本研究以數值方法比較三種不同幾何形狀之模組形成之二維平面蜿蜒微流道中流體的混合，這三種模組為基於鋸齒狀流道的準Ω形模組、Ω形模組與準方波形模組。網格法在高培克萊特數下，會因數值擴散高估混合度，且無法完全避免，因此本研究以蒙地卡羅法驗證反向粒子追跡綜合近似擴散模式模擬，並以後者之模擬結果討論雷諾數對流體流動與混合影響。模擬結果顯示在低雷諾數時流體混合以分子擴散為主，因模組中心線長度相同，流體在不同形狀模組之蜿蜒流道擴散的時間也大致相同，所得之混合度差異不大；在高雷諾數Re=81下，混合效率因模組幾何外形差異而有所不同，因其混合機制由對流主導，而渦流可拉伸摺疊流體，增加流體交界面之接觸面積，促使混合效率提升。各模組之蜿蜒流道流體混合效果的差異在於其流道結構是否容易引發渦流，準Ω形模組因流道具銳角轉彎較容易引發狄恩與分離渦流，因此以準Ω形模組組成的微混合器有最佳的流體混合結果。

We compare the flow and mixing in planar serpentine micromixers composed of different modules by numerical methods, and the three modules considered are the quasi-omega-shaped modules based on zigzag shaped channels, the omega-shaped modules, the quasi-square-wave-shaped modules. The degrees of mixing obtained by the grid method are overestimated due to numerical diffusion at the high Peclet number. Thus, we use particle-tracking simulation with ADM (approximate diffusion model) to investigate the flow and mixing at different Reynolds numbers. The results obtained by the Monte Carlo method verifies the results of particle-tracking simulation with ADM. The simulation results show that fluid mixing is dominated by molecular diffusion, and the degrees of mixing obtained for different microchannels are nearly identical at lower Reynolds numbers. At high Reynolds number, the flow starts to develop vortices, which stretch and distort the interface between fluid streams to enhance fluid mixing. The serpentine micromixer with the quasi-omega-shaped modules that have sharp bends may induce Dean vortices and separation vortices in the mixing channel, and so the mixing performance of this micromixer is the best among the micromixers considered at Re=81.

[1] A. Manz, N. Graber, and H. M. Windmer, “Miniaturized total chemical analysis system: a novel concept for chemical sensing,” Sensors and Actuators, B: Chemical, vol. 1, no. 1-6, pp. 244-248, 1990.
[2] A. Maha, D. O. Barrett, D. E. Nikitopoulosa, S. A. Soper, and M. C. Murphy, “Simulation and design of micromixers for microfluidic devices,” Proceedings SPIE 5345, Microfluidics, BioMEMS, and Medical Microsystems II, pp. 183-194. San Jose, California, United States, Dec. 23, 2003.
[3] R. Miyake, T. S. Lammerink, M. Elwenspoek, and J. H. Fluitman, “Micro mixer with fast diffusion,” presented at the 6th IEEE International Micro Electro Mechanical Systems Conference (MEMS 93), pp. 248-253. Fort Lauderdale, Florida, United States, Aug. 06, 1993.
[4] D. Gobby, P. Angeli, and A. Gavriilidis, “Mixing characteristics of T-type microfluidic mixers,” Journal of Micromechanics and Microengineering, vol. 11, no. 2, pp. 126-132, 2001.
[5] R. H. Liu, M. A. Stremler, K. V. Sharp, M. G. Olsen, J. G. Santiago, R. J. Adrian, H. Aref, and D. J. Beebe, “Passive mixing in a three-dimensional serpentine microchannel,” Journal of Microelectromechanical Systems, vol. 9, no. 2, pp. 190-197, 2000.
[6] R. A. Vijayendran, K. M. Motsegood, D. J. Beebe, and D. E. Leckband, “Evalution of a three-dimensional micromixer in a surface-based biosensor,” Langmuir, vol. 19, no. 5, pp. 1824-1828, 2003.
[7] A. D. Strook, S. K. Dertinger, A. Ajdari, I. Mezic, H. A. Stone, and G. M. Whitesides, “Chaotic mixer for microchannels,” Science, vol. 295, pp. 647-651, 2002.
[8] S. H. Wong, P. Bryant, M. Ward, C. Wharton, “Investigation of mixing in a cross-shaped micromixer with static mixing elements for reaction kinetics studies,” Sensors and Actuators B: Chemical, vol. 95, no. 1-3, pp. 414-424, 2003.
[9] N. Kockmann, C. Föll, and P. Woias, “Flow regimes and mass transfer characteristics in static micro mixers,” Proceedings of International Society for Optics and Photonics, vol. 4982, pp. 319-330, 2003.
[10] D. Bothe, C. Stemich, and H.-J. Warnecke, “Fluid mixing in a T-shaped micro-mixer,” Chemical Engineering Science, vol. 61, no. 9, pp. 2950-2958, 2006.
[11] F. Jiang, K. S. Drese, S. Hardt, M. Küpper, and F. Schöfeld, “Helical flows and chaotic mixing in curved micro channel,” AIChE Journal, vol. 50, no. 9, pp. 2297-2305, 2004.
[12] S. Hossain, M. A. Ansari, and K. Y. Kim, “Evaluation of the mixing performance of the three passive micromixers,” Chemical Engineering Journal, vol. 150, pp. 492-501, 2009.
[13] M.-Y. Kuo, “A backward random-walk Monte Carlo simulation of liquid mixing in a serpentine microchannel,” M.S. thesis, Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, 2015.
[14] A. A. Al-Halhouli, A. Alshare, M. Mohsen, M. Matar, A. Dietzel, and S. Büttgenbach, “Passive micromixers with interlocking semi-circle and omega-shaped modules: Experiments and simulations,” Micromachines, vol. 6, no. 7, pp. 953-968, 2015.
[15] K.-C. Hsu, “Numerical simulation for mixing of fluids in a serpentine microchannel with Z-type and reversed Z-type modules,” M.S. thesis, Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, 2017.
[16] R.-T. Tsai and C.-Y. Wu, “Multidirectional vortices mixing in three-stream micromixers with two inlets,” Microsystem Technologies, vol. 18, no. 6, pp. 779-786, 2012.
[17] J. D. Hoffman and S. Frankel, Numerical methods for engineers and scientists.
New York, United States : CRC Press, pp. 504-514, 2001.
[18] T. Matsunaga, H. J. Lee, and K. Nishino, “An approach for accurate simulation of liquid mixing in a T-shape micromixer,” Lab on a Chip, vol. 13, no. 8, pp. 1515-1521, 2013.
[19] V. Özceyhan and M. Sen, “Particle-tracking random-walk computation of high-Peclet-number convection,” Numerical Heat Transfer, Part A: Applications, vol. 50, no. 7, pp. 607-622, 2006.
[20] P. Szymczak and A. J. C. Ladd “Boundary conditions for stochastic solutions of the convection-diffusion equation,” Physical Review E, vol. 68, no. 3, pp. 036704, 2003.
[21] C. Galletti, M. Roudgar, E. Brunazzi, and R. Mauri, “Effect of inlet conditions on the engulfment pattern in a T-shaped micro-mixer, ” Chemical Engineering Science, vol. 185, pp. 300-313, 2012.
[22] S. A. Rani, B. Pitts, and P. S. Steward, “Rapid diffusion of fluorescent tracers into staphylococcus epidermidis biofilms visualized by time lapse microscopy,” Antimicrobial Agents Chemotherapy, vol. 49, no. 2, pp. 728-732, 2005.
[23] T. Matsunaga, K. Shibata, K. Murotani and S. Koshizuka, “Hybrid grid-particle method for fluid mixing simulation,” Computational Particle Mechanics, vol. 2, no. 3, pp. 233-246, 2015.
[24] H. Bockhorn, D. Mewes, W. Peukert, and H. J. Warnecke, Micro and Macro Mixing. Berlin, Germany : Springer, 2010.
[25] Z. Wu, N.-T. Nguyen, and X. Huang, “Nonlinear diffusive mixing in microchannels: theory and experiments,” Journal of Micromechanics and Microengineering, vol. 14, no. 4, pp. 604-611, 2004.
[26] A. Nealen, “An as-short-as-possible introduction to the least squares,weighted least squares and moving least squares methods for scattered data approximation and interpolation,” Technische Universität Darmstadt, Tech. Rep., 2004, URL:http://www.nealen.com/projects.
[27] J. L. Bentley, “Multidimensional binary search trees used for associative searching,” Communications of the Association for Computing Machinery, vol. 18, no. 9, pp. 509-517, 1975.
[28] S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman, and A. Y. Wu. “An optimal algorithm for approximate nearest neighbor searching fixed dimensions,” Journal of the Association for Computing Machinery, vol. 45, no. 6, pp. 891–923, 1998.
[29] T. Most and C. Bucher, “A moving least squares weighting function for the element-free Galerkin method which almost fulfills essential boundary conditions,” Structural Engineering and Mechanics, vol. 21, no. 3, pp. 315-332, 2005.