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In order to understand the evolution and flow structure within vertical vibrating droplets on hydrophobic surfaces, a three-dimensional model of the vibrating droplet is developed, and the dynamic contact angle of the vibrating droplet is considered. The numerical simulations are performed for the droplet attached to the vertical vibrating plane by the VOF-CSF method, and the four resonance modes of the droplets are obtained. The evolution of modes (2, 4, 6, and 8), internal flow structures and the variation of the dynamic contact angle are predicted. With the change of the vibration acceleration, the droplet can express a wealth of modes, and the specific mode depends on the frequency of the vibrating acceleration. Based on this model, in this paper the internal flow field structure of the droplet is further analyzed. In mode 2 and mode 4, a Y-shaped flow is generated from the bottom of the droplet, while in mode 6 and mode 8, there is a symmetrical eddy flow. And the higher the order of the resonance mode, the larger the average value of the internal velocity of the droplet is. The dynamic contact angle of the vibrating droplet obviously deviates from the static contact angle, indicating the necessity to consider the dynamic contact angle in simulating the vertical vibrating of droplet. The simulation results are compared with the experimental results from the literature, showing that they are in good agreement with each other.
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Keywords:
- vibration droplet /
- modes /
- micro-flow /
- numerical simulation
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图 1 三维模拟计算区域(3 mm × 3 mm × 3 mm)
Figure 1. 3D computational domain (3 mm × 3 mm × 3 mm).
图 4 在2, 4, 6和8模态下液滴内部的三维流场图
Figure 4. The three-dimensional flow field inside the droplet at modes 2, 4, 6, and 8.
图 6 在2, 4, 6和8共振模态下液滴内部平均速度随时间的变化
Figure 6. The variations of the velocity with time at modes 2, 4, 6, and 8.
图 8 在2, 4, 6和8共振模态下液滴动态接触角随时间的变化
Figure 8. The variations of the dynamic contact angle with time at modes 2, 4, 6, and 8.
图 9 在2, 4, 6和8共振模态下液滴润湿面积随时间的变化
Figure 9. The variations of the wetting area with time at modes 2, 4, 6, and 8.
表 1 共振频率的理论值和实验值对比
Table 1. Comparisons of theoretical and experimental results for resonance frequency of a 5
${\text{μ}}\mathrm{L}$ droplet.Mode n Rayleigh方程
计算值 f/Hz共振频率
实验值 f/Hz2 96 85 4 288 226 6 526 469 8 804 635 
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[1] Singhal V, Garimella S V, Raman A 2004 Appl. Mech. Rev. 57 191
Google Scholar
[2] Vukasinovic B, Smith M K, Glezer A 2004 Phys. Fluid 16 306
Google Scholar
[3] Nisisako T, Torri T 2007 Adv. Mater. 19 1489
Google Scholar
[4] Shan Y G, Wang Y L, Coyle T 2013 Appl. Therm. Eng. 51 690
Google Scholar
[5] 高超, 袁俊杰, 曹进军, 杨荟楠, 单彦广 2019 68 140204
Google Scholar
Gao C, Yuan J J, Cao J J, Yang H N, Shan Y G 2019 Acta Phys. Sin. 68 140204
Google Scholar
[6] 张永建 2015 博士学位论文 (西安: 西北工业大学)
Zhang Y J 2015 Ph. D. Dissertation (Xian: Northwestern Polytechnical University) (in Chinese)
[7] Kabi P, Chattopadhyay B, Bhattacharyya S, Chaudhuri S, Basu S 2018 Langmuir 34 12642
Google Scholar
[8] Kelvin L 1882 Mathematical and Physical Papers (London: Cambridge University Press) pp178−181
[9] Rayleigh L 1879 Proc. R. Soc. London 29 71
Google Scholar
[10] Lamb H 1932 Hydrodynamics (London: Cambridge University press) p606
[11] Strani M, Sabetta F 1984 J. Fluid Mech. 141 233
Google Scholar
[12] Ko S H, Lee S J, Kang K H 2009 Appl. Phys. Lett. 94 194102
Google Scholar
[13] 邵学鹏, 解文军 2012 61 134302
Google Scholar
Shao X P, Xie W J 2012 Acta Phys. Sin. 61 134302
Google Scholar
[14] Brunet P, Eggers J, Deegan R D 2009 Eur. Phys. J-Spec. Top. 166 11
Google Scholar
[15] Noblin X, Buguin A, Brochard-Wyart F 2009 Eur. Phys. J. E 166 7
Google Scholar
[16] Dong L, Chaudhury A, Chaudhury M K 2006 Eur. Phys. J. E 21 231
Google Scholar
[17] 周建臣, 耿兴国, 林可君, 张永建, 臧渡洋 2014 63 216801
Google Scholar
Zhou J C, Geng X G, Lin K J, Zhang Y J, Zang D Y 2014 Acta Phys. Sin. 63 216801
Google Scholar
[18] Noblin X, Buguin A, Brochard-Wyart F 2004 Eur. Phys. J. E 14 395
Google Scholar
[19] Shin Y S, Lim H C 2014 Eur. Phys. J. E 37 1
Google Scholar
[20] Kim H, Lim H C 2015 J. Phys. Chem. B 119 6740
Google Scholar
[21] Park C S, Kim H, Lim H C 2016 Exp. Therm. Fluid Sci. 78 112
Google Scholar
[22] Ramos S M M 2008 Nucl. Instrum. Methods Phys. Res., Sect. B 266 3143
Google Scholar
[23] Ehrhorn J, Semke W 2013 Folia Parasit. 5 243
Google Scholar
[24] Li Y and Umemura A 2014 Int. J. Multiphase Flow 60 64
Google Scholar
[25] James A J, Smith M K and Glezer A 2003 J. Fluid Mech. 476 29
Google Scholar
[26] 王瑞金, 张凯, 王刚 2007 Fluent技术基础与应用实例 (北京: 清华大学出版社) 第136−150页
Wang R J, Zhang K, Wang G 2007 Fluent Technology Foundation and Application Examples (Beijing: Tsinghua University Press) pp136−150 (in Chinese)
[27] Brackbill J U, Kothe D B, Zemach C 1992 J. Comput. Phys. 100 335
Google Scholar
[28] Chernova A A, Kopysov S P, Tonkov L E 2016 IOP Conference Series: Mater. Sci. Eng. 158 1
Google Scholar
[29] Kistler S F 1993 Wettability (New York: Marcel Dekker) pp311−429
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