3D Lattice Boltzmann Simulation of Droplet Formation in a Cross-Junction Microchannel

Transcription

1 3D Lattce Boltzmann Smulaton of Droplet Formaton n a Cross-Juncton Mcrochannel SURESH ALAPATI, SANGMO KANG AND YONG KWEON SUH * Department of Mechancal Engneerng Dong-A Unversty 840 Hadan-dong, Saha-gu, Busan SOUTH KOREA Abstract: - Ths study descrbes the numercal smulaton of three-dmensonal droplet formaton and the followng moton n a cross-juncton mcrochannel by usng the Lattce Boltzmann Method (LBM). Our am s to develop the three-dmensonal bnary fluds model, consstng of two sets of dstrbuton functons to represent the total flud densty and the densty dfference, whch ntroduces the repulsve nteracton consstent wth a free-energy functon between two fluds. We valdated the LBM code wth the velocty profle n a 3-dmentonal rectangular channel. Then, we appled our code to the numercal smulaton of a bnary flud flow n a cross-juncton channel focusng on the nvestgaton of the droplet formulaton. Due to the pressure and nterfacal-tenson effect, one component of the fluds whch s njected from one nlet s cut off nto many droplets perodcally by the other component whch s njected from the other nlets. We consdered the effect of the boundary condtons for densty dfference (order parameter) on the wettng of the droplet to the sde s. Key-Words: - Lattce Boltzmann Method, Free-Energy Model, Cross-Juncton Mcrochannel, Droplet Formaton 1 Introducton Most of the numercal works for flud flow smulatons n mcrochannel are based on solvng the contnuum Naver Stokes (N S) uatons. Recently, consderable attenton has been gven to the lattce Boltzmann method (LBM) among flud dynamc researchers [1, ]. It solves the lattce Boltzmann uaton (LBE) knetcally on a regular lattce where a number of fcttous partcles evolve accordng to the laws representng the physcal prncples of mass, momentum and energy conservaton. Snce, n LBM, the number of partcles dstrbuted n the computatonal feld s not drectly related to the actual number of molecules, LBM s computatonally more effcent than other partcle dynamc technques such as Molecular Dynamcs (MD) [3] and Drect Smulaton Monte Carlo (DSMC) [4]. Besdes, LBM solver s based on a smple Bhatnagar-Gross-Krook (BGK) collson approxmaton [5], and so we are free from solvng complcated uatons needed for the full BE. Furthermore, smulatng complex flows such as multphase/multcomponent flows has always been a challenge to conventonal CFD because of the movng and deformable nterfaces. More fundamentally, the nterfaces between dfferent phases (lqud and vapor) or components (e.g, ol and water) orgnate from the specfc nteractons among flud molecules. Therefore t s dffcult to mplement such mcroscopc nteractons nto the macroscopc Naver-Stokes uaton. However, n LBM, the partculate knetcs provdes a relatvely easy and consstent way to ncorporate the underlyng mcroscopc nteractons by modfyng the collson operator. In LBM phase separatons are generated automatcally from the partcle dynamcs and no specal treatment s needed to manpulate the nterfaces as n tradtonal CFD methods. Successful applcatons of multphase/multcomponent LBM models can be found n varous complex flud systems, ncludng nterface nstablty, bubble/droplet dynamcs, wettng on sold surfaces, nterfacal slp, and droplet electrohydrodynamc deformatons, etc. In ths present work, LBM s used as the 3D numercal tool to study the droplet formaton and ts moton n a cross-juncton mcrochannel. Ultmately, we must establsh a code beng robust, accurate and stable n smulaton of the droplet formaton wth the lattce Boltzmann method that reproduces the expermental results for mcrofludc devces. In ths paper, we frst dscuss the theoretcal background of ISBN: page 150 ISSN:

2 the lattce Boltzmann method, ncludng the mplementaton of the multphase model. Then, benchmark of the LBM code s done by comparng the results wth those obtaned wth the analytcal treatment for sngle-phase flow through a channel wth the same geometry. In the next secton, smulatons of droplet formaton n a cross-juncton are presented, and the results are shown for dfferent cases of wettng boundary condtons. Lattce Boltzmann Smulaton In the LBE approach, one solves the knetc uaton for the partcle velocty dstrbuton functon (PVDF), f ( xt, ) n whch x s the spatal poston vector and t s the tme. The macroscopc quanttes (such as mass densty ρ and momentum densty ρu) can then be obtaned by evaluatng the hydrodynamc moments of the dstrbuton functon f [6]. In ths paper we presented 3D lattce Boltzmann BGK Model based on D3Q19 (3 dmensonal and 19 veloctes) lattce. A square lattce s used n whch each ste s connected wth nearest and next-to-nearest neghbors. The horzontal and vertcal lnks have the length Δx, the dagonal lnks Δ x, where Δx s the spatal step. Δt s defned as the smulaton tme step, and c represents the lattce velocty vectors havng the magntude c =Δx/ Δt c for = 1-6 and c = c for = through the mesh because of chemcal potental gradents. The nterface does not have to be tracked separately. However, the nterface s dffuse (.e., the transton of one phase to another s not abrupt but somewhat gradual). Because a physcal nterface s qute sharp, the dffuse nterface s a numercal artfact. For modelng droplet formaton and detachment, as presented n ths artcle, we have developed a multphase 3D code based on ths Gnzburg-Landau free-energy approach for bnary flow n a mcrochannel. 3 Numercal Smulaton Method Our smulatons are based on the lattce Boltzmann scheme developed by Swft et al. [10]. Ths scheme s based on a free-energy functonal. Actually the free-energy model does consder the thermodynamcs of the problem. The approach to ulbrum s governed by the free energy whch enters the model through the ulbrum dstrbuton functon. The model therefore may be used for dong the smulaton of multple mmscble fluds. 3.1 Lattce Boltzmann Scheme In ths method, the dynamcs s defned by the velocty dstrbuton functons f ( xt, ) used to model the total densty, ρ = ρ a + ρb and g ( xt, ) used to model the order parameter or densty dfference φ = ρa ρb, defned at each lattce ste x at each tme t. The evoluton of both dstrbuton functons are governed by the sngle relaxaton tme Boltzmann uatons f ( r+ cδ t, t+δt) f ( r, t) = 1 [ (, ) f r t f ( r, t )], (1) τ g ( r+ cδ t, t+δt) g ( r, t) = Fg. 1 D3Q19 (3D wth 19 velocty vectors) lattce Several models are known n the lterature to descrbe multphase systems wth the lattce Boltzmann approach. The frst multphase LBE model was ntroduced by Gunstensen et al. [7] whch s known as color model. The model based on nter partcle potentals s ntroduced by Shan and Chen [8]. The model based on a Gnzburg-Landau free-energy approach for phase transton was developed by Swft and co-workers [9, 10]. An advantage of the Gnzburg-Landau free-energy approach s that the dffuse nterface evolves 1 [ (, ) g r t g ( r, t )], () τ ϕ where f ( rt, ) and g ( rt, ) are local ulbrum dstrbutons, τ and τ ϕ are ndependent relaxaton parameters, and represents the lnk number of lattce velocty vector. The dstrbuton functons are related to the total densty ρ, to the mean flud velocty u, and to the order parameterφ through ρ = f, ρu = f c, φ = g. (3) These quanttes are locally conserved n any ISBN: page 151 ISSN:

3 collson process. In order to obtan the contnuum uatons to a bnary flud mxture, we need to derve the hgher order moments of local ulbrum dstrbuton functons as follows. f c c = c P +ρu u, 4 α β αβ α β ( ) ( ) f c c c = ρc ( u δ + u δ + u δ ), 5 α β γ s α βγ β αγ γ αβ α g c = φu α, (6) α β = ΓΔ μδαβ + ϕ α β g c c c u u. (7) where Γ s a coeffcent related to the moblty of the flud, c s s the speed of sound, δ αβ s the Kronecker delta, P αβ s the complete pressure tensor, and Δ μ s the chemcal potental dfference between the two fluds, whch s responsble for phase separaton. The local ulbrum dstrbuton functons can be expressed as an expanson at the second order n the velocty u f0 = A0 + C0u, f = Ak + Bkuαcα + Cku + Dkuαuβcαcβ + Gkαβ cα cβ, (8) g0 = a0 + c0u, g = a + b u c + c u + d u u c c. (9) k k α α k k α β α β The constants A, B, C, D, G, a, b, c, and d take dfferent values for the nearest ( = 1-6; k=1) and next-nearest ( = 7-18; k=) vectors. 3. Free-energy model The free-energy functonal F generally used for studes on a bnary system s gven by [11] 1 b 4 κ F = dr[ ρln( ρ) + φ + φ + ( φ )], (10) 3 a 4 where φ s the order parameter that descrbes the normalzed dfference n densty of the two fluds. The thermodynamc propertes of the fluds follow drectly from the free energy. The functonal dervatve of (10) gves the chemcal potental dfference between the two fluds δ F 3 Δ μ = = a φ + b φ κ φ. (11) δφ scalar part p 0, of the pressure tensor s gven by F F p0 = δ δ φ ρ h( ρ, φ) δφ + δρ 1 a 3b 4 κ = ρ + φ + φ κφ( φ) ( φ). (1) 3 4 In ths uaton, h( ρφ, ) ) s the free-energy densty, whch s the ntegrand of (10). The complete pressure tensor, P αβ n (4) can be derved from the Gbbs-Duhem relaton [1] P αβ = φ μ, (13) and now the complete pressure tensor becomes P = p 0 δ + κ φ φ. (14) αβ αβ α β The parameter values of b and κ are always postve. When a s negatve, phase separaton occurs; when a s postve, there s no phase separaton. In ths artcle, a negatve value of a s used because we want to model two mmscble phases. The value of κ corresponds to the nterfacal propertes of the nterface between the two fluds. Further we set b= a so that the ulbrum values for the order parameter become the predetermned ones ( φ =± 1). We use fnte dfference scheme for the calculaton of φ and φ terms n the (11) and (1). The dmensonless surface tenson at the nterface s gven by a σ = aκ. (15) 3b The nterfacal wdth s gven by κ ξ =. (16) a The dmensonless knematc vscosty s gven by τ 1 υ =. (17) 6 4 Valdaton of code Our numercal code s verfed by applyng t to the sngle phase flow n a rectangular channel. Fully developed flow n a long channel wth a flat rectangular cross secton, of wdth W and heght H (s at x = ± W and z =± H ) has a specfc flow profle dependng on the values of W and H. The general soluton for flow n a rectangular cross secton channel s gven below [13] Also, the pressure can be derved from (10). The ISBN: page 15 ISSN:

4 ΔPW x 3 n vy ( x, z) = 1 ( 1) ( ) 3 3 8μ W + n= 1 n 1 π cosh[(n 1) π z/ W] cos[( n 1) π x / W ]. (19) cosh[(n 1) π H / W] where μ s the dynamc vscosty, Δ P s the pressure drop over the channel. We compare the (ulbrum) velocty profle obtaned by LBM wth that gven by the analytcal soluton for dfferent aspect ratos. Fgure shows the result for the aspect rato 1. Fg. Comparson of the smulaton velocty profle wth the analytcal soluton Fg. 3 3D cross-juncton mcrochannel used n LBM smulaton of droplet formaton. In dong the smulaton, there are some free parameters to be set n the program. A sutable choce of the parameters are ρa = ρb = 1.0, a= b= 0.04, κ = 0.04 and the dmensonless relaxaton tme for total densty and densty dstrbuton functons s set to be ual,.e. τ = τ φ = 0.8. The maxmum values of nlet parabolc profles at man channel and sde channels are V n =0.003 and W n =0.018 respectvely. The smulatons are performed on a lattce of sze, Nx Ny Nz, where N x = 16, N y = 71 and N z = 159. It s seen that the agreement s almost exact ndcatng that our numercal code s relable. 5 Numercal Results for 3D Droplet Formaton n a Mcrochannel In order to study the droplet dynamcs of bnary flud model, a standard cross-juncton channel s consdered, whch s shown n fgure 3. The wdth of man channel s set at 00μm and that of two sde channels at100μm. The depth of each channel s taken as100μm. Flud A s njected through the horzontal channel wth maxmum velocty V n. Flud B s njected through sde channels wth maxmum velocty W n. The rato of two velocty magntudes, W n /V n s fxed as 6. Snce the flow rate of flud A s 3 tmes smaller than that of B, The droplet s expected to be composed of the flud A only. Fg. 4 Intal condton of order parameter Fgure 4 shows the ntal condton of the smulaton regardng the order parameter. The flud A and B are denoted by red and blue colors, respectvely. For the s or other sold obstacles that have specfc wettng propertes, we have to assgn a certan value of the order parameter (φ ), to the sold lattce ste next to the. However we do not have specfc gudes as to the rght choce of the boundary condton dependng on the wettng propertes. Therefore we smulate the droplet formaton by consderng dfferent cases of boundary condtons for order parameter (φ ) at s of mcrochannel. We consder three cases for order parameter at s. ISBN: page 153 ISSN:

5 5.1 Case1: Zero gradent of order parameter at s In ths case, the gradent of order parameter normal to s set to be zero,.e. φ / n = 0. The results of smulaton for ths case at dfferent tme steps are shown n fgure 5. We plotted the results for man channel only. We observe from the results that the droplet of flud A s formed at the juncton and movng through the channel. It can also be seen that the droplet s wettng to the sde s of mcrochannel after t s detached from the upstream bulk of lqud A. T=500 T=3500 T= Case : Zero value of order parameter at s In ths case the value of order parameter at s set to be zero ( φ = 0.0 ).If the value of the order parameter assgned to the sold lattce stes s ual to that of flud A( φ a = 1), then the flud A close to these lattce stes wll spread on the surface, and the flud B phase wll not spread on ths surface. However, f the value of the order parameter assgned to the sold lattce stes s ual to that of φ = ), then flud B wll spread on the surface and flud A wll not. For neutral wettng, the order parameter of the sold lattce stes should be exactly between the order parameters of the flud A flud B ( 1 b phase and the flud B phase ( φ = 0 ). The results of smulaton for ths case at dfferent tme steps are shown n fgure 6. T=4500 T=5000 T=6000 T=500 T=3500 T=4000 T=7500 T=8500 T=5000 T=6500 T=8000 T=10500 T=10000 T=11500 T=1000 T=1000 Fg.5 Droplet generaton and moton n the crossjuncton mcrochannel wth zero gradent of order parameter at s. T=13500 Fg.6 Droplet generaton and moton wth zero value of order parameter at s. ISBN: page 154 ISSN:

6 From ths result t can be seen that the droplet s of a roller shape, the generaton of droplet s more fruent than compared to case1 and partal wettng of droplet s takng place on sde s of mcrochannel. We also observed that the sze of droplet s reducng, whle t s movng through the channel. 5.3 Case 3: Order parameter value of -1.0 at s In ths case the value of order parameter at s s set to be -1.0 ( φ = 1.0 ), whch s the same value of flud B. We set ths value because flud B s njectng through sde channels and the flow rate of flud B s 3 tmes faster than the flow rate of flud A. So the spreadng of flud B s expected to be more domnant at the s than flud A. The results of smulaton for ths case at dfferent tme steps are shown n fgure 7. From ths results t can seen that the shape of droplet s sphercal roller and no wettng of droplet s takng place. T=500 T=3500 T=4500 T=1000 Fg.7 Droplet generaton and moton wth value of order parameter s -1.0 at s. As a summary, we show n Table 1 the shape of droplet and wettng phenomena for dfferent cases of order parameter boundary condtons. Table 1. Shape of droplet and wettng phenomena for dfferent boundary condtons for the order parameter. φ value φ =0 (Exactly between flud A and flud B) φ = 1.0 (Equal to the flud B) Droplet Pattern Roller shape Sphercal roller shape Wettng Phenomena At s Partal wettng s takng place Not wettng T=5500 T=6000 φ n = 0.0 (Gradent normal to the s zero) - Completely wettng takng place on the sde s T=7500 T= Concluson FORTRAN code s developed for lattce-boltzmann method to smulate the droplet formaton n a mcrochannel. From the numercal results we can conclude that the free-energy model of the LBE method can be successfully used for the smulaton of the droplet moton. The droplets are smoothly generated n the juncton area of mcrochannel. The boundary condtons of order parameter at s of mcrochannel s mportant, because the wettng phenomena of droplet at s s depend on the surface propertes of materal whch s used for mcrochannel. Studyng on the effect of parameters ISBN: page 155 ISSN:

7 such as abκ,, and Γ on the smulaton results as well as the stablty s a bg task. In the future, we are plannng to parallelze our LBM code and conduct the stablty test for dfferent value of parameters. Acknowledgment Ths work was supported by the Korea Scence and Engneerng Foundaton (KOSEF) through Natonal Research Laboratory Program funded by Mnstry of Scence and Technology (No ). mcrochannel usng the lattce Boltzmann method, J. Mech. Sc. Tech., Vol. 1, No. 1, 007, pp [1] S. van der Graaf, T. Nssako, C. G. P. H. Schroen, R. G. M. van der Sman and R. M. Boom, Lattce Boltzmann Smulatons of Droplet Formaton n a T-Shaped Mcrochannel, Langmur, Vol., 006, pp [13] P. Gondret, N. Rakotomalala, M. Rabaud, D. Saln and P. Watzky, Vscous parallel flows n fnte aspect rato Hele-Shaw cell: Analytcal and numercal results, Phys. Fluds, Vol. 9, No. 6, 1997, pp References: [1] R. Benz, S. Succ and M. Vergassola, The lattce Boltzmann uaton: theory and applcatons. Phys. Reports Rev. Sec. Phys. Lett., Vol., 199, pp [] S. Chen and G.D. Doolen, Lattce Boltzmann method for flud flows. Annu Rev. Flud Mech., Vol.30, 1998, pp [3] J. Koplk and J.R. Banavar, Contnuum deductons from molecular hydrodynamcs. Annu Rev. Flud Mech., Vol.7, 1995, pp [4] G. Brd, Molecular gas dynamcs and the drect smulaton of gas flows. Oxford Sc. Pub., Oxford, [5] P.L. Bhatnagar, E.P. Gross and M. Krook, A model for collson processes n gases. I. Small ampltude processes n charged and neutral onecomponent systems. Phys. Rev., Vol.94, 1954, pp [6] D. Yu, R. Me, L.S. Luo and W. Shyy, Vscous flow computatons wth the method of lattce Boltzmann uaton, Prog. Aero. Sc., Vol. 39, 003, pp [7] A. Gunstensen and D.Rothman, Lattce Boltzmann studes of mmscble two phase flows through porous meda, Phys. Rev., Vol. A 43, 1991, pp [8] X. Shan and H. Chen, Lattce Boltzmann model for smulatng flows wth multple phases and components, Phys. Rev., Vol. E 47, 1993, pp [9] M.R. Swft, E. Orlandn, W.R. Osborn and J.M. Yeomans, Lattce Boltzmann smulaton of lqud gas and bnary flud systems, Phys. Rev. Lett., Vol. E54, 1996, pp [10] M.R. Swft, W.R. Osborn and J.M. Yeomans, Lattce Boltzmann smulaton of nondeal fluds, Phys. Rev. Lett., Vol.75, No.5, 1995, pp [11] Z. L, J. Kang, J.H. Park and Y.K. Suh, Numercal smulaton n a cross-juncton ISBN: page 156 ISSN: