Structural stability analysis of spherical bubble clusters in acoustic cavitation fields

Liu Rui; Huang Chen-Yang; Wu Yao-Rong; Hu Jing; Mo Run-Yang; Wang Cheng-Hui

doi:10.7498/aps.73.20232008

Abstract

The upwelling growth and evolution of spherical bubble clusters appearing at one-quarter wavelength from the water surface in ultrasonic cavitation fields at frequencies of 28 kHz and 40 kHz are studied by high-speed photography. Due to the interactions among bubbles, the stable bubble aggregation occurs throughout the rise of the bubble cluster, whose vertical pressure difference leads to a more significant spreading in the upper part of the cluster in the standing-wave field. At 28 kHz, the rising speed is about 0.6 m/s, controlled by the primary acoustic field. After a violent collapse of the bubble clusters, the aggregating structure begins to hover near the water surface. The size and stability of the structure are affected by the frequency and pressure of the primary acoustic field. If two clusters are close to each other, the clusters deviate from the spherical shape, even trailing off, and eventually merge into a single bubble cluster. By considering the influence of water-air boundary, based on the mirror principle, a spherical bubble cluster model is developed to explore the structure stability of the clusters, and the modified dynamics equations are obtained. The effects of driving acoustic pressure amplitude, bubble number density, water depth, and bubble equilibrium radius on the optimal stable radius of the spherical bubble cluster are numerically analyzed by using the equivalent potentials at 28 kHz and 40 kHz. The results show that the optimal stabilizing radius of spherical bubble cluster is in a range of 1–2 mm, and it tends to decrease slightly with the increase of the driving acoustic pressure and bubble number density. It is worth noting that the nonlinearity is enhanced by increasing acoustic pressure, which may promote the stability of the cluster structure. The smaller the unstable equilibrium radius, the easier it is to grow, and the stable size at 40 kHz is slightly smaller than that at 28 kHz. Generally, spherical clusters first appear in a high-pressure region and then move to a low-pressure region. If the acoustic pressure drops below a certain critical value, bubble clusters disappear. The theoretical analysis is in good agreement with the experimental observation. The analysis of the growth and structural stability of spherical bubble cluster is helpful in understanding the behavioral modulation of bubbles.

Keywords:

Authors and contacts

Institute of Shaanxi Key Laboratory of Ultrasonics, Shaanxi Normal University, Xi’an 710062, China

Corresponding author: Wang Cheng-Hui, wangld001@snnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12374441, 11974232) and the Science and Technology Bureau of Yulin, China (Grant No. CXY-2022-178).

[1]	应崇福 2007 中国科学: 物理学力学天文学 37 129 Google Scholar Ying C F 2007 Sci. Sin. Phys. Mech. Astron. 37 129 Google Scholar
[2]	程效锐, 张舒研, 房宁 2018 应用化工 47 1753 Google Scholar Cheng X R, Zhang S Y, Fang N 2018 Appl. Chem. Ind. 47 1753 Google Scholar
[3]	Ehsan Z S, Christian G, Helmut P, Christian J, Michael B, Cheistian B, Niels J, Christoph F D 2022 Ultrasound Med. Biol. 48 598 Google Scholar
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[5]	陈伟中 2014 声空化物理 (北京: 科学出版社) 第231—236页 Chen W Z 2014 Acoustic Cavitation Physics (Beijing: Science Press) pp231–236
[6]	Keller B J, Miksis M 1998 J. Acoust. Soc. Am. 68 628 Google Scholar
[7]	Rasoul S B, Nastaran R, Homa E, Mona M 2010 Phys. Rev. E 82 016316 Google Scholar
[8]	Bai L X, Xu W L, Deng J J, Gao Y D 2014 Ultrason. Sonochem. 21 1696 Google Scholar
[9]	Yu A 2011 Phys. Rev. E 83 066313 Google Scholar
[10]	Zhang W J, An Y 2013 Phys. Rev. E 87 053023.Google Scholar
[11]	Hansson I, Mctrch K A 1980 J. Appl. Phys. 51 4651 Google Scholar
[12]	Wu P F, Bai L X, Lin W J, Yan J C 2017 Ultrason. Sonochem. 38 75 Google Scholar
[13]	Li F, Zhang X M, Tian H, Hu J, Chen S, Wang C H, Guo J Z, Mo R Y 2022 Acta Phys. Sin. 71 084303 [李凡, 张先梅, 田华, 胡静, 陈时, 王成会, 郭建中, 莫润阳 2022 物理学报 71 084303]Google Scholar Li F, Zhang X M, Tian H, Hu J, Chen S, Wang C H, Guo J Z, Mo R Y 2022 Acta Phys. Sin. 71 084303 Google Scholar
[14]	Parlitz O, Lauterbor W 1994 J. Acoust. Soc. Am. 96 3627 Google Scholar
[15]	Akhatov I, Parlitz U, Lauterborn W 1996 Phys. Rev. E 54 4992 Google Scholar
[16]	Appel J, Koch P, Mettin R, Krefting D, Lauterborn W 2004 Ultrason. Sonochem. 11 39 Google Scholar
[17]	Li F, Zhang X M, Tian H, Hu J, Chen S, Mo R Y, Wang C H, Guo J Z 2022 Ultrason. Sonochem. 87 106057 Google Scholar
[18]	Li F, Huang C Y, Zhang X M, Wang C H, Guo J Z, Lin S Y, Tian H 2023 Ultrasonics 132 106992 Google Scholar
[19]	Xu K, Xu L, Zhou G P 2021 Acta Phys. Sin. 70 194301 [徐珂, 许龙, 周光平 2021 物理学报 70 194301]Google Scholar Xu K, Xu L, Zhou G P 2021 Acta Phys. Sin. 70 194301 Google Scholar
[20]	Nasibullaevaa E S, Akhatovb I S 2012 J. Acoust. Soc. Am. 133 3727 Google Scholar
[21]	Wang C H, Mo R Y, Hu J, Chen S 2015 Acta Phys. Sin. 64 234301 [王成会, 莫润阳, 胡静, 陈时 2015 物理学报 64 234301]Google Scholar Wang C H, Mo R Y, Hu J, Chen S 2015 Acta Phys. Sin. 64 234301 Google Scholar
[22]	Elwin W V, Christopher F 2021 J. Acoust. Soc. Am. 149 2477 Google Scholar
[23]	Joseph B, Keller, Ignace I, Kolodner 2004 J. Appl . Phys. 27 1152
[24]	Christian V, Cleofé C P 2012 Ultrason. Sonochem. 19 217 Google Scholar
[25]	Kyuichi Y, Yasuo I, Toru T, Teruyuki K, Atsuya T 2008 Phys. Rev. E 77 016609 Google Scholar
[26]	Fabian R, Sergey L, Khadija A-B, Gunther B, Robert M 2019 Ultrason. Sonochem. 55 383 Google Scholar
[27]	Mettin R, Akhatov I, Parlitz U, Ohl C-D, Lauterborn W 1997 Phys. Rev. E 56 2924 Google Scholar

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图 1 实验装置　(a) 原型图; (b) 示意图

Figure 1. Experimental setup: (a) Experimental devices; (b) schematic diagram.

DownLoad: Full-Size Img PowerPoint

图 2 驱动频率为28 kHz球状泡群上浮并向层状结构演化的过程, 清洗槽输入电功率360 W

Figure 2. Evolution of a spherical bubble cluster to a layer structure at a driving frequency of 28 kHz, with an input power of 360 W to the cleaning tank.

DownLoad: Full-Size Img PowerPoint

图 3 驱动频率为28 kHz声场中球状泡团形状和上浮位置变化　(a) 泡团半径; (b) 泡团相对水面深度

Figure 3. Variation of the shape and uplift position of a spherical bubble cluster in the 28 kHz sound field: (a) Radius of the cluster; (b) depth of the cluster below the water surface.

DownLoad: Full-Size Img PowerPoint

图 4 驱动频率为40 kHz球状泡群上浮并向层状结构演化的过程, 清洗槽输入电功率为360 W

Figure 4. Evolution of a spherical bubble cluster to a layer structure at 40 kHz, with an input power of 360 W to the cleaning tank.

DownLoad: Full-Size Img PowerPoint

图 5 驱动频率为40 kHz声场中球状泡团形状和上浮位置变化　(a) 泡团半径; (b) 泡团相对水面深度

Figure 5. Variation of the shape and uplift position of a spherical bubble cluster in the 40 kHz sound field: (a) Radius of the cluster; (b) depth of the cluster below the water surface.

DownLoad: Full-Size Img PowerPoint

图 6 驱动频率为40 kHz, 不同清洗槽输入电功率条件下球状泡群上浮过程代表帧照片　(a) 360 W; (b) 324 W; (c) 288 W

Figure 6. Photographs of representative frames of the upwelling process of spherical bubble cluster under different power conditions with driving frequency of 40 kHz: (a) 360 W; (b) 324 W; (c) 288 W.

DownLoad: Full-Size Img PowerPoint

图 7 模型示意图

Figure 7. Schematic diagram of the model.

DownLoad: Full-Size Img PowerPoint

图 8 泡团中心处气泡半径随时间变化曲线, n = 9×10¹². f = 28 kHz, a = 11 mm, R₀ = 6 μm　(a)考虑边界; (b)不考虑边界. f = 40 kHz, a = 9 mm, R₀ = 5 μm　 (c)考虑边界; (d)不考虑边界

Figure 8. Bubble radius at the center of the bubble cluster versus time, n = 9×10¹²: (a) With and (b) without considering the impacts of soft boundary, f = 28 kHz, a = 11 mm, R₀ = 6 μm; (c) with and (d) without considering the impacts of soft boundary, f = 40 kHz, a = 9 mm, R₀ = 5 μm.

DownLoad: Full-Size Img PowerPoint

图 9 等效势随球状泡群半径变化曲线, R₀ = 5 μm, n = 9×10¹², P_a = 150 kPa　(a) 声波频率28 kHz; (b) 声波频率40 kHz

Figure 9. Equivalent potential versus radius of the bubble cluster, R₀ = 5 μm, n = 9×10¹², P_a = 150 kPa : (a) f = 28 kHz; (b) f = 40 kHz

DownLoad: Full-Size Img PowerPoint

图 10 驱动声压幅值对等效势的影响 (n = 9×10¹²)　(a) f = 28 kHz, R₀ = 6 μm, a = 11 mm; (b) f = 40 kHz, R₀ = 5 μm, a = 9 mm

Figure 10. Effect of driving sound pressure amplitude on equivalent potential (n=9×10¹²): (a) f = 28 kHz, R₀ = 6 μm, a = 11 mm; (b) f = 40 kHz, R₀ = 5 μm, a = 9 mm.

DownLoad: Full-Size Img PowerPoint

图 11 气泡团数密度对等效势的影响(P_a = 150 kPa)　(a) f = 28 kHz, R₀ = 6 μm, a = 11 mm; (b) f = 40 kHz, R₀ = 5 μm, a = 9 mm

Figure 11. Effect of bubble cluster number density on equivalent potential (P_a=150 kPa): (a) f = 28 kHz, R₀ = 6 μm, a = 11 mm; (b) f = 40 kHz, R₀ = 5 μm, a = 9 mm.

DownLoad: Full-Size Img PowerPoint

图 12 球状泡团内气泡平衡半径对等效势的影响 (P_a = 150 kPa)　(a) f = 28 kHz, a = 11 mm, 泡团内气泡占空比为0.0081; (b) f = 40 kHz, a = 9 mm, 泡团内气泡占空比为0.0047

Figure 12. Influence of the equivalent radii of small bubbles within the cluster on the equivalent potential (P_a = 150 kPa): (a) f = 28 kHz, a = 11 mm, and void ratio of the cluster is 0.0081; (b) f = 40 kHz, a = 9 mm, and void ratio of the cluster is 0.0047

DownLoad: Full-Size Img PowerPoint

[1]	应崇福 2007 中国科学: 物理学力学天文学 37 129 Google Scholar Ying C F 2007 Sci. Sin. Phys. Mech. Astron. 37 129 Google Scholar
[2]	程效锐, 张舒研, 房宁 2018 应用化工 47 1753 Google Scholar Cheng X R, Zhang S Y, Fang N 2018 Appl. Chem. Ind. 47 1753 Google Scholar
[3]	Ehsan Z S, Christian G, Helmut P, Christian J, Michael B, Cheistian B, Niels J, Christoph F D 2022 Ultrasound Med. Biol. 48 598 Google Scholar
[4]	Plesset S M, Prosperetti A 1977 Ann. Rev. Fluid Mech. 85 145
[5]	陈伟中 2014 声空化物理 (北京: 科学出版社) 第231—236页 Chen W Z 2014 Acoustic Cavitation Physics (Beijing: Science Press) pp231–236
[6]	Keller B J, Miksis M 1998 J. Acoust. Soc. Am. 68 628 Google Scholar
[7]	Rasoul S B, Nastaran R, Homa E, Mona M 2010 Phys. Rev. E 82 016316 Google Scholar
[8]	Bai L X, Xu W L, Deng J J, Gao Y D 2014 Ultrason. Sonochem. 21 1696 Google Scholar
[9]	Yu A 2011 Phys. Rev. E 83 066313 Google Scholar
[10]	Zhang W J, An Y 2013 Phys. Rev. E 87 053023.Google Scholar
[11]	Hansson I, Mctrch K A 1980 J. Appl. Phys. 51 4651 Google Scholar
[12]	Wu P F, Bai L X, Lin W J, Yan J C 2017 Ultrason. Sonochem. 38 75 Google Scholar
[13]	Li F, Zhang X M, Tian H, Hu J, Chen S, Wang C H, Guo J Z, Mo R Y 2022 Acta Phys. Sin. 71 084303 [李凡, 张先梅, 田华, 胡静, 陈时, 王成会, 郭建中, 莫润阳 2022 物理学报 71 084303]Google Scholar Li F, Zhang X M, Tian H, Hu J, Chen S, Wang C H, Guo J Z, Mo R Y 2022 Acta Phys. Sin. 71 084303 Google Scholar
[14]	Parlitz O, Lauterbor W 1994 J. Acoust. Soc. Am. 96 3627 Google Scholar
[15]	Akhatov I, Parlitz U, Lauterborn W 1996 Phys. Rev. E 54 4992 Google Scholar
[16]	Appel J, Koch P, Mettin R, Krefting D, Lauterborn W 2004 Ultrason. Sonochem. 11 39 Google Scholar
[17]	Li F, Zhang X M, Tian H, Hu J, Chen S, Mo R Y, Wang C H, Guo J Z 2022 Ultrason. Sonochem. 87 106057 Google Scholar
[18]	Li F, Huang C Y, Zhang X M, Wang C H, Guo J Z, Lin S Y, Tian H 2023 Ultrasonics 132 106992 Google Scholar
[19]	Xu K, Xu L, Zhou G P 2021 Acta Phys. Sin. 70 194301 [徐珂, 许龙, 周光平 2021 物理学报 70 194301]Google Scholar Xu K, Xu L, Zhou G P 2021 Acta Phys. Sin. 70 194301 Google Scholar
[20]	Nasibullaevaa E S, Akhatovb I S 2012 J. Acoust. Soc. Am. 133 3727 Google Scholar
[21]	Wang C H, Mo R Y, Hu J, Chen S 2015 Acta Phys. Sin. 64 234301 [王成会, 莫润阳, 胡静, 陈时 2015 物理学报 64 234301]Google Scholar Wang C H, Mo R Y, Hu J, Chen S 2015 Acta Phys. Sin. 64 234301 Google Scholar
[22]	Elwin W V, Christopher F 2021 J. Acoust. Soc. Am. 149 2477 Google Scholar
[23]	Joseph B, Keller, Ignace I, Kolodner 2004 J. Appl . Phys. 27 1152
[24]	Christian V, Cleofé C P 2012 Ultrason. Sonochem. 19 217 Google Scholar
[25]	Kyuichi Y, Yasuo I, Toru T, Teruyuki K, Atsuya T 2008 Phys. Rev. E 77 016609 Google Scholar
[26]	Fabian R, Sergey L, Khadija A-B, Gunther B, Robert M 2019 Ultrason. Sonochem. 55 383 Google Scholar
[27]	Mettin R, Akhatov I, Parlitz U, Ohl C-D, Lauterborn W 1997 Phys. Rev. E 56 2924 Google Scholar

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