Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 126 (2015) 507 – 511
7th International Conference on Fluid Mechanics, ICFM7
Phase-field-based finite volume method for simulating thermocapillary flows Long Qiaoa,b, Zhong Zeng*a,b,c, Haiqiong Xiea,b a
Department of Engineering Mechanics, College of Aerospace Engineering, Chongqing University, Chongqing 400044, China b State Key Laboratory of Coal Mine Disaster Dynamics and Control (Chongqing University), China c State Key Laboratory of Crystal Material (Shandong University), China
Abstract Based on the phase-field interface-capturing scheme, a novel and common simulation strategy is put forward with the finite volume method to study the thermocapillary flows, which avoids the fourth-order derivative efficaciously and adopts the mean parameter method to preserve the convergence in the large density ratio flows. Through simulating the thermocapillary migration of bubble, the results show that the new strategy can successfully describe the performance of bubble with different density ratios, i.e. 0.1 and 0.01, and dimensionless parameters Re=1.5×10-2 and 1.5×102. When Re is 1.5×102, the bubble has an apparent deformation, and a larger migration velocity of the bubble is observed with a larger temperature gradient along the surface. © Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license ©2015 2015The TheAuthors. Authors. Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Chinese Society of Theoretical and Applied Mechanics (CSTAM). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM) Keywords: Phase-field method; Thermocapillary flows; Finite volume method; Bubble migration
1. Introduction Thermocapillary force, due to the temperature gradient in the two-phase interface, is the most decisive influence under some special conditions, for instance, microgravity environment and microfluidic devices, and the research on thermocapillary flows goes on apace. In the experiments under microgravity environment, thermocapillary migration of air bubbles has been carried out by Hadland et al. [1] and Kang et al. [2], and their results are similar with the theoretical analyses [3]. In recent years, thermocapillary force has been applied to manipulate the droplet or bubble migration in microfluidic devices [4,5] because of the ratio of large surface area to volume in micro scale, and it is
* Corresponding author. Tel.: +86-23-65111813; fax: +86-23-65111067. E-mail address:
[email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)
doi:10.1016/j.proeng.2015.11.292
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Long Qiao et al. / Procedia Engineering 126 (2015) 507 – 511
also adopted to manufacture advanced material [6]. In the meantime, numerical simulation, as a preferential method, is also widely used to investigate the thermocapillary flows, and some capturing methods have been developed, such as the phase-field [7] and front-tracking [8] et al. In this paper, a new and common phase-field-based finite volume method is proposed, which avoids the fourth-order derivative in the conventional phase-field equation and adopts the mean parameter method to improve the convergence in the cases with large density ratio. In addition, some results from the simulation of bubble thermocapillary migration are presented with the novel method. 2. Theoretics and numerical strategies 2.1. Phase-field theory and numerical evolution Phase-field theory, a kind of diffuse-interface model, is introduced from the free energy, and it is gradually applied to capture the interface between two fluids in the numerical simulation. Normally, the order parameter ϕ is adopted to describe the composition distribution of two immiscible fluids, and it is presented in the computational domain continuously ranging from -1 to 1. Under the hypothesis of the equilibrium interface profiles given by Van der Waals, Cahn-Hilliard equation
It u I (M P )
(1)
is developed to model the variation of order parameter with time, where u is the fluid velocity and M is the nonnegative mobility [9]. Generally, the mobility M is also introduced as a function of order parameter, i.e., M = Mc(1ϕ2), where Mc is a constant parameter. The parameter μ represents the chemical potential, defined as
P I 3 I 22I ,
(2)
where denotes the thickness of interface layer. From Eq. (1) and Eq. (2), the fourth-order derivative of the order parameter is presented. One kind of solving strategies is obtaining the chemical potential firstly, then solving the order parameter through Eq. (1), where the diffusion caused by chemical potential is treated as a source term. In this paper, a new solving strategy is put forward to decrease the order of derivative by reforming conventional Cahn-Hilliard equation as follows:
0 2I SI and SI
2
(I 3 I P ) ,
0 ( M Pˆ ) SP and SP
It u I ,
(3) (4)
where Sϕ and Sμ are the sources of the transport equations of chemical potential and order parameter respectively. In addition, the artificial free energy [10] is introduced to obstruct the order parameter less than the minimum value 1.0 in this paper. The effective chemical potential is defined as Pˆ P wEA wI , where EA is modifier of free energy and it is stated as Eq.(5)
E (1 I )2 , if I 1, EA (I ) ® A , if I t 1. ¯0 ,
(5)
In these forms, the order parameter is firstly predicted by Eq. (3), where the former chemical potential and order parameter are used; subsequently, the chemical potential is updated using the predicted order parameter using the Eq. (4), and the time evolution and convective transport of order parameter are introduced. These two solving steps are repeated until the final tolerance criterion is reached.
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2.2. Mean parameter method and governing equations for thermocapillary flow In general, the incompressible governing equations are adopted to describe the incompressible fluid flows. But in phase-field model, the density is defined as a linear function with the order parameter, i.e., ρ=(1-ϕ)ρ1/2+(1+ϕ)ρ2/2, which causes an evident change of density in the interface zone. For two fluids with a large density ratio, the influence caused by density change cannot be ignored and the compressible governing equations are utilized. In this paper, a mixed form of governing equations in proposed using the mean parameter method, which is incompressible but contains the effect induced by the density change. The amended governing equations for thermocapillary flow of two immiscible fluids are listed as follows
u 0,
(6)
Um (ut (uu)) p [K (u uT )] FS FI ,
(7)
[m (Tt (uT )) (kT ) ST ,
(8)
where p and T represent the pressure and temperature of the mixture fluid, respectively; ρm is the mean density of two fluids, i.e., ρm=ρ1/2+ρ2/2; ξm is the mean value of the product of specific heat and density for two fluids; η and k are dynamic viscosity and thermal conductivity of mixture respectively, and they have the same linear relationship as density, i.e., η=(1-ϕ)η1/2+(1+ϕ)η2/2 and k=(1-ϕ)k1/2+(1+ϕ)k2/2. In addition, the subscript 1 and 2 represent fluid 1 and fluid 2. Fϕ and ST are sources in momentum and energy equations respectively, which are caused by the mean parameter method including the parts determined by the order parameter, i.e.,
FI
( UIIu)t ( UIIuu) ,
(9)
ST
([IIT )t ([IITu) ,
(10)
where ρϕ and ξϕ are the changing coefficients of ρ and ξ for the order parameter, i.e., ρϕ=ρ1/2-ρ2/2 and ξϕ=ξ1/2-ξ2/2. FS is the interface force which under the continuous surface force model (CSF model). In this paper, the interface force abides by the following form
FS
3 2
4
[V (| I |2 I II )] ,
(11)
where σ is the surface tension which has a linear relation with the temperature, i.e., σ=σref+σT(T-Tref). And, σref is the reference surface tension at the reference temperature Tref and σT is the changing ratio of surface tension for temperature. 3. Numerical experiments 3.1. Problem statement For confirming the practicability of the new strategies, two-dimensional thermocapillary migration of bubble with the above theories is studied. The bubble, filled with fluid 1, is located at the center of a rectangular box full of fluid 2 and it is driven by the uniform temperature gradient field moving from bottom to top. The top and bottom are no-slip walls, while the periodic condition is adopted for left and right side walls. In this paper, the properties of two fluids and other important parameters, including the sizes of bubbles and box and the temperature condition etc., are introduced from Ref. [9]. Where, the density ρ2, dynamic viscosity η2,
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specific heat cp2 and thermal conductivity k2 of fluid 2 are 1.0, 0.2, 5.0 and 0.2 respectively. The properties of fluid 1 are defined as ρ1=ρ0ρ2, η1=η0η2, cp1=cp0cp2, k1=k0k2, and ρ0, η0, cp0 k0 are the ratios of parameter between fluid 1 and fluid 2. The radius of the bubble R=30, the size of box is 8R×16R where the distance from bottom to top is 16R. In addition, the top wall and bottom wall are kept constant temperature Th=32 and Tc=0 respectively. The reference temperature Tref=16, and the surface tension under this temperature σref=2.5×10-3. The surface tension coefficient σT=-1.0×10-5. The mobility constant Mc=0.05 and the thickness of the interface =21/2. In this paper, the size of mesh used for simulating is 1×1 and the time-step size is 1.0. The SIMPLE arithmetic is adopted for solving the couple of pressure field and velocity field. In addition, two dimensionless numbers are defined to describe the heat/mass transport of different parameters. The non-dimensional velocity U, Reynolds number Re and Marangoni number are defined as follows:
U
V T Tf R K2 , Re
RU Q 2 , Ma
U2c p 2 RU k2 ,
where |Tf|=(Th-Tc)/16R is the far-field temperature gradient and ν η2/ρ2 is the kinematic viscosity. 3.2. Results and discussions Based on the novel scheme for simulating the thermocapillary flows, bubble migration with different physical property ratios is studied. Firstly, the smaller density ratio problem is discussed, where the ratios of property parameters are ρ0=η0=0.1, k0=0.2, cp0=0.4. The dimensionless numbersare Re=1.5×10-2 and Ma=7.5×10-2. Fig. 1 depicts the relative vector distribution and temperature contour around the bubble. Where the purple line represents the initial position of bubble while the blue line is the position at 30000 time-steps. Moreover, two counter vortexes in the bubble are observed and the temperature profile is significantly affected by the bubble. The results are comparable with the results in the Ref. [9] which is a three-dimensional simulation.
(a)
(b)
Fig. 1 At 30000 time-steps, the vector distribution in a reference frame moving with the bubble (a) and temperature contour (b) around the bubble with Re=1.5×10-2, Ma=7.5×10-2 and density ratio 0.1.
Subsequently, the validity of the new theory for larger density ratio is tested. The parameter ratios are taken as ρ0=0.01, η0=0.1, k0=0.02, cp0=0.4. Additionally, dynamic viscosity η2 is reduced to 0.002, and the Reynolds number and Marangoni number are Re=1.5×102 and Ma=7.5. At 50000 time-steps, the relative vector distribution and temperature contour near the bubble are plotted in Fig. 2. The shape of bubble, represented by the blue line, has been changed apparently by the vortex-ring behind the bubble resulted by the pressure difference between the top and bottom of the bubble. The temperature distribution also has a new pattern, in which a larger gradient emerges along the interface and provides a power for the bubble migration.
Long Qiao et al. / Procedia Engineering 126 (2015) 507 – 511
(a)
(b)
Fig. 2 At 50000 time-steps, the vector distribution in a reference frame moving with the bubble (a) and temperature contour (b) around the bubble with Re=1.5×102, Ma=7.5 and density ratio 0.01.
4. Conclusions In this paper, a new solving strategy is put forward to avoid the fourth-order derivative, and the mean parameter method is applied to improve the simulation convergence for the thermocapillary flows with large density ratio. The bubble migration driving by the thermocapillary force is obtained by the proposed method, and the results indicate that it is feasible. This paper also presents the deformation of bubble, the distribution of velocity and temperature with different physical parameters. Acknowledgements This work is supported by Fundamental Research Funds for the Central Universities (No. CDJZR13248801), Program for Changjiang Scholars and Innovative Research Team in University (No IRT13043), and Research Fund for the Doctoral Program of Higher Education of China (No. 20110191110037). References [1] P.H. Hadland, R. Balasubramaniam, G. Wozniak, R.S. Subramanian, Thermocapillary migration of bubbles and drops at moderate to large Marangoni number and moderate Reynolds number in reduced gravity, Experiments in Fluids, 26 (1999) 240-248. [2] Q. Kang, H.L. Cui, L. Hu, L. Duan, On-board experimental study of bubble thermocapillary migration in a recoverable satellite, Microgravity Sci. Technol, 20 (2008) 67-71. [3] R. Sun, W. Hu, Planar thermocapillary migration of two bubbles in microgravity environment, Physics of Fluids, 15 (2003) 3015-3027. [4] M.R. de Saint Vincent, J.-P. Delville, Thermocapillary migration in small-scale temperature gradients: Application to optofluidic drop dispensing, Physical Review E, 85 (2012) 026310. [5] J. Rodrigo Velez-Cordero, A.M. Velazquez-Benitez, J. Hernandez-Cordero, Thermocapillary flow in glass tubes coated with photoresponsive layers, Langmuir, 30 (2014) 5326-5336. [6] S.H. Jin, S.N. Dunham, J. Song, X. Xie, J.-h. Kim, C. Lu, A. Islam, F. Du, J. Kim, J. Felts, 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 thermocapillary flows to create arrays of purely semiconducting single-walled carbon nanotubes, Nat Nano, 8 (2013) 347-355. [7] H. Liu, A.J. Valocchi, Y. Zhang, Q. Kang, Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows, Physical Review E, 87 (2013) 013010. [8] Z. Yin, L. Chang, W. Hu, Q. Li, H. Wang, Numerical simulations on thermocapillary migrations of nondeformable droplets with large Marangoni numbers, Physics of Fluids, 24 (2012). [9] H. Liu, A.J. Valocchi, Y. Zhang, Q. Kang, Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows, Physical Review E, 87 (2013) 013010. [10] T. Lee, L. Liu, Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces, Journal of Computational Physics, 229 (2010) 8045-8063.
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