal sysem free energy F can be wren as Z F D np./ f dv C NX Z f p ds, (19) where f p s he specfc parcle free energy ha depends only on he concenraon a he parcle and np./ s he flud doman volume bounded wh a parcle / D S N and f s defned n Equaon (1). The surface negral erm n Equaon (19) represens he conrbuon of sold flud neracons. A equlbrum, F s a s mnmum. Mnmzng F usng mehods of varaonal calculus subjec o a naural boundary condon gves he followng boundary condon on he @n p D, where n s he drecon normal and f p s he specfc parcle free energy. For f p we use he form proposed by Jacqmn [31], whch reads, f p D c c3, (21) 3 where s assumed o be a consan and referred o as he weng poenal. I can be made o vary spaally o ndcae chemcal heerogeney of he parcle surface. Wh f p of he form p =@c evaluaed a c B s zero, so a equlbrum he parcle surface s no enrched n one of he fluds and depleed n he oher. Equaons (2) and (21) are nondmensonalzed usng he dmensonless varables defned n Secon 2.1 wh he addon of lv as he characersc scale for he specfc parcle free energy o C 1 c 2 D, (22) f p D c c3, (23) 3 where D.c B /=. lv / s he dmensonless weng poenal and C s he Cahn number defned n Equaon (13). Usng Young s equaon, whch connecs he conac angle wh he nerfacal ensons lv, sv and sl cos D sv sl lv, (24) whch s also shown n Fgure 1. The parameer can be relaed o he (equlbrum) conac angle o yeld cos D 4. (25) 3 From (25) s concluded ha for a conac angle equal o 9 ı, s zero and he mxed boundary condon (22) reduces o he naural boundary D.

6 Y. J. CHOI AND P. D. ANDERSON Flud 2 Flud 1 Parcle Fgure 1. Defnon of conac angle. 3. NUMERICAL METHODS 3.1. Weak form In dervng he weak form of he governng equaons for a flud parcle sysem, we follow he combned equaon of moon approach [17], n whch he hydrodynamc forces and orques acng on parcles are elmnaed from he equaon of moon because hey are nernal. The no-slp boundary condon on he parcle surface s mposed by usng consrans mplemened wh Lagrangan C u rc C M.rr, r/ D, (26) e, c ˇc 3 ".re, rc/ C.e, / D, (27).rv/ T, 2D.r v, g/ C.v.V C.x X //, D.v, / N C V F ex, C T ex, C.v, rc/, (28).q, ru/ D, (29)., u.u C!.x X D, (3) for all admssble es funcons r, e, v, q,, V and.., /,., / N,and., are proper nner producs on he flud doman np./, on he Neumann boundary N and on he parcle respecvely Tme dscrezaon of he dffuse-nerface model We solve he governng Equaons (26) (3) n a decoupled way. Frs, he concenraon c and chemcal poenal are solved smulaneously from Equaons (26) and (27) wh approprae boundary condons. Then, we solve he Sokes-ype flow problem by reang he addonal rc erm as a forcng. For he me dscrezaon of he evoluon equaon of he concenraon (Equaon (26)), we use a second-order Gear scheme r,! 3 2 cnc1 2c n C 1 2 cn 1 C Ou nc1 rc nc1 C M rr, r nc1 D, (31) e, c nc1 ˇ.c nc1 / 3 " re, rc nc1 C e, nc1 D, (32) where s he me sep and Ou nc1 D 2.u n u n m /.un 1 u n 1 m /. Here u m represens a mesh velocy because of a mesh movemen scheme ha wll be defned n Secon 3.4. Because hese equaons are nonlnear, we solve hem by Pcard eraon a each me level D nc1

7 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS r, 3 c C1 C Ou nc1 rc C1 C M.rr, r C1 / D 2! r, 2cn 1 2 cn 1, (33) e,. ˇc 2 /c C1 ".re, rcc1 / C.e, C1 / D, (34) for D, 1, :::unl convergence, wh c D c n. For he frs me sep D, we use an mplc Euler scheme nsead of Equaon (31) r, cnc1 c n C.u n u n m / rcnc1 C M rr, r nc1 D. (35) The complee me negraon seps are gven n Secon XFEM formulaon For he drec numercal smulaon of flows wh freely suspended parcles, we use he XFEM combned wh he ALE scheme, whch was presened by Cho e al. [26]. Here we brefly revew he mehod. For a dealed descrpon of he mehod, please see [26] and he references heren. In he XFEM conex, he velocy, pressure, concenraon, and chemcal poenal can be dscrezed as u h.x/ D X.x/H.s/u, (36) g h.x/ D X.x/H.s/g, (37) c h.x/ D X.x/H.s/c, (38) h.x/ D X.x/H.s/, (39) where ² C1 f s >, H.s/ D f s<, (4) s defned by a level se funcon s 8 < s.x/ : D f x s /, > f x s n np./, < f x s n P./. (41) We use a bquadrac Q 2 nerpolaon ( ) for he velocy u, concenraon c, and chemcal poenal ; a blnear Q 1 nerpolaon ( ) for he modfed pressure g. For elemens nerseced by he surface of a parcle, he no-slp boundary condon (Equaon (18)) s mposed by usng consrans mplemened wh Lagrangan mulplers as shown n Equaons (28) and (3). The nner produc., s he sandard nner produc n L 2.@P.//: Z., u.u C!.x X D.u.U C!.x X /// ds. For he dscrezaon of Equaon (42), we use a lnear shape funcon P 1 for he dscrezaon of Lagrangan mulplers and a quadrac shape funcon P 2 for he geomercal shape of each elemen.

8 Y. J. CHOI AND P. D. ANDERSON 3.4. Temporary ALE scheme For a movng parcle problem, he feld varable a he prevous me levels, such as u n, u n 1, c n and c n 1, can become undefned near he boundary of he parcle because here may have been no flud flow a me level n. To overcome hs problem, we use a ALE scheme, whch defnes a mappng of feld varables a he prevous me levels on he curren me level [26]. In hs mehod, mesh nodes near a parcle follow he moon of he parcle, whereas, mesh nodes far away from he parcle are saonary. A mesh velocy feld u m s solved usng Laplace s equaon: r 2 u m D n, (43) u m D on, (44) u m D U C!.x X / (45) For a crcular parcle, Equaon (45) can be replaced by u m D U. Noe ha Equaons (43) (45) are solved on an Euleran mesh, ncludng he parcle doman P./, by usng a smlar echnque as n he fcous doman mehod [15 17]. Equaon (45) s realzed by usng a consran mplemened wh Lagrangan mulplers. An ALE mesh a he prevous me level D n, x n ALE, s consruced usng a predcor correcor mehod x m D xn 1 m x n ALE D xn 1 m C 1 2 C u m.x n 1 m, n 1 / (predcor) (46) um.x m, n / C u m.xm n 1 / (correcor) (47) Then, he ALE mesh a he curren me level, x nc1 ALE, can be consruced usng a second-order Adams Bashforh mehod (AB2): 3 x nc1 ALE D xn ALE C 2 u m.x n ALE, n / 1 2 u m.xm n 1, n 1 / (AB2) (48) Equaons (46) and (47) defne he mappng and Equaon (48) defnes he mappng ˆ. The mappngs and ˆ are represened n Fgure 2. The feld varables a prevous me levels are mapped along wh he ALE meshes (Fgure 3): c n D c.ˆ 1.x/, n /, (49) u n D u.ˆ 1.x/, n /, (5) u n m D um.ˆ 1.x/, n /, (51) c n 1 D c. 1 ı ˆ 1.x/, n 1 /, (52) u n 1 D u. 1 ı ˆ 1.x/, n 1 /, (53) u n 1 m D u m. 1 ı ˆ 1.x/, n 1 /. (54) Noe ha he unknowns a he curren me level, such as c nc1 and nc1, are compued on he fxed Euleran mesh Tme negraon A he nal me D we solve he flow equaons whou he rgh-hand sde erm rc o oban an nal flow soluon. Also, he nal concenraon feld c D c. D / should be specfed. Then, we apply he followng procedure a every me sep: Sep 1. Consruc a emporary ALE mesh usng Equaons (46) (48) for he nerpolaon of feld varables a prevous me levels. A he frs me sep, we use a frs-order scheme gven by x nc1 ALE D xn m C u m.x n m, n /. (55)

9 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS a = n 1 a = n a = n +1 (a) n 1 (d) n +1 x n m (b) x m xm = x n m = x n m 1 (c) xn ALE (e) x n +1 ALE Fgure 2. Consrucon of ALE meshes xale n and xnc1 ALE usng a second-order scheme, whch defnes he mappngs and ˆ. (a)xm n 1,(b)xm n,(c)xn ALE,(d)xnC1 m D xm n D xn 1 m,and(e)x nc1 ALE. 1 x x n 1 1 x n (a) = n 1 (b) = n (c) = n +1 Fgure 3. Feld varables a prevous me levels are mapped along wh he ALE meshes. The advecon of he ALE meshes defnes he mappngs and ˆ. (a) D n 1,(b) D n,and(c) D nc1. Sep 2. Updae he parcle confguraon by negrang he knemac equaons n Equaons (16) and (17) usng he explc second-order Adams Bashforh mehod (AB2) X nc1 3 D X n C 2 U n 1 2 U n 1, (56) nc1 D n C 3 2!n 1 2!n 1. (57) For crcular parcles, he updae of angular roaons s no necessary. A he frs me sep, we use an explc Euler mehod X nc1 D X n C U n. (58)

10 Y. J. CHOI AND P. D. ANDERSON Sep 3. Modfy he compuaonal mesh o avod very small negraon areas [26]. Frs, compue a mesh velocy feld Ou m r 2 Oum D n, (59) Ou m D on, (6) Ou m D n (61) where n s he ouwardly-dreced un normal vecor on he parcle surface. Then, each mesh pon x m moves accordng o he followng advecon equaons ² dx m Oum f x D m 2 np./, (62) d m f x m 2 P./, x m. m D / D x m, (63) where x m, s he nal poson of mesh pon. In our smulaons, we use a hrdorder Adams Bashforh mehod (AB3), and nodal pons are moved unl each area of negraon s larger han.5% of he elemen area. Sep 4. Compue c nc1 and nc1 by solvng Equaons (33) and (34) eravely. Sep 5. Compue u nc1, g nc1, nc1, U nc1, and! nc1 from he momenum balance and connuy equaon.rv/ T, 2D.u nc1 / rv, g nc1 C v.v C.x X nc1 //, nc1 / D.v, / N C V F ex, C T ex, C v, nc1 rc nc1, (64) q, ru nc1 D, (65), u nc1.u nc1 C! nc1.x X nc1 // D. nc1 / Equaons (33) and (34) are solved by usng a drec solver HSL MA41, and Equaons (64) (66) are solved by usng a drec symmerc solver HSL MA57 [32]. 4. PARTICLE AT A FLUID FLUID INTERFACE 4.1. Problem descrpon As a model problem, a parcle s placed a a flud flud nerface, confned beween wo parallel plaes. Inally, we assume he seady sae condon, ha s, he fluds are saonary and he parcle s a res n he mddle of he flud flud nerface (see Fgure 4). The parcle radus s denoed by a, he hckness of he flud flud nerface by, and he vscosy of lower and upper fluds are 1 and 2, respecvely. The effecve vscosy s assumed o have a lnear relaonshp gven by Equaon (7). A D, we dsurb he flow by applyng an exernal force F D., F y / on he parcle n he y drecon for a ceran me duraon F.For> F, he exernal force F on he parcle s removed, and he parcle freely moves o s equlbrum poson as a resul of he acng surface enson forces. Noe ha dependen on he value of F he conac poson of he nerface wh he parcle may change, bu as long as he parcle remans a he nerface for < F wll reurn o s equlbrum poson. In hs problem, he scalng of equaons by dmensonless groups gven n Equaon (13) s no rval because a characersc velocy U s unknown pror o solvng he flud and parcle veloces. Insead of sang he dmensonless groups gven n Equaon (13), we provde he acual values used n our smulaons, and n prncple one could deermne he characersc velocy U o esmae he magnude of he characersc dmensonless groups. We fx he channel hegh H D 1 and channel lengh L D 1, assumng ha he op and boom walls are saonary and he flow s perodc n he x drecon. Oherwse saed, he parcle radus equals a D.15, he vscoses 1 D 2 D1,

11 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS c = 1 y H a x 2 1 F c =+1 L Fgure 4. Geomery for a parcle a a flud flud nerface a D. he magnude of he exernal force F y D 1,and D 1, ˇ D 1, " D.1, M D.1, D 1 for maeral parameers used n governng Equaons (2), (4), and (8). Noe ha he nerfacal hckness D p "= D.1 for he gven values. Fgure 5 shows he concenraon profle a D.1 for forcng me duraon F D.1 wh F y D 1, where he parcle poson s lowes for he gven forcng condon. Afer valdaon of he compuaonal scheme, we wll nvesgae he parcle moon o s equlbrum poson for varous parameers - F, a, 2 = 1, and so on Convergence es Before we sudy he effec of he dfferen maeral and process parameers on he dynamcs of he parcle, we frs demonsrae he convergence of he mehod by mesh and me sep refnemens. Four meshes are defned wh decreasng elemen sze M1 (77 elemens), M2 (11 elemens), M3 ( elemens), and M4 (1515 elemens), and he mesh parameers are summarzed n Table I. Fgure 6 shows he poson of he cener of he parcle as a funcon of Fgure 5. Concenraon conours a D.1 wh forcng me F D.1 and F y D 1.

12 Y. J. CHOI AND P. D. ANDERSON Table I. Meshes used for he smulaons. Mesh Number of elemens Number of nodes M ,881 M2 1, 4,41 M3 15,625 63,1 M4 22,5 9, y M1 M2 M3 M Fgure 6. Mesh convergence showng he poson of he parcle as a funcon of me 6 5 wh forcng me F D.1. me for he case of forcng me duraon F D.1 wh he exernal force F y D 1. The me sep D.1 s used for all meshes. The resul of M1 shows nonmonoonc behavor of parcle movemen because of he unresolved mesh resoluon; he resuls of M3 and M4 are fully overlapped and canno be dsngushed. The me sep convergence s checked by usng mesh M3 as demonsraed a fully resolved mesh resoluon. Fgure 7 shows he hsores of he parcle poson obaned by usng varous me seps. For D.1, he parcle moon s predced slghly slower han he oher cases. For all oher cases, 6.2, he parcle posons fully overlap, bu f we use a me sep >.2, he smulaon becomes unsable. Hence, for all our smulaons n he paper, we use he mesh M3 n combnaon wh he me sep D.1. y Δ =.1 Δ =.2 Δ =.1 Δ = Fgure 7. Tme convergence showng he poson of he parcle as a funcon of me 6 5.

13 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS Fnally, o oban more nformaon abou he nduced flow by he reracng parcle, we sudy he vorcy of he flow as a funcon of me. Noe ha he vorcy s defned (67) where u and v are he x-dreconal and y-dreconal velocy of he fluds, respecvely. Fgure 8 shows he vorcy n he doman a mes D 1. and D 5. wh forcng me F D.1. The fgure shows ha he magnude of he vorcy s larger below he parcle compared wh above he parcle and a flow s nduced o push he parcle owards s equlbrum poson. Fgure 9 shows he correspondng pressure plos a he same me levels; hese fgures also show a hgher pressure below he parcle ha pushes he parcle n he drecon back o equlbrum. Fgure 1 shows he maxmum of he absolue value of he vorcy n he flud doman j!j max as a funcon of me. Durng 6 6 F, he maxmum vorcy ncreases because of exernal force F, hen decreases o zero as me goes on. The maxmum vorcy obaned by usng M1 and M2 shows flucuaons n me especally when we pull down he parcle durng 6 6 F.ByfurherrefnngoM3and M4, we can oban mesh convergence for maxmum vorcy n me. (a) (b) Fgure 8. Vorcy conours a me (a) D 1. and (b) D 5. wh forcng me F D.1. (a) (b) Fgure 9. Pressure conours a me (a) D 1. and (b) D 5. wh forcng me F D.1.

14 Y. J. CHOI AND P. D. ANDERSON M1 M2 M3 M4 1 max Fgure 1. Maxmum of absolue vorcy n he flud doman as a funcon of me 6 1 wh forcng me F D Tme duraon of appled exernal force We nvesgae he effec of he me duraon of appled exernal force on he parcle. For me 6 6 F, he parcle moves downward because of he acon of exernal force F (see Fgure 11), hen he parcle moves freely under he nfluence of he surface enson. Evenually, he parcle y F =.1 F =.7 F =.5 F =.3 F = Fgure 11. The poson of he parcle as a funcon of me 6 1 for dfferen forcng mes F. -.5 y F =.1 F =.7 F =.5 F =.3 F = Fgure 12. The poson of he parcle as a funcon of me 6 5 for dfferen forcng mes F.

15 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS reaches s equlbrum poson n he mddle of he channel as shown n Fgure 12, as long as he parcle says n beween he flud flud nerface durng he exernal dsurbance Inerfacal hckness In he dffuse-nerface model, he nerfacal hckness s defned by D p "=, as explaned n Secon 2. The nondmensonal measure for he nerface hckness s he Cahn number, defned by C D =H. Because we scaled every lengh wh channel hegh H,has,H D 1, he nerfacal hckness s already nondmensonal ( D C ). We change he nerfacal hckness by changng " values whle keepng he " D M 2 relaonshp. Noe ha C D D p " D M because we fx D 1. Fgure 13 shows he nerface of Cahn number.1 and.4, whch means he nerfacal hckness s 1% and4% of he channel hegh, respecvely. Noe ha he parcle radus s a D.15. Fgure 14 shows he hsores of he parcle poson for varous Cahn numbers where F D.5 wh F y D 1. As he Cahn number ncreases, ha s, he nerfacal hckness ncreases, he parcle moves rapdly owards he equlbrum poson. Here we vared " values o change he nerfacal hckness. However, hs changes no only he Cahn number bu also he Capllary number and Pécle number (Equaon (13)). Hence, Fgure 14 s manfesed by he combnaon of hese nondmensonal parameers. In he Cahn Hllard model, s no rval o change he Cahn number only whou affecng oher parameers. H (a) (b) Fgure 13. Inerfacal hckness for he Cahn number (a) C D.1 and (b) C D y C =.1 C =.2 C =.3 C = Fgure 14. Effec of he Cahn number on he poson of he parcle as a funcon of me 6 2.

16 Y. J. CHOI AND P. D. ANDERSON 4.5. Surface enson The nondmensonal measure of he surface enson s descrbed by he capllary number Ca D U="cB 2, whch conans a characersc velocy U. Because he flud and parcle veloces are unknowns and par of he soluon, s no rval o defne a characersc velocy U, pror o solvng he gven problem. In our smulaons, we change he value of o change he surface enson, whle fxng he oher values D.1, D 1., " D 1 4,andM D 1 2. By ncreasng, he surface enson ncreases (he capllary number Ca decreases). Fgure 15 shows he hsores of parcle poson for varous values where F D.5 wh F y D 1. As ncreases, ha s, he surface enson ncreases, he parcle moves rapdly owards he equlbrum poson. The resuls are que nuve; because he drvng force pullng he parcle back o s orgnal equlbrum poson s he surface enson, he parcle wll reurn faser under hgher surface enson Parcle sze Fgure 16 shows he hsores of he parcle poson for varous parcle rad where F D.5 wh F y D 1. As he parcle radus a decreases, he parcle moves furher downward when exernal force F s appled because smaller parcle experences less drag han larger one. Afer he exernal force s released, smaller parcle moves faser han larger one, whch can be seen n Fgure 17 where he y-dreconal ranslaonal velocy of he parcle V s shown. -.5 = = 2 = 3 y= Fgure 15. Effec of he surface enson on he poson of he parcle as a funcon of me 6 2. As ncreases, he surface enson ncreases. -.5 y a =.1 a =.125 a =.15 a =.175 a = Fgure 16. Effec of he parcle sze on he poson of he parcle as a funcon of me 6 2.

17 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS a =.1 a =.125 a =.15 a =.175 a =.2 V Fgure 17. Effec of he parcle sze on he velocy of he parcle as a funcon of me Vscosy rao We now consder he effec of dfferen vscoses of upper and lower fluds. We defne he vscosy rao D 2 = 1. Fgure 18 shows he hsores of he parcle poson for dfferen vscosy raos where F D.5 wh F y D 1. As he vscosy rao decreases, ha s, he upper flud s hnner han he lower flud, he parcle moves furher downward when exernal force F s appled. Afer he exernal force s released, he parcle moves faser for lower vscosy raos, evenually he parcle reaches he equlbrum poson earler. The fas movemen of he parcle for lower vscosy raos s clearly seen n Fgure 19, where he y-dreconal ranslaonal velocy of he parcle V s ploed afer he exernal force F s released. 5. MULTILAYER CONFIGURATION In hs secon, we show he dynamcs of a parcle passng hrough mulple layers of fluds, confned beween wo parallel plaes. A schemac descrpon of he problem s shown n Fgure 2. The lengh of he channel s L D 1, he posons of he uppermos and lowes fluds are H 1 D H 5 D.6 and he posons of he fluds n-beween are H 2 D H 3 D H 4 D.3. The vscosy of he flud layers s chosen D 1, bu he model can handle dfferen vscosy raos as shown n he prevous secon. A parcle of radus a D.15 s suspended a he nal poson X D., 1.8/ and s sedmenng downward as a resul of a consan exernal force F D., 1/ acng on he parcle. The upper and lower walls are saonary and he flow s assumed o be -.5 y =.7 =.9 =1.1 = Fgure 18. Effec of he vscosy rao on he poson of he parcle as a funcon of me 6 2.

18 Y. J. CHOI AND P. D. ANDERSON =.7 =.9 =1.1 =1.3.2 V Fgure 19. Effec of he vscosy rao on he velocy of he parcle as a funcon of me 6 1. H 1 c = +1 F H 2 c = 1 H 3 c = +1 H 4 c = 1 c = +1 H 5 y x Fgure 2. Geomery for a parcle n mulple layers of fluds. L perodc n he x drecon. The maeral parameers used n governng Equaons (2), (4), and (8) are D 1, ˇ D 1, " D.1, M D.1, and D 1. Noe ha he nerfacal hckness D p "= D.1.

19 CAHN HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS The problem s solved usng a mesh wh 1 21 elemens because provdes an accurae soluon wh manageable compuaonal coss. The me sep D.1 s used for he smulaon. Fgure 21 shows he evoluon of he srucure of he mullayered morphology caused by he sedmenng parcle. As he parcle passes hrough he mulple layers of fluds, a flud layer breaks (a) = 1.5 (b) = 3. (c) = 4. (d) = 5. (e) = 5.5 (f) = 6. Fgure 21. Snapshos of a parcle passng hrough mulple layers of fluds V Fgure 22. Translaonal velocy of he parcle passng hrough mulple layers of fluds as a funcon of me.

20 Y. J. CHOI AND P. D. ANDERSON up no several drops, hen he drops merge wh oher layers of he flud. Fgure 22 shows he ranslaonal velocy of he parcle n he y drecon as a funcon of me. The proposed mehod can provde fully resolved velocy felds for he parcle and fluds, assocaed wh he evoluon of he nherenly complex morphology. 6. CONCLUSIONS We presened a combned model of he Cahn Hllard heory and XFEM for he dynamcs of parcles suspended n wo-phase flows. In he dffuse-nerface model of Cahn Hllard, he nerface s consdered o have a small bu fne hckness. The nerface profle and hckness are deermned by governng equaons ha couple hermodynamc and hydrodynamc forces. For he drec numercal smulaon of flows wh suspended parcles, we use XFEM, whch decouples he flud and parcle domans whle usng a compuaonal mesh ncludng boh fluds and parcles. To cope wh he movemen of parcles, a ALE scheme s used o defne a mappng of feld varables a prevous me levels ono he compuaonal mesh a he curren me level. The no-slp boundary condon a he parcle surface s mposed by usng a consran mplemened wh Lagrangan mulplers. By combnng he dffuse nerface model and XFEM, he parcle moon a a flud flud nerface can be smulaed on a fxed Euleran mesh whou any need of re-meshng. We presen he moon of a sngle parcle a an nerface beween wo fluds. Inally, he fluds and parcle are saonary. The nal equlbrum sae s dsurbed by applyng a consan force on he parcle for a ceran me duraon. Then he exernal force on he parcle s released, and he parcle moves freely o s equlbrum poson under he acon of surface enson. As long as he parcle says n beween wo fluds durng exernal dsurbance, always comes back o s nal equlbrum poson. We nvesgaed he effec of nerfacal hckness, surface enson, parcle sze, and vscosy rao of he wo fluds on he parcle s movemen owards s equlbrum poson. As nerfacal hckness ncreases, surface enson ncreases, parcle sze decreases or vscosy rao decreases; he parcle moves rapdly owards s equlbrum poson afer he exernal force s released. We also demonsraed he wde applcably of he mehod and deermned he moon of a sedmenng parcle passng hrough mulple layers of fluds and he correspondng morphology change of he fluds. The proposed mehod s general and s applcable o more complex problems, such as mulple parcles n phase separang fluds and srucure formaon of parcles a a flud flud nerface. Also, he mehod can be easly exended o hree-dmensonal smulaons whou any loss of generaly, only requrng heaver compuaonal load. Fuure work wll be focused on hese problems. REFERENCES 1. Pckerng SU. Emulsons. Journal of he Chemcal Socey, Transacons 197; 91: van der Waals JD. The hermodynamc heory of capllary under he hypohess of a connuous densy varaon. Journal of Sascal Physcs 1979; 2(2): Cahn JW, Hllard JE. Free energy of a nonunform sysem. I. Inerfacal energy. Journal of Chemcal Physcs 1958; 28(2): Anderson DM, McFadden GB, Wheeler AA. Dffuse-nerface mehods n flud mechancs. Annual Revew of Flud Mechancs 1998; 3: Prusy M, Keesra J, Goossens JGP, Anderson PD. Expermenal and compuaonal sudy on srucure developmen of PMMA/SAN blends. Chemcal Engneerng Scence 27; 62(6): Wse SM, Lowengrub JS, Crsn V. An adapve mulgrd algorhm for smulang sold umor growh usng mxure models. Mahemacal and Compuer Modellng 211; 53(1 2): Khaavkar VV, Anderson PD, Mejer HEH. Capllary spreadng of a drople n he parally weng regme usng a dffuse-nerface model. Journal of Flud Mechancs 27; 572: Khaavkar VV, Anderson PD, Duneveld PC, Mejer HEH. Dffuse-nerface modellng of drople mpac. Journal of Flud Mechancs 27; 581: Khaavkar VV, Anderson PD, Duneveld PC, Mejer HEH. Dffuse nerface modelng of drople mpac on a pre-paerned sold surface. Macromolecular Rapd Communcaons 25; 26(4): Tufano C, Peers GWM, Mejer HEH, Anderson PD. Effecs of paral mscbly on drop wall and drop drop neracons. Journal of Rheology 21; 54(1):

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