在微流體的應用開發中，了解溫度梯度來控制與操縱液滴之運動行為是很重要的。在本研究中，採用數值計算方式來探討微流道與毛細管中的熱毛細遷移現象。透過COMSOL Multiphysics軟件開發的有限元素方法，以two-phase level set技術來求解Navier-Stokes 與能量方程式，並以conservative level set的方法，配合arbitrary Lagrangian Eulerian（ALE）運動描述法和連續的表面力學法（CSF）來追蹤液體/氣體界面，並確保該自由界面附近良好的分辨率。在液氣界面上考慮兩種力量的作用，如界面曲率法線方向上的毛細力以及作用在自由表面切線方向上的熱毛細力。
在微通道的矽液滴模擬中，微通道的下壁為假設於均勻的溫度梯度下，同時上壁則是在絕熱、等於環境溫度或高於下壁溫度的固定值。當上壁被設定為絕熱時，液滴內最初會有一對非對稱熱毛細對流渦旋產生，但當時間夠長時最後會形成只有一個熱毛細對流渦旋。在等於環境溫度的情況下，一對非對稱熱毛細對流渦旋總是維持在液滴的內部。當上壁面溫度高於下壁面時，兩對熱毛細渦流產生熱毛細動量啟動液滴的運動，當達到一定時間後，熱毛細渦流反而阻擋液滴的流動。而液滴在絕熱和等於環境溫度情況下，一開始會被加速，在絕熱情況下，液滴速度降低比等於環境溫度條件快。因此在較熱的上壁面狀況下，液滴在一開始會被加速然後速度降到零，要提高液滴的移動速度可藉由更高的溫度梯度與減少微通道高度或更小的接觸角來達成。
毛細管內矽塞遷移現象中，矽塞的運動，主要由靠近後退端氣液界面因溫度梯度所形成的熱毛細作用，及矽塞兩端之間的溫度差所產生的毛細力來驅動。當時間足夠長時，由毛細力來驅動，於靠近管壁處，下流體主要從熱側往冷側進行水平移動，然後返回到管的中心熱側，一個較小的順時針迴流在後退接觸角附近產生，這流動型態造成矽塞內部的等溫線扭曲及提高管內的溫度梯度。液體運動在起步階段迅速加速然後減速，達到最大速度。在遷移過程中，後退接觸角總是比前接觸角大。當增加輸入的熱通量時，由於提高沿管壁的溫度梯度而導致更高的遷移速度。當初始接觸角較小時，由於具備較高的毛細力，導致移動速度更快。而在較低粘度的矽液體中，由於較低的粘滯力亦會導致運動加快。液滴之數值模擬結果與過去之實驗結果具有一致性的趨勢。

An understanding of the transport behavior of a liquid droplet controlled and manipulated by the thermal gradient is very important for the development of microfluidic applications. In this study, a numerical computation is utilized to investigate the thermocapillary actuation behavior of a liquid with two different physical problems: microchannel and capillary tube. The finite element method with the two-phase level set technique, developed by Comsol Multiphysics, is used to solve the incompressible Navier-Stokes equations coupled with the energy equation. The conservative level set method, the arbitrary Lagrangian Eulerian (ALE), and the continuum surface force (CSF) method are used to track the liquid/gas interface and ensure good resolution near the free interface. Two forces are considered at the liquid/gas interface such as the capillary force acting in the normal direction, and the thermocapillary force acting in the tangential direction to the free surface.
For modeling of a liquid droplet in microchannel, the lower wall of the microchannel is subjected to a uniform temperature gradient, while the upper one is adiabatic, isothermal or heated wall. When the upper wall is set to be adiabatic, a pair of asymmetric thermocapillary convection vortices initially occurs inside the droplet but these turn into a sole thermocapillary vortex once enough time has passed. For the isothermal case, a pair of asymmetric thermocapillary convection vortices always appears inside the droplet. For the case of the upper wall temperature higher than the bottom one, the net thermocapillary momentum generated by two pairs of thermocapillary vortices assists the droplet migration during the initial stage. When time reaches a certain value, it turns to go against the droplet migration. The droplet initially accelerates for all cases. The droplet velocity then decreases dramatically for the adiabatic case while it decreases slowly for the isothermal one. For the heated upper wall case, the droplet velocity decelerates to zero velocity after it gets the maximal value. The actuation velocity of the droplet is affected by temperature gradients, contact angles and microchannel heights for adiabatic, isothermal or heated wall cases.
For a silicone plug migration inside a capillary tube, flow motion is affected by the thermocapillary effect generated by the temperature gradient along the gas-liquid interface near the receding side and the capillary force caused by the temperature difference between the ends of the liquid plug. When time is long enough, the flow mainly moves horizontally from the hot side to the cold side near the tube wall and then returns to the hot side near the center of the tube due to the capillary force effect. There is a smaller clockwise circulation near the receding contact angle caused by the thermocapillary convection. The flow motion causes significant distortion of the isotherms inside the silicone plug. The temperature gradient along the tube is enhanced by the flow motion inside the capillary tube. The liquid plug accelerates rapidly in the initial stage and then decelerates after it reaches the maximum speed. During the migration process, the receding contact angle is always greater than the advancing one. An increase in the input heat flux leads to a higher migration velocity due to the higher temperature gradient along the tube wall. When the initial contact angle is smaller, the migration velocity moves faster due to the higher capillary force. A liquid plug with a lower viscosity moves faster owing to the lower viscous force. The numerical simulation results are in good agreement with the results from previous experiments.

[1] S. Haeberle and R. Zengerle, “Microfluidic platforms for lab-on-a-chip applications,” Lab Chip 7 (2007) 1094-1110.
[2] H. A. Stone, A. D. Stroock, A. Ajdari, “Engineering Flows in small devices: Microfluidics towards a lab-on-a-chip,” Annu. Rev. Fluid Mech. 36 (2004) 381-411.
[3] E. Lauga, M.P. Brenner, and H.A. Stone, “Handbook of experimental Fluid Dynamics,” edited by C. Tropea, J. Foss, and A. Yarin. New York: Springer, 2005.
[4] H. Bruus, Theoretical Microfluidics: Oxford University Press, 2008.
[5] A. Manz, N. Graber, H.M. Widmer, “Miniaturized total chemical-analysis systems-a novel concept for chemical sensing,” Sens. Actuators B: Chem. 1 (1990) 244-248.
[6] H. Song, M.R. Bringer, J.D. Tice, C.J. Gerdts, R.F. Ismagilov, “Experimental test of scaling of mixing by chaotic advection in droplets moving through microfluidic channels,” Applied Physics Letter 83 (2003) 4664-4666.
[7] N.T. Nguyen and S.T. Wereley, “Fundamentals and applications of microfluidics,” Artech House, Boston, 2006.
[8] N. Damean, P.P.L. Regtien, M. Elwenspoek, “Heat transfer in a MEMS for microfluidics,” Sensors and Actuators A: Physical 105 (2003) 137-149.
[9] L. Lofdahl and M. Gad-El-Hak, “MEMS applications in turbulence and flow control,” Prog. Aeosp. Sci. 35 (1999) 101-203.
[10] Z. Jiao, X. Huang, N.T. Nguyen, P. Abgrall, “Thermocapillary actuation of droplet in a planar microchannel,” Microfluidic Nanofluid (2008) 205-214.
[11] M.G. Pollack, R.B. Fair and A.D. shenderov, “Electrowetting-based actuation of droplets for integrated microfluidics,” Appl. Phys. Lett 77 (2000) 1725-1726.
[12] N.T. Nguyen, K. Meng Ng, and X. Huang, “Manipulation of ferrofluid droplets using planar coils,” Appl. Phys. Lett. 89 (2006) 052509.
[13] K. Ichimura, S.K. Oh, and M. Nakagawa, “Light-driven motion of liquids on a photoresponsive surface,” Science 288 (2000) 1624-1626.
[14] Z.C. Che, T.N. Wong, N.T. Nguyen, “An analytical model for plug flow in microcapillaries with circular cross section,” Int. J. Heat Fluid Flow 32 (2011) 1005-1013.
[15] R. Gupta, D.F Fletcher, B.S. Haynes, “Taylor flow in microchannels: A review of experimetal and computational work,” The Journal of Computational Multiphase Flow 2 (2010) 1-31.
[16] A.L. Yarin, W. Liu and D.H. Reneker, “Motion of droplets along thin fibers with temperature gradient,” J. Appl. Phys. 91(2002), 4751-4760.
[17] W. Qu, M. Mala, D. Li, “Pressure-driven water flows in trapezoidal silicone microchannels,” Int. J. Heat Mass Transfer 43 (2000) 353–364.
[18] T.S. Sammarco, M.A. Burns, “Thermocapillary pumping of discrete drops in microfabricated analysis devices,” AIChE J. 45 (1999) 350-366.
[19] Z. Yu, O. Hemminger, L.S. Fan, “Experiment and lattice Boltzmann simulation of two-phase gas-liquid flows in microchannels,” Chem. Eng. Sci. 62 (2007), 7172-7183.
[20] A.K. Bajpai, S. Khandekar, “Thermal transport behavior of a liquid plug moving inside a dry capillary tube,” Heat Pipe Sci. Technol. 3(2-4) (2012) 97-124.
[21] P.S. Glockner, G.F. Naterer, “Surface tension and frictional resistance of thermocapillary pumping in a closed microchannel,” Int. J. Heat Mass Transfer 49 (2006) 4424-4436.
[22] Z. Jiao, N.T. Nguyen, X. Huang, “Thermocapillary actuation of liquid plugs using a heater array,” Sensors and Actuators A 140 (2007) 145-155.
[23] N.T. Nguyen, X. Huang, “Thermocapillary effect of a liquid plug in transient temperature fields,” J. J. of Appl. Phys. 44 (2005) 1139-1142.
[24] N.T. Nguyen, W.W. Pang, X. Huang, “Sample transport with thermocapillary force for microfluidics,” J. Phys. 34 (2006) 967-972.
[25] C. Song, K. Kim, K. Lee, and H. K. Pak, “Thermochemical control of oil droplet motion on a solid substrate,” Appl. Phys. Letters 93 (2008) 084102-1-3.
[26] V. Pratap, N. Moumen, and R.S. Subramanian, “Thermocapillary motion of a liquid drop on a horizontal solid surface,” Langmuir 24 (2008) 5185-5193.
[27]. M. L. Ford and A. Nadim, "Thermocapillary migration of an attached drop on a solid surface," Phys. Fluids 6 (1994) 3183-3185.
[28] J. Z. Chen, S. M. Troian, A. A. Darhuber and S. Wagner, “Effect of contact angle hysteresis on thermocapillary droplet actuation,” J. App. Phys. 97 (2005) 014906.
[29] M.K. Smith, “Thermocapillary migration of a two-dimensional liquid droplet on a solid surface,” J. Fluid Mech. 294 (1995) 209-230.
[30] J. M. Gomba, G. M. Homsy, “Regimes of thermocapillary migration of droplets under partial wetting conditions,” J. Fluid Mech. 647 (2010) 125-142.
[31] G. Karapetsas, K. C. Sahu, and Omar K. Matar, “Effect of contact line dynamics on the thermocapillary motion of a droplet on an inclined plate,” Langmuir 29 (2013) 8892-8906.
[32] H.-B.Nguyen and J.-C.Chen, “A numerical study of thermocapillary migration of a small liquid droplet on a horizontal solid surface,” Phys. Fluid 22 (2010) 062102.
[33] H.-B.Nguyen and J.-C.Chen, “Numerical study of a droplet migration induced by combined thermocapillary-bouyancy convection,” Phys. Fluid 22 (2010) 122101.
[34] H.-B.Nguyen and J.-C.Chen, “Effect of slippage on the thermocapillary migration of a small droplet,” Biomicrofluidics 6 (2012) 012809.
[35] T.-L. Le, J.-C. Chen, B.-C. Shen, F.-S. Hwu and H.-B. Nguyen, “Numerical investigation of the thermocapillary actuation behavior of a droplet in a microchannel,” Int. J. Heat Mass Transfer 83 (2015) 721-730.
[36] T.-L. Le, J.-C. Chen, F.-S. Hwu, H.-B. Nguyen, “Numerical study of a small droplet migration in microchannel under a heated upper wall,” Twelfth International Conference on Flow Dynamics (ICFD2015).
[37] T.-L. Le, J.-C. Chen, H.-B. Nguyen, and F.-S. Hwu, “Numerical study of thermocapillary migration of a silicone plug inside the capillary tube,” In Proceedings of 2014 International Conference on Machining, Materials and Mechanical Technologies (IC3MT).
[38] T.-L. Le, J.-C. Chen, F.-S. Hwu, H.-B. Nguyen, “Numerical study of the migration of a silicone plug inside a capillary tube subjected to an unsteady wall temperature gradient,” International Journal of Heat and Mass Transfer 97 (2016) 439-449.
[39] H. Bouasse, Capillarité: phé nomènes superficiels: Delagrave, 1924.
[40] L. Shui, J.C.T. Eijkel, A.V.D. Berg, “Multiphase flow in micro- and nanochannels,” Sensors and Actuators B 121 (2007) 263-276.
[41] J.-C. Chen, C.-W. Kuo, and G.P. Neitzel, “Numerical simulation of thermocapillary nonwetting,” Int. J. Heat Mass Transfer 49 (2006) 4567-4576.
[42] R.S Subramanian, “Thermocapillary motion of bubbles and drops,” Microgravity Fluid Mechanics, International Union of Theoretical and Applied Mechanics (1992) 393-403.
[43] T. Young, “An essay on the cohesion of fluids,” Philos. Trans. R. Soc. London 95 (1805) 65-87.
[44] S. Goldstein, “Fluid mechanics in first half of this century,” Ann. Rev. Fluid Mech. 1 (1969) 1-28.
[45] M.E. O'Neill, K.B. Ranger, and H. Brenner, "Slip at the surface of a translating–rotating sphere bisected by a free surface bounding a semi-infinite viscous fluid: Removal of the contact-line singularity," Phys. Fluids 29 (1986) 913-924.
[46] V.S.J. Craig, C. Neto, and D.R.M. Williams, "Shear-Dependent Boundary Slip in an Aqueous Newtonian Liquid," Phys. Rev. Lett. 87 (2001) 054504-1-4.
[47] D.C. Tretheway and C.D. Meinhart, "Apparent fluid slip at hydrophobic microchannel walls," Phys. Fluids 14 (2002) L9-L12.
[48] Y. Zhu and S. Granick, "Limits of the Hydrodynamic No-Slip Boundary Condition," Phys. Rev. Lett. 88 (2002) 106102-1-4.
[49] C.-H. Choi, K.J.A. Westin, and K.S. Breuer, "Apparent slip flows in hydrophilic and hydrophobic microchannels," Phys. Fluids 15 (2003) 2897-2902.
[50] P. Joseph and P. Tabeling, "Direct Measurement of the Apparent Slip Length," Phys. Rev. E 71 (2005) 035303-1-4.
[51] P. Tabeling, "Investigating slippage, droplet breakup, and synthesizing microcapsules in microfluidic systems," Phys. Fluids 22 (2010) 021302-1-7.
[52] J. Koplik, J. R. Banavar, and J. F. Willemsen, “Molecular dynamics of fluid flow at solid surfaces,” Phys. Fluids A 1 (1989) 781-794.
[53] C.L.M.H. Navier, "Memoire sur les lois du mouvement des fluides," Mem. Acad. Sci. Inst. 6 (1823) 389-440.
[54] P.A. Thompson, S.M. Troian, “A general boundary condition for liquid flow at sodlid surface,” Nature 389 (1997) 360-362.
[55] C. Neto, D.R. Evans, E. Bonaccurso, H. Butt, and V.S.J. Craig, "Boundary slip in Newtonian liquids: a review of experimental studies " Reports Prog. Phys. 68 (2005) 2859.
[56] E.B. Dussan V., "The moving contact line: the slip boundary condition," J. Fluid Mech. 77 (1976) 665-684.
[57] E.B. Dussan V., "On the Spreading of Liquids on Solid Surfaces: Static and Dynamic Contact Lines," Annu. Rev. Fluid Mech. 11 (1979) 371-400.
[58] Y.T. Tseng, F.G. Tseng, Y.F. Chen, and C. C. Chieng, "Fundamental studies on micro-droplet movement by Marangoni and capillary effects," Sensors & Actuators: A. Physical 114 (2004) 292-301.
[59] J.B. Brzoska, F. Brochard-Wyart, and F. Rondelez, "Motions of droplets on hydrophobic model surfaces induced by thermal gradients," Langmuir 9 (1993) 2220-2224, 1993.
[60] A.A. Darhuber, J. M. Davis, S. M. Troiana, and W. W. Reisner, "Thermocapillary actuation of liquid flow on chemically patterned surfaces," Phys. Fluids (2003) 1295-1304.
[61] C. Song, K. Kim, K. Lee, and H.K. Pak, "Themochemical control of oil droplet motion on a solid substrate," Appl. Phys. Lett. 93 (2008) 084102-1-3.
[62] E. Olsson, G. Kreiss, “A conservative level set method for two phase flow,” Journal of Computational Physics 210 (2005) 225-246.
[63] E. Olsson, G. Kreiss, S. Zahedi, “A conservative level set method for two phase flow II,” J. Comp. Phys. 225 (2007) 785-807.
[64] S. Zahedi, K. Gustavsson, and G. Kreiss, "A conservative level set method for contact line dynamics," J. Comp. Phys. 228 (2009) 6361-6375.
[65] T.W.H. Shue, C.H. Yu, and P.H. Chiu, "Development of a dispersively accurate conservative level set scheme for capturing interface in two-phase flows," J. Comp. Phys. 228 (2009) 661-686.
[66] J.A. Sethian and P. Smereka, "Level Set Methods for Fluid Interfaces," Annu. Rev. Fluid Mech. 35 (2003) 341-372.
[67] J.A. Sethian, “Level Set Methods and Fast Marching Methods,” Cambridge University Press, 1999.
[68] S. Osher, J.A. Sethian, “Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79 (1988) 12-49.
[69] Comsol Multiphysics User’s Guide, Comsolab, 2008
[70] J. Donea, A. Huerta, J.P. Ponthot and A. Rodriguez-Ferran, “Arbitrary Lagrangian-Eulerian methods,” Encyc. Comp. Mech. (2004) 1-38
[71] T. Uchiyama, "ALE finite element method for gas-liquid two-phase flow including moving boundary based on an incompressible two-fluid model," Nuclear Engineering and Design 205 (2001) 69-82.
[72] F. Duarte, R. Gormaz, and S. Natesan, "Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries," Computer Methods in Appl. Mech. Eng. 193 (2004) 4819-4836.
[73] J. U. Brackbill, D. B. Kothe, C. Zemach, “A continuum method for modeling surface tension,” J. Comp. Phys. 100 (1991) 335-354.
[74] F.P. Incropera and D.P. Dewitt, “Fundamentals of Heat and Mass Transfer,” New York: John Wiley and Sons, 2002.
[75] H. Nagai, F. Rossignol, Y. Nakata, T. Tsurue, M. Suzuki, and T. Okutani, “Thermal conductivity measurement of liquid materials by a hot-disk method in short-duration microgravity environments,” Mater. Sci. Eng. A 276 (2000) 117-123.
[76] Power Chemical Corporation, “Characteristics PCC silicone oil,” (2006).
[77] J. W. Naughton and M. Sheplak, “Modern development in shear-stress measurement,” Progress in Aerospace Sciences 38 (2002) 515-570.
[78] A. Sommers, A.M. Jacobi, “Calculating the volume of water droplets on topographically-modified, mirco-grooved aluminum surface,” Int. Refrigeration and Air Conditioning Conference (2008) paper 931.
[79] “Mathamatica is Wolfram’s original, flagship product-primarily aimed at technical computing for R&D and education”.