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Modeling and simulation of thermocapillary flows using lattice Boltzmann method. (English) Zbl 1426.76607
Summary: To understand how thermocapillary forces manipulate droplet motion in microfluidic channels, we develop a lattice Boltzmann (LB) multiphase model to simulate thermocapillary flows. The complex hydrodynamic interactions are described by an improved color-fluid LB model, in which the interfacial tension forces and the Marangoni stresses are modeled in a consistent manner using the concept of a continuum surface force. An additional convection – diffusion equation is solved in the LB framework to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. A stress-free boundary condition is also introduced to treat outflow boundary, which can conserve the total mass of an incompressible system, thus improving the numerical stability for creeping flows.
The model is firstly validated against the analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of three-dimensional deformable droplet at various Marangoni numbers, and its accuracy is once again verified against the theoretical prediction in the limit of zero Marangoni number. Finally, we numerically investigate how the localized heating from a laser can block the microfluidic droplet motion through the induced thermocapillary forces. The droplet motion can be completely blocked provided that the intensity of laser exceeds a threshold value, below which the droplet motion successively undergoes four stages: constant velocity, deceleration, acceleration, and constant velocity. When the droplet motion is completely blocked, four steady vortices are clearly visible, and the droplet is fully filled by two internal vortices. The external vortices diminish when the intensity of laser increases.

##### MSC:
 76M28 Particle methods and lattice-gas methods 76D45 Capillarity (surface tension) for incompressible viscous fluids 80A20 Heat and mass transfer, heat flow (MSC2010)
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##### References:
 [1] Song, H.; Chen, D.L.; Ismagilov, R.F., Reactions in droplets in microfluidic channels, Angew. chem. int. ed., 45, 44, 7336-7356, (2006) [2] Mohr, S.; Zhang, Y.; Macaskill, A.; Day, P.J.R.; Barber, R.W.; Goddard, N.J.; Emerson, D.R.; Fielden, P.R., Numerical and experimental study of a droplet-based PCR chip, Microfluid. nanofluid., 3, 611-621, (2007) [3] Zhang, Y.; Ozdemir, P., Microfluidic DNA amplification-a review, Analytica chimica acta, 638, 115-125, (2009) [4] Song, H.; Ismagilov, R.F., Millisecond kinetics on a microfluidic chip using nanoliters of reagents, J. am. chem. soc., 125, 47, 14613-14619, (2003) [5] Roach, L.S.; Song, H.; Ismagilov, R.F., Controlling nonspecific protein adsorption in a plug-based microfluidic system by controlling interfacial chemistry using fluorous-phase surfactants, Anal. chem., 77, 785-796, (2005) [6] Huebner, A.; Srisa-Art, M.; Holt, D.; Abell, C.; Hollfelder, F.; deMello, A.J.; Edel, J.B., Quantitative detection of protein expression in single cells using droplet microfluidics, Chem. commun., 1218-1220, (2007) [7] Lau, B.T.C.; Baitz, C.A.; Dong, X.P.; Hansen, C.L., A complete microfluidic screening platform for rational protein crystallization, J. am. chem. soc., 129, 3, 454-455, (2007) [8] Li, L.; Ismagilov, R.F., Protein crystallization using microfluidic technologies based on valves, droplets, and slipchip, Ann. rev. biophys., 39, 139-158, (2010) [9] Hung, L.-H.; Choi, K.M.; Tseng, W.-Y.; Tan, Y.-C.; Shea, K.J.; Lee, A.P., Alternating droplet generation and controlled dynamic droplet fusion in microfluidic device for cds nanoparticle synthesis, Lab chip, 6, 174-178, (2006) [10] Hatakeyama, T.; Chen, D.L.; Ismagilov, R.F., Microgram-scale testing of reaction conditions in solution using nanoliter plugs in microfluidics with detection by MALDI-MS, J. am. chem. soc., 128, 8, 2518-2519, (2006) [11] Cheow, L.F.; Yobas, L.; Kwong, D.-L., Digital microfluidics: droplet based logic gates, Appl. phys. lett., 90, 054107, (2007) [12] Prakash, M.; Gershenfeld, N., Microfluidic bubble logic, Science, 315, 5813, 832-835, (2007) [13] Yang, C.-G.; Xu, Z.-R.; Wang, J.-H., Manipulation of droplets in microfluidic systems, Trend. anal. chem., 29, 2, 141-157, (2010) [14] Thorsen, T.; Roberts, R.W.; Arnold, F.H.; Quake, S.R., Dynamic pattern formation in a vesicle-generating microfluidic device, Phys. rev. lett., 86, 18, 4163-4166, (2001) [15] Garstecki, P.; Fuerstman, M.J.; Stone, H.A.; Whitesides, G.M., Formation of droplets and bubbles in a microfluidic T-junction-scaling and mechanism of break-up, Lab chip, 6, 437-446, (2006) [16] Anna, S.L.; Mayer, H.C., Microscale tipstreaming in a microfluidic flow focusing device, Phys. fluids, 18, 121512, (2006) · Zbl 1146.76314 [17] Pollack, M.G.; Fair, R.B.; Shenderov, A.D., Electrowetting-based actuation of liquid droplets for microfluidic applications, Appl. phys. lett., 77, 1725, (2000) [18] Schwartz, J.A.; Vykoukal, J.V.; Gascoyne, P.R.C., Droplet-based chemistry on a programmable micro-chip, Lab chip, 4, 11-17, (2004) [19] Chatterjee, D.; Shepherd, H.; Garrell, R.L., Electromechanical model for actuating liquids in a two-plate droplet microfluidic device, Lab chip, 9, 1219-1229, (2009) [20] Darhuber, A.A.; Valentino, J.P.; Davis, J.M.; Troian, S.M.; Wagner, S., Microfluidic actuation by modulation of surface stresses, Appl. phys. lett., 82, 657, (2003) [21] Chen, J.Z.; Troian, S.M.; Darhuber, A.A.; Wagner, S., Effect of contact angle hysteresis on thermocapillary droplet actuation, J. appl. phys., 97, 014906, (2005) [22] Franke, T.; Abate, A.R.; Weitz, D.A.; Wixforth, A., Surface acoustic wave (SAW) directed droplet flow in microfluidics for PDMS devices, Lab chip, 9, 2625-2627, (2009) [23] Okochi, M.; Tsuchiya, H.; Kumazawa, F.; Shikida, M.; Honda, H., Droplet-based gene expression analysis using a device with magnetic force-based-droplet-handling system, J. biosci. bioeng., 109, 2, 193-197, (2010) [24] Zhang, Y.; Park, S.; Liu, K.; Tsuan, J.; Yang, S.; Wang, T.-H., A surface topography assisted droplet manipulation platform for biomarker detection and pathogen identification, Lab chip, 11, 398-406, (2011) [25] Di Leonardo, R.; Ruocco, G.; Leach, J.; Padgett, M.J.; Wright, A.J.; Girkin, J.M.; Burnham, D.R.; McGloin, D., Parametric resonance of optically trapped aerosols, Phys. rev. lett., 99, 1, 010601, (2007) [26] McGloin, D.; Burnham, D.R.; Summers, M.D.; Rudd, D.; Dewar, N.; Anand, S., Optical manipulation of airborne particles: techniques and applications, Faraday discuss., 137, 335-350, (2008) [27] Delville, J.-P.; de Saint Vincent, M.R.; Schroll, R.D.; Chraïbi, H.; Issenmann, B.; Wunenburger, R.; Lasseux, D.; Zhang, W.W.; Brasselet, E., Laser microfluidics: fluid actuation by light, J. opt. A-pure appl. opt., 11, 3, 034015, (2009) [28] Sciven, L.E.; Sternling, C.V., The Marangoni effects, Nature, 187, 186-188, (1960) [29] Baroud, C.N.; Delville, J.-P.; Gallaire, F.; Wunenburger, R., Thermocapillary valve for droplet production and sorting, Phys. rev. E, 75, 4, 046302, (2007) [30] Young, N.O.; Goldstein, J.S.; Block, M.J., The motion of bubbles in a vertical temperature gradient, J. fluid mech., 6, 03, 350-356, (1959) · Zbl 0087.19902 [31] Shankar, N.; Shankar Subramanian, R., The Stokes motion of a gas bubble due to interfacial tension gradients at low to moderate Marangoni numbers, J. colloid interface sci., 123, 2, 512-522, (1988) [32] Chen, J.; Lee, Y., Effect of surface deformation on thermocapillary bubble migration, Aiaa j., 30, 993-998, (1992) [33] Oliver, D.; Witt, K.D., Transient motion of a gas bubble in a thermal gradient in low gravity, J. colloid interface sci., 164, 2, 263-268, (1994) [34] Subramanian, R.; Balasubramaniam, R., The motion of bubbles and drops in reduced gravity, (2001), Cambridge University Press London · Zbl 0982.83001 [35] Subramanian, R.; Balasubramaniam, R.; Wozniak, G., Physics of fluids in microgravity, (2002), Taylor & Prancis London, Ch. Fluid mechanics of bubbles and drops, pp. 149-177 [36] Yin, Z.; Gao, P.; Hu, W.; Chang, L., Thermocapillary migration of nondeformable drops, Phys. fluids, 20, 082101, (2008) · Zbl 1182.76854 [37] Rybalko, S.; Magome, N.; Yoshikawa, K., Forward and backward laser-guided motion of an oil droplet, Phys. rev. E, 70, 046301, (2004) [38] Kotz, K.T.; Noble, K.A.; Faris, G.W., Optical microfluidics, Appl. phys. lett., 85, 2658, (2004) [39] Baroud, C.N.; de Saint Vincent, M.R.; Delville, J.-P., An optical toolbox for total control of droplet microfluidics, Lab chip, 7, 1029-1033, (2007) [40] de Saint Vincent, M.R.; Wunenburger, R.; Delville, J.-P., Laser switching and sorting for high speed digital microfluidics, Appl. phys. lett., 92, 154105, (2008) [41] Shyy, W.; Smith, R.W.; Udaykumar, H.S.; Rao, M.M., Computational fluid dynamics with moving boundaries, (1996), Taylor & Francis London · Zbl 0887.76059 [42] Succi, S., The lattice Boltzmann equation for fluid dynamics and beyond, (2001), Oxford University Press Oxford · Zbl 0990.76001 [43] He, X.; Luo, L.-S., A priori derivation of the lattice Boltzmann equation, Phys. rev. E, 55, R6333-R6336, (1997) [44] Dupin, M.M.; Halliday, I.; Care, C.M., Simulation of a microfluidic flow-focusing device, Phys. rev. E, 73, 055701, (2006) [45] Kusumaatmaja, H.; Léopoldès, J.; Dupuis, A.; Yeomans, J.M., Drop dynamics on chemically patterned surfaces, Europhys. lett., 73, 5, 740, (2006) [46] Liu, H.; Zhang, Y., Phase-field modeling droplet dynamics with soluble surfactants, J. comput. phys., 229, 24, 9166-9187, (2010) · Zbl 1427.76187 [47] Liu, H.; Zhang, Y., Droplet formation in microfluidic cross-junctions, Phys. fluids, 23, 082101, (2011) [48] Wang, W.; Liu, Z.; Jin, Y.; Cheng, Y., LBM simulation of droplet formation in micro-channels, Chem. eng. J., 173, 3, 828-836, (2011) [49] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for previous modeling surface tension, J. comput. phys., 100, 335-354, (1992) · Zbl 0775.76110 [50] Latva-Kokko, M.; Rothman, D.H., Diffusion properties of gradient-based lattice Boltzmann models of immiscible fluids, Phys. rev. E, 71, 5, 056702, (2005) [51] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Ann. rev. fluid mech., 30, 1, 329-364, (1998) · Zbl 1398.76180 [52] Gunstensen, A.K.; Rothman, D.H.; Zaleski, S.; Zanetti, G., Lattice Boltzmann model of immiscible fluids, Phys. rev. A, 43, 8, 4320-4327, (1991) [53] Shan, X.; Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. rev. E, 47, 3, 1815, (1993) [54] Swift, M.R.; Osborn, W.R.; Yeomans, J.M., Lattice Boltzmann simulation of nonideal fluids, Phys. rev. lett., 75, 5, 830-833, (1995) [55] Swift, M.R.; Orlandini, E.; Osborn, W.R.; Yeomans, J.M., Lattice Boltzmann simulations of liquid-gas and binary fluid systems, Phys. rev. E, 54, 5, 5041-5052, (1996) [56] Yuan, P.; Schaefer, L., Equations of state in a lattice Boltzmann model, Phys. fluids, 18, 042101, (2006) · Zbl 1185.76872 [57] Sbragaglia, M.; Benzi, R.; Biferale, L.; Succi, S.; Sugiyama, K.; Toschi, F., Generalized lattice Boltzmann method with multirange pseudopotential, Phys. rev. E, 75, 2, 026702, (2007) [58] Kupershtokh, A.; Medvedev, D.; Karpov, D., On equations of state in a lattice Boltzmann method, Comput. math. appl., 58, 5, 965-974, (2009) · Zbl 1189.76413 [59] van der Sman, R.; van der Graaf, S., Emulsion droplet deformation and breakup with lattice Boltzmann model, Comput. phys. commun., 178, 492-504, (2008) · Zbl 1196.65168 [60] Lishchuk, S.V.; Care, C.M.; Halliday, I., Lattice Boltzmann algorithm for surface tension with greatly reduced microcurrents, Phys. rev. E, 67, 036701, (2003) [61] Halliday, I.; Law, R.; Care, C.M.; Hollis, A., Improved simulation of drop dynamics in a shear flow at low Reynolds and capillary number, Phys. rev. E, 73, 5, 056708, (2006) [62] Guo, Z.; Zheng, C.; Shi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. rev. E, 65, 046308, (2002) · Zbl 1244.76102 [63] Halliday, I.; Hollis, A.P.; Care, C.M., Lattice Boltzmann algorithm for continuum multicomponent flow, Phys. rev. E, 76, 026708, (2007) [64] Peng, Y.; Shu, C.; Chew, Y.T., Simplified thermal lattice Boltzmann model for incompressible thermal flows, Phys. rev. E, 68, 026701, (2003) [65] LADD, A.J.C., Numerical simulations of particulate suspensions via a discretized Boltzmann equation. part 1: theoretical foundation, J. fluid mech., 271, 309-311, (1994) [66] Inamuro, T.; Yoshino, M.; Ogino, F., A non-slip boundary condition for lattice Boltzmann simulations, Phys. fluids, 7, 2928, (1995) · Zbl 1027.76631 [67] Chen, S.; Martnez, D.; Mei, R., On boundary conditions in lattice Boltzmann methods, Phys. fluids, 8, 2527, (1996) · Zbl 1027.76630 [68] Zou, Q.; He, X., On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys. fluids, 9, 1591, (1997) · Zbl 1185.76873 [69] Guo, Z.-L.; Zheng, C.-G.; Shi, B.-C., Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method, Chinese phys., 11, 4, 366, (2002) [70] Liu, C.-H.; Lin, K.-H.; Mai, H.-C.; Lin, C.-A., Thermal boundary conditions for thermal lattice Boltzmann simulations, Comput. math. appl., 59, 7, 2178-2193, (2010) · Zbl 1193.80027 [71] Pendse, B.; Esmaeeli, A., An analytical solution for thermocapillary-driven convection of superimposed fluids at zero Reynolds and Marangoni numbers, Int. J. therm. sci., 49, 7, 1147-1155, (2010) [72] Welch, S.W., Transient thermocapillary migration of deformable bubbles, J. colloid interface sci., 208, 2, 500-508, (1998) [73] Wang, Y.; Lu, X.; Zhuang, L.; Tang, Z.; Hu, W., Numerical simulation of drop Marangoni migration under microgravity, Acta astronaut., 54, 5, 325-335, (2004) [74] Hill, M.J.M., On a spherical vortex, Proc. R. soc. lond., 55, 219-224, (1894) [75] Gallaire, F.; Baroud, C.N.; Delville, J.-P., Thermocapillary manipulation of microfluidic droplets: theory and applications, Int. J. heat technol., 26, 161-166, (2008)
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