微通道中流体扩散和混合机理及其微混合器的研究
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摘要
本文是以数值模拟为主对影响微通道中流体扩散和混合规律的因素进行研
究,并在此基础上设计出高效率的微混合器。
首先,在介绍微通道在流体扩散和混合规律的应用背景和研究状况的基础
上,利用流体力学的基本方程,对通道宽度在50-500微米的微通道中流体扩散
和混合进行数值模拟和实验研究,模拟结果显示:Reynolds数对扩散具有决定性
的影响,Reynolds数越大,扩散越小,混合越不充分。此外,T-sensor的结构参
数对扩散的效率有很大影响,如扩散通道宽度小时扩散快,入流角度大时扩散快,
弯曲通道的扩散和混合效果比直通道要好得多。扩散尺寸/通道深度(d/w)对扩
散也有一定的影响,当(d/w)较大时,沿w的浓度分布明显为抛物线,此时上
下壁面的影响是主要的;反之,当d/w较小时,沿d、w二个方向的速度分布都
为抛物线,上下左右壁面影响都不可忽略。实验研究结果部分验证了数值模拟结
果。
其次,在介绍和了解微混合器的原理和混沌混合理论的基础上,根据马蹄型
混沌对流循环的理论设计了一种新型的螺旋式微混合器,为说明其混合效果,对
不同组分流体在分合式微通道中的扩散和混合进行了数值模拟,结果发现在小
Reynolds数下,由于流体分层增加了接触面积,大大加快了流体混合的速度,其
混合效果不仅比直通道,而且比蛇形通道明显好。随着Reynolds数的增大,对
流作用得到加强,分合式通道的混合效果与直通道相比提高更明显,但与蛇形通
道相差无几,所得结果为微流芯片的被动式混合提供了一种新的形式。同时还设
计了一种主动式磁性混合器,分析了影响其混合效果的主要因素。
第三,为了解和解决毛细管电泳通道接头带来的ζ势的变化和管径变化对
毛细管电泳分离效率和采样的影响,分别利用流体力学的全N-S方程(把电场
力考虑为体积力加到方程内),结合电势方程,对毛细管内的流场进行数值模拟,
结果显示:在ζ势较小的情况下,由于速度剖面的内凹不明显,这时ζ和尺寸
的增大对速度剖面的畸变影响不大,但是在ζ势较大的情况下,由于沿程反向压
This thesis is mainly related to the numerical study on the diffusion and mixing in the micro-channel flow. Based on it, the micro-mixers with high efficiency are designed. The thesis can be divided into three parts.In the first part, the numerical simulation of the diffusion-based mixing of the species in the micro-channel with the width of 50-500 microns and some experiments were carried out on the basis of the introduction of the research background about this theme, using the basic equations of the fluid dynamics. The numerical results show that the Reynolds number crucially influences the diffusion and mixing in the micro-channel, the lower the Reynolds is, the wider the species diffuse, furthermore, the dimension and shape of the micro-channel of T-sensor also play an important role in the diffusion of the flow. The smaller the channel width is, the more obviously the species diffuse, and the inlet angle has a reversed impact on the diffusion process. The ratio of the diffusion dimension (d) to the channel width (w) also plays a role in diffusion, the velocity distribution along diffusion dimension is almost the same, and the velocity contribution along channel width is obviously parabola when the ratio d/w is large, so the influence of upper and lower wall is primary. Conversely, when d/w is small, the velocity contribution along both directions are parabola, thereby the influence of all walls (upper and lower, right and left) could not be ignored. The experimental results validate partially the numerical results.In the second part, after introducing the various mixing principles and chaotic mixing theory, a new helical micro-mixer with horseshoe map was designed. In order to illuminate the mixing efficiency of the helical micro-mixer, the diffusion and mixing of the multi-species in the helical micro-channel were simulated numerically, the numerical results show that the mixing efficiency is much better than that in the straight micro-channel and obviously better than that in the serpentine micro-channel because of the layered structure and more contact areas when the Reynolds number is
low. The mixing efficiency is also much better than that in the straight channel, but no obvious difference from that in serpentine micro-channel when the Reynolds number is high because the convection dominates the flow. A new micro-channel type with good passive mixing efficiency is provided. An active micro-mixer with magnetic bead and external magnetic field was designed, and then the factors impacting on the mixing efficiency for the micro-mixer were analysed.In the third part, the numerical simulation of Electroosmosis flow in a micro-channel was carried out by using the full Navier-Stokes equation (the electric force is considered as the volume force in this equation) in order that the separating efficiency and sampling quality can be improved when the zeta potential along the micro-channel wall becomes non-uniform because of the joint of different capillaries with different material and dimension. The numerical results indicate that a step change in ^-potential causes a significant variation in the velocity profile and in the pressure distribution. A step change both in ^-potential and dimension will result in more violent variation of the velocity profile near the joint. This variation will reduce the separation efficiency and sampling quality. However, the small difference between the maximum velocity and minimum velocity is benefit to the separation efficiency of the capillary electrophoresis when narrowing channel with bigger step-changed ^-potential.
究,并在此基础上设计出高效率的微混合器。
首先,在介绍微通道在流体扩散和混合规律的应用背景和研究状况的基础
上,利用流体力学的基本方程,对通道宽度在50-500微米的微通道中流体扩散
和混合进行数值模拟和实验研究,模拟结果显示:Reynolds数对扩散具有决定性
的影响,Reynolds数越大,扩散越小,混合越不充分。此外,T-sensor的结构参
数对扩散的效率有很大影响,如扩散通道宽度小时扩散快,入流角度大时扩散快,
弯曲通道的扩散和混合效果比直通道要好得多。扩散尺寸/通道深度(d/w)对扩
散也有一定的影响,当(d/w)较大时,沿w的浓度分布明显为抛物线,此时上
下壁面的影响是主要的;反之,当d/w较小时,沿d、w二个方向的速度分布都
为抛物线,上下左右壁面影响都不可忽略。实验研究结果部分验证了数值模拟结
果。
其次,在介绍和了解微混合器的原理和混沌混合理论的基础上,根据马蹄型
混沌对流循环的理论设计了一种新型的螺旋式微混合器,为说明其混合效果,对
不同组分流体在分合式微通道中的扩散和混合进行了数值模拟,结果发现在小
Reynolds数下,由于流体分层增加了接触面积,大大加快了流体混合的速度,其
混合效果不仅比直通道,而且比蛇形通道明显好。随着Reynolds数的增大,对
流作用得到加强,分合式通道的混合效果与直通道相比提高更明显,但与蛇形通
道相差无几,所得结果为微流芯片的被动式混合提供了一种新的形式。同时还设
计了一种主动式磁性混合器,分析了影响其混合效果的主要因素。
第三,为了解和解决毛细管电泳通道接头带来的ζ势的变化和管径变化对
毛细管电泳分离效率和采样的影响,分别利用流体力学的全N-S方程(把电场
力考虑为体积力加到方程内),结合电势方程,对毛细管内的流场进行数值模拟,
结果显示:在ζ势较小的情况下,由于速度剖面的内凹不明显,这时ζ和尺寸
的增大对速度剖面的畸变影响不大,但是在ζ势较大的情况下,由于沿程反向压
This thesis is mainly related to the numerical study on the diffusion and mixing in the micro-channel flow. Based on it, the micro-mixers with high efficiency are designed. The thesis can be divided into three parts.In the first part, the numerical simulation of the diffusion-based mixing of the species in the micro-channel with the width of 50-500 microns and some experiments were carried out on the basis of the introduction of the research background about this theme, using the basic equations of the fluid dynamics. The numerical results show that the Reynolds number crucially influences the diffusion and mixing in the micro-channel, the lower the Reynolds is, the wider the species diffuse, furthermore, the dimension and shape of the micro-channel of T-sensor also play an important role in the diffusion of the flow. The smaller the channel width is, the more obviously the species diffuse, and the inlet angle has a reversed impact on the diffusion process. The ratio of the diffusion dimension (d) to the channel width (w) also plays a role in diffusion, the velocity distribution along diffusion dimension is almost the same, and the velocity contribution along channel width is obviously parabola when the ratio d/w is large, so the influence of upper and lower wall is primary. Conversely, when d/w is small, the velocity contribution along both directions are parabola, thereby the influence of all walls (upper and lower, right and left) could not be ignored. The experimental results validate partially the numerical results.In the second part, after introducing the various mixing principles and chaotic mixing theory, a new helical micro-mixer with horseshoe map was designed. In order to illuminate the mixing efficiency of the helical micro-mixer, the diffusion and mixing of the multi-species in the helical micro-channel were simulated numerically, the numerical results show that the mixing efficiency is much better than that in the straight micro-channel and obviously better than that in the serpentine micro-channel because of the layered structure and more contact areas when the Reynolds number is
low. The mixing efficiency is also much better than that in the straight channel, but no obvious difference from that in serpentine micro-channel when the Reynolds number is high because the convection dominates the flow. A new micro-channel type with good passive mixing efficiency is provided. An active micro-mixer with magnetic bead and external magnetic field was designed, and then the factors impacting on the mixing efficiency for the micro-mixer were analysed.In the third part, the numerical simulation of Electroosmosis flow in a micro-channel was carried out by using the full Navier-Stokes equation (the electric force is considered as the volume force in this equation) in order that the separating efficiency and sampling quality can be improved when the zeta potential along the micro-channel wall becomes non-uniform because of the joint of different capillaries with different material and dimension. The numerical results indicate that a step change in ^-potential causes a significant variation in the velocity profile and in the pressure distribution. A step change both in ^-potential and dimension will result in more violent variation of the velocity profile near the joint. This variation will reduce the separation efficiency and sampling quality. However, the small difference between the maximum velocity and minimum velocity is benefit to the separation efficiency of the capillary electrophoresis when narrowing channel with bigger step-changed ^-potential.
引文
1.王琪民.微型机械导论.合肥:中国科学技术大学出版社,2003
2.周兆英等.微型机电系统技术.中国机械工程,11(1),163-168(2000)
3.丁衡高.中国科技前沿——微机电系统.北京:高等教育出版社,1998
4. Nadim Maluf. An Introduction To Microeletromechanical Systems Engineering. Artech House, 2000
5. T. Fukuda and W. Menz. Micro Mechanical System. Elsevier, 1998
6. Ilene J. Busch-Vishnies. Electromechanical Sensors and Actuators. Springer, 1998
7. Stemme G. et al. A Sub-micron particles filter in Silicon channels. Sensors and Actuators, 431(4), 21-23(1990)
8. Liwei Lin. Curriculum Development in Microeletromechanical Systems in Mechanical Engineering. IEEE Transactions on Education, 44(1), 1-10 (2001)
9.朱静等.纳米材料和器件.北京:清华大学出版社,2003
10. Moser L., Jobst G. et al. Miniaturized thin flim gultamate and gulamine biosensors. Biosens Bioelectron, 10, 527-532 (1995)
11. Colyer C. L., T. Tang et al. Clinical potential of microchip capillary electrophoresis systems. Electrophoresis, 18, 1733-1741 (1997)
12. Osbourn, Damon M.; Lunte, Craig E.. On-Column Electrochemical Detection for Microchip Capillary Electrophoresis. Analytical Chemistry, 75(11), 2710-14 (2003)
13. Arnett, Stacy D; Lunte, Craig E. Investigation of the mechanism of pH-mediated stacking of anions for the analysis of physiological samples by capillary electrophoresis. Electrophoresis, 24(11), 1745-1752 (2003)
14. Oki A, Adachi S, Takamura Y, Ishihara K. Electroosmosis injection of blood serum into biocompatible microcapillary chip fabricated on quartz plate. Electrophoresis. 22(2), 341-347 (2001)
15.仲武,陈云飞.毛细管电渗流微泵的流体动力学的数值仿真.机械工程学报,40(2),73-78(2004)
16.陈令新,关亚风等.电渗泵中电渗流的控制.分析化学,31(5),619-623(2003)
17. Mohamed Gad-el-Hak. The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture. Journal of Fluids Engineering, 121 (3), 5-38 (1999)
18. Service R. F.. Coming soon: the pocket DNA sequencer . Science, 282, 399-401 (1998)
19. Burns M. A. et al.. An integrated nanoliter DNA analysis device. Science, 282, 484-490 (1998)
20. Ehrlich D. J. and Matsudaira P.. Micro fluidic devices for DNA analysis. Trends Biotechnol., 17, 315-319 (1999)
21. Khandurina J., Mcknight T. E. et al. Integrated system for rapid PCR-based DNA analysis inmicrofluidicdevices. Anal. Chem., 72, 2995-3000 (2000)
22. Weigl B. and Yager P.. Micro fluidic diffusion-based separation and detection. Science, 283, 346-347 (1999)
23. Kamholz A. E. and Yager P. Theoretical analysis of molecular diffusion in pressure-driven flow in micro fluidic channels. Biophys. J., 80, 155-160 (2001)
24. Kamholz A. E., Schilling E. A. and Yager P. Optical Measurement of Transverse Molecular Diffusion in a Microchannel. Biophys J., 80(4), 1967-1972 (2001)
25. Ruijin Wang, Jianzhong Lin, and Zhi-hua Li. Study on the Impacting Factors of Transverse Diffusion in the Micro-Channels of T-Sensors. Journal of Nanoscience andNanotechnology, 5(7), 1-5 (2005)
26. Robin H L, Mark A S, Kendra V S et al. Passive mixing in a three-dimensional serpentine microchannel. Journal of Microelectromec. Sys., 9(2), 190-197 (2000)
27. Mengeaud, V., Josserand, J. and Girault, H. H. Mixing Processes in a Zigzag Microchannel: Finite Element Simulations and Optical Study. Anal. Chem., 74, 4279-4286 (2002)
28. Wang, H., Iovenitti, P., Harvey, E. and MasoodS. Optimizing layout of obstacles for enhanced mixing in microchannels. Smart Mater. Struct., 11, 662-667(2002)
29. StroockA. D., DertingerS. K. W., AjdariA., MezicI., Stone H. A., Whitesides G. M. Chaotic Mixer forMicrochannels. Science, 295, 647-651 (2002)
30. Johnson T. J., Ross D. and Locasciol. E. Rapid Microfluidic Mixing. Anal. Chem., 74, 45-51 (2002)
31. Y. K. Lee et al.. Chaotic Mixing in Electrokinetically and Pressure Driven Micro Flows. Proc. MEMS Conf., 2001, 483-486
32. Hiroaki Suzuki, Nobuhide Kasagi. Chaotic mixing of magnetic beads in microcell seoarator. Proc. 3rd Int. Symp. Turbulence and Shear Flow Phenomena Sendai, Japan, June 24-27, 2003, 817-823
33. M. K. Jeonl, J-H. Kim. Design of a recycle micromixer. 7th International Conference on Miniaturized Chemical and Biochemical Analysts Systems, Squaw Valley, California USA October 5-9, 2003, 109-112
34. Qian, S. and Bau, H. H.. A Chaotic Electroosmotic Stirrer. Anal. Chem. 74, 3616-362 (2002)
35. I. Meisel and P. Ehrhard. Simulation of electrically excited flows in microchannels for mixing application. Nanotech, 1, 62-65 (2002)
36. Joanne Deval. A Dielectrophoretic Chaotic Mixer. IEEE 0-7803-7185-2/02 (2002)
37. Hiroaki Suzuki. Particle tracking velocimetry measurement of chaotic mixing in a micromixer. The International Symposium on Micro-Mechanical Engineering December 1-3, 2003, 212-218
38. R. Rong, J. W. Choi, and C. H. Ahn. A Novel Magnetic Chaotic Mixer for in-flow mixing of magnetic beads. 7th International Conference on Miniaturized Chemical and Biochemical Analysis Systems, Squaw Valley, California USA, October 5-9, 2003, 335-339
39. Ajay A. Deshmukh, Dorian Liepmann, and Albert P. Pisano. Continuous micromixer with pulsatile micropump. In Proceedings of the 2000 Solid-State Sensor and Actuator Workshop, Transducers Research Foundation, Cleveland OH, IEEE,73-76 (2000).
40. S. D. Muller, I. Mezi, J. H. Walther, and P. Koumoutsakos. Transverse momentum micromixer optimization with evolution strategies. Computers and Fluids, 33, 521-531 (2004)
41. Yi-Kuen Lee. Chaotic mixing in eleltrokinetically and pressure driven micro flows. The 14th IEEE Workshop on MEMS Interlaken, Switzerland, Jan., 2001, 483-487
42. Arash Dodge a, Marie-Caroline Jullien. An example of a chaotic micromixer: the cross-channel micromixer. C. R. Physique, 5, 557-563(2004)
43. Jens Ducree, Thilo Brenner, Thomas Glatzel, Roland Zengerle. Ultrafast micromixer by Coriolis-induced multi-lamination of centrifugal flow. ACTUATOR2004, 9th International Conference on New Actuators, Bremen, Germany, 14-16 June 2004, 533-537
44. Yi, M., Bau, H. H. and Hu, H. Peristaltically induced motion in a closed cavity with two vibrating walls. Physics of Fluids, 14(1), 184-197(2002)
45. Julio M. Ottino and Stephen Wiggins. Designing Optimal Micromixers. Science, 305(23), 485-486 (2004)
46. Lu, L. H., Ryu, K. S. and Liu, C. A Magnetic Microstirrer and Array for Microfluidic Mixing. Journal of MicroelectromechanicalSystems, 11(5), 462-470(2002)
47. Kee Suk Ryu, Kashan Shaikh. Micro magnetic stir-bar mixer integrated with parylene microfluidic channels. Lab Chip, 4, 608-613 (2004)
48. V. Vivek, Y. Zeng; E. S. Kim. Novel acoustic-wave micromixer. Micro electro mechanical systems, MEMS 2000, 13th annual international conference, 668-673
2.周兆英等.微型机电系统技术.中国机械工程,11(1),163-168(2000)
3.丁衡高.中国科技前沿——微机电系统.北京:高等教育出版社,1998
4. Nadim Maluf. An Introduction To Microeletromechanical Systems Engineering. Artech House, 2000
5. T. Fukuda and W. Menz. Micro Mechanical System. Elsevier, 1998
6. Ilene J. Busch-Vishnies. Electromechanical Sensors and Actuators. Springer, 1998
7. Stemme G. et al. A Sub-micron particles filter in Silicon channels. Sensors and Actuators, 431(4), 21-23(1990)
8. Liwei Lin. Curriculum Development in Microeletromechanical Systems in Mechanical Engineering. IEEE Transactions on Education, 44(1), 1-10 (2001)
9.朱静等.纳米材料和器件.北京:清华大学出版社,2003
10. Moser L., Jobst G. et al. Miniaturized thin flim gultamate and gulamine biosensors. Biosens Bioelectron, 10, 527-532 (1995)
11. Colyer C. L., T. Tang et al. Clinical potential of microchip capillary electrophoresis systems. Electrophoresis, 18, 1733-1741 (1997)
12. Osbourn, Damon M.; Lunte, Craig E.. On-Column Electrochemical Detection for Microchip Capillary Electrophoresis. Analytical Chemistry, 75(11), 2710-14 (2003)
13. Arnett, Stacy D; Lunte, Craig E. Investigation of the mechanism of pH-mediated stacking of anions for the analysis of physiological samples by capillary electrophoresis. Electrophoresis, 24(11), 1745-1752 (2003)
14. Oki A, Adachi S, Takamura Y, Ishihara K. Electroosmosis injection of blood serum into biocompatible microcapillary chip fabricated on quartz plate. Electrophoresis. 22(2), 341-347 (2001)
15.仲武,陈云飞.毛细管电渗流微泵的流体动力学的数值仿真.机械工程学报,40(2),73-78(2004)
16.陈令新,关亚风等.电渗泵中电渗流的控制.分析化学,31(5),619-623(2003)
17. Mohamed Gad-el-Hak. The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture. Journal of Fluids Engineering, 121 (3), 5-38 (1999)
18. Service R. F.. Coming soon: the pocket DNA sequencer . Science, 282, 399-401 (1998)
19. Burns M. A. et al.. An integrated nanoliter DNA analysis device. Science, 282, 484-490 (1998)
20. Ehrlich D. J. and Matsudaira P.. Micro fluidic devices for DNA analysis. Trends Biotechnol., 17, 315-319 (1999)
21. Khandurina J., Mcknight T. E. et al. Integrated system for rapid PCR-based DNA analysis inmicrofluidicdevices. Anal. Chem., 72, 2995-3000 (2000)
22. Weigl B. and Yager P.. Micro fluidic diffusion-based separation and detection. Science, 283, 346-347 (1999)
23. Kamholz A. E. and Yager P. Theoretical analysis of molecular diffusion in pressure-driven flow in micro fluidic channels. Biophys. J., 80, 155-160 (2001)
24. Kamholz A. E., Schilling E. A. and Yager P. Optical Measurement of Transverse Molecular Diffusion in a Microchannel. Biophys J., 80(4), 1967-1972 (2001)
25. Ruijin Wang, Jianzhong Lin, and Zhi-hua Li. Study on the Impacting Factors of Transverse Diffusion in the Micro-Channels of T-Sensors. Journal of Nanoscience andNanotechnology, 5(7), 1-5 (2005)
26. Robin H L, Mark A S, Kendra V S et al. Passive mixing in a three-dimensional serpentine microchannel. Journal of Microelectromec. Sys., 9(2), 190-197 (2000)
27. Mengeaud, V., Josserand, J. and Girault, H. H. Mixing Processes in a Zigzag Microchannel: Finite Element Simulations and Optical Study. Anal. Chem., 74, 4279-4286 (2002)
28. Wang, H., Iovenitti, P., Harvey, E. and MasoodS. Optimizing layout of obstacles for enhanced mixing in microchannels. Smart Mater. Struct., 11, 662-667(2002)
29. StroockA. D., DertingerS. K. W., AjdariA., MezicI., Stone H. A., Whitesides G. M. Chaotic Mixer forMicrochannels. Science, 295, 647-651 (2002)
30. Johnson T. J., Ross D. and Locasciol. E. Rapid Microfluidic Mixing. Anal. Chem., 74, 45-51 (2002)
31. Y. K. Lee et al.. Chaotic Mixing in Electrokinetically and Pressure Driven Micro Flows. Proc. MEMS Conf., 2001, 483-486
32. Hiroaki Suzuki, Nobuhide Kasagi. Chaotic mixing of magnetic beads in microcell seoarator. Proc. 3rd Int. Symp. Turbulence and Shear Flow Phenomena Sendai, Japan, June 24-27, 2003, 817-823
33. M. K. Jeonl, J-H. Kim. Design of a recycle micromixer. 7th International Conference on Miniaturized Chemical and Biochemical Analysts Systems, Squaw Valley, California USA October 5-9, 2003, 109-112
34. Qian, S. and Bau, H. H.. A Chaotic Electroosmotic Stirrer. Anal. Chem. 74, 3616-362 (2002)
35. I. Meisel and P. Ehrhard. Simulation of electrically excited flows in microchannels for mixing application. Nanotech, 1, 62-65 (2002)
36. Joanne Deval. A Dielectrophoretic Chaotic Mixer. IEEE 0-7803-7185-2/02 (2002)
37. Hiroaki Suzuki. Particle tracking velocimetry measurement of chaotic mixing in a micromixer. The International Symposium on Micro-Mechanical Engineering December 1-3, 2003, 212-218
38. R. Rong, J. W. Choi, and C. H. Ahn. A Novel Magnetic Chaotic Mixer for in-flow mixing of magnetic beads. 7th International Conference on Miniaturized Chemical and Biochemical Analysis Systems, Squaw Valley, California USA, October 5-9, 2003, 335-339
39. Ajay A. Deshmukh, Dorian Liepmann, and Albert P. Pisano. Continuous micromixer with pulsatile micropump. In Proceedings of the 2000 Solid-State Sensor and Actuator Workshop, Transducers Research Foundation, Cleveland OH, IEEE,73-76 (2000).
40. S. D. Muller, I. Mezi, J. H. Walther, and P. Koumoutsakos. Transverse momentum micromixer optimization with evolution strategies. Computers and Fluids, 33, 521-531 (2004)
41. Yi-Kuen Lee. Chaotic mixing in eleltrokinetically and pressure driven micro flows. The 14th IEEE Workshop on MEMS Interlaken, Switzerland, Jan., 2001, 483-487
42. Arash Dodge a, Marie-Caroline Jullien. An example of a chaotic micromixer: the cross-channel micromixer. C. R. Physique, 5, 557-563(2004)
43. Jens Ducree, Thilo Brenner, Thomas Glatzel, Roland Zengerle. Ultrafast micromixer by Coriolis-induced multi-lamination of centrifugal flow. ACTUATOR2004, 9th International Conference on New Actuators, Bremen, Germany, 14-16 June 2004, 533-537
44. Yi, M., Bau, H. H. and Hu, H. Peristaltically induced motion in a closed cavity with two vibrating walls. Physics of Fluids, 14(1), 184-197(2002)
45. Julio M. Ottino and Stephen Wiggins. Designing Optimal Micromixers. Science, 305(23), 485-486 (2004)
46. Lu, L. H., Ryu, K. S. and Liu, C. A Magnetic Microstirrer and Array for Microfluidic Mixing. Journal of MicroelectromechanicalSystems, 11(5), 462-470(2002)
47. Kee Suk Ryu, Kashan Shaikh. Micro magnetic stir-bar mixer integrated with parylene microfluidic channels. Lab Chip, 4, 608-613 (2004)
48. V. Vivek, Y. Zeng; E. S. Kim. Novel acoustic-wave micromixer. Micro electro mechanical systems, MEMS 2000, 13th annual international conference, 668-673