Available online at ScienceDirect. Procedia Engineering 126 (2015 )

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1 Available online a ScienceDirec Procedia Engineering 16 (015 ) h Inernaional Conference on Fluid Mechanics, ICFM7 Phase-field-based finie volume mehod for simulaing hermocapillary flows Long Qiao a,b, Zhong Zeng* a,b,c, Haiqiong Xie a,b a Deparmen of Engineering Mechanics, College of Aerospace Engineering, Chongqing Universiy, Chongqing , China b Sae Key Laboraory of Coal Mine Disaser Dynamics and Conrol (Chongqing Universiy), China c Sae Key Laboraory of Crysal Maerial (Shandong Universiy), China Absrac Based on he phase-field inerface-capuring scheme, a novel and common simulaion sraegy is pu forward wih he finie volume mehod o sudy he hermocapillary flows, which avoids he fourh-order derivaive efficaciously and adops he mean parameer mehod o preserve he convergence in he large densiy raio flows. Through simulaing he hermocapillary migraion of bubble, he resuls show ha he new sraegy can successfully describe he performance of bubble wih differen densiy raios, i.e. 0.1 and 0.01, and dimensionless parameers Re= and When Re is , he bubble has an apparen deformaion, and a larger migraion velociy of he bubble is observed wih a larger emperaure gradien along he surface. 015 The Auhors. Published by by Elsevier Ld. Ld. This is an open access aricle under he CC BY-NC-ND license (hp://creaivecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibiliy of Chinese Sociey of Theoreical and Applied Mechanics (CSTAM). Peer-review under responsibiliy of The Chinese Sociey of Theoreical and Applied Mechanics (CSTAM) Keywords: Phase-field mehod; Thermocapillary flows; Finie volume mehod; Bubble migraion 1. Inroducion Thermocapillary force, due o he emperaure gradien in he wo-phase inerface, is he mos decisive influence under some special condiions, for insance, micrograviy environmen and microfluidic devices, and he research on hermocapillary flows goes on apace. In he experimens under micrograviy environmen, hermocapillary migraion of air bubbles has been carried ou by Hadland e al. [1] and Kang e al. [], and heir resuls are similar wih he heoreical analyses [3]. In recen years, hermocapillary force has been applied o manipulae he drople or bubble migraion in microfluidic devices [4,5] because of he raio of large surface area o volume in micro scale, and i is * Corresponding auhor. Tel.: ; fax: address: zzeng@cqu.edu.cn The Auhors. Published by Elsevier Ld. This is an open access aricle under he CC BY-NC-ND license (hp://creaivecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibiliy of The Chinese Sociey of Theoreical and Applied Mechanics (CSTAM) doi: /j.proeng

2 508 Long Qiao e al. / Procedia Engineering 16 ( 015 ) also adoped o manufacure advanced maerial [6]. In he meanime, numerical simulaion, as a preferenial mehod, is also widely used o invesigae he hermocapillary flows, and some capuring mehods have been developed, such as he phase-field [7] and fron-racking [8] e al. In his paper, a new and common phase-field-based finie volume mehod is proposed, which avoids he fourh-order derivaive in he convenional phase-field equaion and adops he mean parameer mehod o improve he convergence in he cases wih large densiy raio. In addiion, some resuls from he simulaion of bubble hermocapillary migraion are presened wih he novel mehod.. Theoreics and numerical sraegies.1. Phase-field heory and numerical evoluion Phase-field heory, a kind of diffuse-inerface model, is inroduced from he free energy, and i is gradually applied o capure he inerface beween wo fluids in he numerical simulaion. Normally, he order parameer ϕ is adoped o describe he composiion disribuion of wo immiscible fluids, and i is presened in he compuaional domain coninuously ranging from -1 o 1. Under he hypohesis of he equilibrium inerface profiles given by Van der Waals, Cahn-Hilliard equaion u ( M ) (1) is developed o model he variaion of order parameer wih ime, where u is he fluid velociy and M is he nonnegaive mobiliy [9]. Generally, he mobiliy M is also inroduced as a funcion of order parameer, i.e., M = M c (1- ϕ ), where M c is a consan parameer. The parameer μ represens he chemical poenial, defined as 3, () where denoes he hickness of inerface layer. From Eq. (1) and Eq. (), he fourh-order derivaive of he order parameer is presened. One kind of solving sraegies is obaining he chemical poenial firsly, hen solving he order parameer hrough Eq. (1), where he diffusion caused by chemical poenial is reaed as a source erm. In his paper, a new solving sraegy is pu forward o decrease he order of derivaive by reforming convenional Cahn-Hilliard equaion as follows: 0 S and 3 S ( ), (3) 0 ( Mˆ ) and S u, (4) S where S ϕ and S μ are he sources of he ranspor equaions of chemical poenial and order parameer respecively. In addiion, he arificial free energy [10] is inroduced o obsruc he order parameer less han he minimum value in his paper. The effecive chemical poenial is defined as ˆ E A, where E A is modifier of free energy and i is saed as Eq.(5) A(1 ), if 1, EA( ) 0, if 1., (5) In hese forms, he order parameer is firsly prediced by Eq. (3), where he former chemical poenial and order parameer are used; subsequenly, he chemical poenial is updaed using he prediced order parameer using he Eq. (4), and he ime evoluion and convecive ranspor of order parameer are inroduced. These wo solving seps are repeaed unil he final olerance crierion is reached.

3 Long Qiao e al. / Procedia Engineering 16 ( 015 ) Mean parameer mehod and governing equaions for hermocapillary flow In general, he incompressible governing equaions are adoped o describe he incompressible fluid flows. Bu in phase-field model, he densiy is defined as a linear funcion wih he order parameer, i.e., ρ=(1-ϕ)ρ 1 /+(1+ϕ)ρ /, which causes an eviden change of densiy in he inerface zone. For wo fluids wih a large densiy raio, he influence caused by densiy change canno be ignored and he compressible governing equaions are uilized. In his paper, a mixed form of governing equaions in proposed using he mean parameer mehod, which is incompressible bu conains he effec induced by he densiy change. The amended governing equaions for hermocapillary flow of wo immiscible fluids are lised as follows u 0, (6) ( u ( uu)) p [ ( uu T )] F F, (7) m S ( T ( ut)) ( kt) S, (8) m T where p and T represen he pressure and emperaure of he mixure fluid, respecively; ρ m is he mean densiy of wo fluids, i.e., ρ m =ρ 1 /+ρ /; ξ m is he mean value of he produc of specific hea and densiy for wo fluids; η and k are dynamic viscosiy and hermal conduciviy of mixure respecively, and hey have he same linear relaionship as densiy, i.e., η=(1-ϕ)η 1 /+(1+ϕ)η / and k=(1-ϕ)k 1 /+(1+ϕ)k /. In addiion, he subscrip 1 and represen fluid 1 and fluid. F ϕ and S T are sources in momenum and energy equaions respecively, which are caused by he mean parameer mehod including he pars deermined by he order parameer, i.e., F ( u) ( uu), (9) S ( T) ( Tu), (10) T where ρ ϕ and ξ ϕ are he changing coefficiens of ρ and ξ for he order parameer, i.e., ρ ϕ =ρ 1 /-ρ / and ξ ϕ =ξ 1 /-ξ /. F S is he inerface force which under he coninuous surface force model (CSF model). In his paper, he inerface force abides by he following form 3 [ ( I )] FS, (11) 4 where σ is he surface ension which has a linear relaion wih he emperaure, i.e., σ=σ ref +σ T (T-T ref ). And, σ ref is he reference surface ension a he reference emperaure T ref and σ T is he changing raio of surface ension for emperaure. 3. Numerical experimens 3.1. Problem saemen For confirming he pracicabiliy of he new sraegies, wo-dimensional hermocapillary migraion of bubble wih he above heories is sudied. The bubble, filled wih fluid 1, is locaed a he cener of a recangular box full of fluid and i is driven by he uniform emperaure gradien field moving from boom o op. The op and boom are no-slip walls, while he periodic condiion is adoped for lef and righ side walls. In his paper, he properies of wo fluids and oher imporan parameers, including he sizes of bubbles and box and he emperaure condiion ec., are inroduced from Ref. [9]. Where, he densiy ρ, dynamic viscosiy η,

4 510 Long Qiao e al. / Procedia Engineering 16 ( 015 ) specific hea c p and hermal conduciviy k of fluid are 1.0, 0., 5.0 and 0. respecively. The properies of fluid 1 are defined as ρ 1 =ρ 0 ρ, η 1 =η 0 η, c p1 =c p0 c p, k 1 =k 0 k, and ρ 0, η 0, c p0 k 0 are he raios of parameer beween fluid 1 and fluid. The radius of he bubble R=30, he size of box is 8R 16R where he disance from boom o op is 16R. In addiion, he op wall and boom wall are kep consan emperaure T h =3 and T c =0 respecively. The reference emperaure T ref =16, and he surface ension under his emperaure σ ref = The surface ension coefficien σ T = The mobiliy consan M c =0.05 and he hickness of he inerface = 1/. In his paper, he size of mesh used for simulaing is 1 1 and he ime-sep size is 1.0. The SIMPLE arihmeic is adoped for solving he couple of pressure field and velociy field. In addiion, wo dimensionless numbers are defined o describe he hea/mass ranspor of differen parameers. The non-dimensional velociy U, Reynolds number Re and Marangoni number are defined as follows: U T T R, Re RU, Ma cpru k, where T =(T h -T c )/16R is he far-field emperaure gradien and νη /ρ is he kinemaic viscosiy. 3.. Resuls and discussions Based on he novel scheme for simulaing he hermocapillary flows, bubble migraion wih differen physical propery raios is sudied. Firsly, he smaller densiy raio problem is discussed, where he raios of propery parameers are ρ 0 =η 0 =0.1, k 0 =0., c p0 =0.4. The dimensionless numbersare Re= and Ma= Fig. 1 depics he relaive vecor disribuion and emperaure conour around he bubble. Where he purple line represens he iniial posiion of bubble while he blue line is he posiion a ime-seps. Moreover, wo couner vorexes in he bubble are observed and he emperaure profile is significanly affeced by he bubble. The resuls are comparable wih he resuls in he Ref. [9] which is a hree-dimensional simulaion. (a) (b) Fig. 1 A ime-seps, he vecor disribuion in a reference frame moving wih he bubble (a) and emperaure conour (b) around he bubble wih Re= , Ma= and densiy raio 0.1. Subsequenly, he validiy of he new heory for larger densiy raio is esed. The parameer raios are aken as ρ 0 =0.01, η 0 =0.1, k 0 =0.0, c p0 =0.4. Addiionally, dynamic viscosiy η is reduced o 0.00, and he Reynolds number and Marangoni number are Re= and Ma=7.5. A ime-seps, he relaive vecor disribuion and emperaure conour near he bubble are ploed in Fig.. The shape of bubble, represened by he blue line, has been changed apparenly by he vorex-ring behind he bubble resuled by he pressure difference beween he op and boom of he bubble. The emperaure disribuion also has a new paern, in which a larger gradien emerges along he inerface and provides a power for he bubble migraion.

5 Long Qiao e al. / Procedia Engineering 16 ( 015 ) (a) (b) Fig. A ime-seps, he vecor disribuion in a reference frame moving wih he bubble (a) and emperaure conour (b) around he bubble wih Re=1.5 10, Ma=7.5 and densiy raio Conclusions In his paper, a new solving sraegy is pu forward o avoid he fourh-order derivaive, and he mean parameer mehod is applied o improve he simulaion convergence for he hermocapillary flows wih large densiy raio. The bubble migraion driving by he hermocapillary force is obained by he proposed mehod, and he resuls indicae ha i is feasible. This paper also presens he deformaion of bubble, he disribuion of velociy and emperaure wih differen physical parameers. Acknowledgemens This work is suppored by Fundamenal Research Funds for he Cenral Universiies (No. CDJZR ), Program for Changjiang Scholars and Innovaive Research Team in Universiy (No IRT13043), and Research Fund for he Docoral Program of Higher Educaion of China (No ). References [1] P.H. Hadland, R. Balasubramaniam, G. Wozniak, R.S. Subramanian, Thermocapillary migraion of bubbles and drops a moderae o large Marangoni number and moderae Reynolds number in reduced graviy, Experimens in Fluids, 6 (1999) [] Q. Kang, H.L. Cui, L. Hu, L. Duan, On-board experimenal sudy of bubble hermocapillary migraion in a recoverable saellie, Micrograviy Sci. Technol, 0 (008) [3] R. Sun, W. Hu, Planar hermocapillary migraion of wo bubbles in micrograviy environmen, Physics of Fluids, 15 (003) [4] M.R. de Sain Vincen, J.-P. Delville, Thermocapillary migraion in small-scale emperaure gradiens: Applicaion o opofluidic drop dispensing, Physical Review E, 85 (01) [5] J. Rodrigo Velez-Cordero, A.M. Velazquez-Beniez, J. Hernandez-Cordero, Thermocapillary flow in glass ubes coaed wih phooresponsive layers, Langmuir, 30 (014) [6] S.H. Jin, S.N. Dunham, J. Song, X. Xie, J.-h. Kim, C. Lu, A. Islam, F. Du, J. Kim, J. Fels, Y. Li, F. Xiong, M.A. Wahab, M. Menon, E. Cho, K.L. Grosse, D.J. Lee, H.U. Chung, E. Pop, M.A. Alam, W.P. King, Y. Huang, J.A. Rogers, Using nanoscale hermocapillary flows o creae arrays of purely semiconducing single-walled carbon nanoubes, Na Nano, 8 (013) [7] H. Liu, A.J. Valocchi, Y. Zhang, Q. Kang, Phase-field-based laice Bolzmann finie-difference model for simulaing hermocapillary flows, Physical Review E, 87 (013) [8] Z. Yin, L. Chang, W. Hu, Q. Li, H. Wang, Numerical simulaions on hermocapillary migraions of nondeformable droples wih large Marangoni numbers, Physics of Fluids, 4 (01). [9] H. Liu, A.J. Valocchi, Y. Zhang, Q. Kang, Phase-field-based laice Bolzmann finie-difference model for simulaing hermocapillary flows, Physical Review E, 87 (013) [10] T. Lee, L. Liu, Laice Bolzmann simulaions of micron-scale drop impac on dry surfaces, Journal of Compuaional Physics, 9 (010)