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In this paper, the interaction between the two bubbles with the effects of mass transfer, heat transfer and diffusion was investigated under ultrasonic action by using the model combining the modified Keller-Miksis equation with Noble-Abel-Stiffend-Gas equation of state. The maximum normalized radius of cavitation the bubbles and the secondary Bjerknes force (coefficient) of the interaction between the bubbles are calculated respectively, and compared with the maximum normalized radius of the bubbles and the secondary Bjerknes force (coefficient) in the two bubbles system without the effects of mass transfer, heat transfer and diffusion given by Mettin et al [Mettin R, Akhatov I, Parlitz U, Ohl C D, Lauterborn W 1997 Phys. Rev. E 56 2924]. The results show that the maximum normalized radius of the bubbles and the secondary Bjerknes forces between the bubbles are larger in the two bubbles system with the effects of mass transfer, heat transfer and diffusion than in the two bubbles system without the effects of mass transfer, heat transfer and diffusion. The main reason for this result is that the effects of mass transfer, heat transfer and diffusion can increase the amount of gas contained in the cavitation bubble. With the increase of the initial radius of a certain the bubble in the two bubbles system, the secondary Bjerknes force will increase and the difference between the secondary Bjerknes forces in the two cases will gradually increase. With the increase of the driving sound pressure amplitude, the secondary Bjerknes force will also increase and the difference between the secondary Bjerknes forces in the two cases will also gradually increase. With the increase of the driving frequency, the secondary Bjerknes force will decrease and the difference between the secondary Bjerknes forces in the two cases will also gradually decrease. Therefore, the effects of mass transfer, heat transfer and diffusion have a significant effect on the secondary Bjerknes force of the interaction between cavitation the bubbles, which is an important factor that cannot be ignored. The research results of this article have positive significance for further understanding of the secondary Bjerknes force of the interaction between the two bubbles under ultrasonic action.
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Keywords:
- cavitation two bubbles /
- secondary Bjerknes force /
- mass transfer and heat transfer effect /
- diffusion effect
[1] Hilgenfeldt S, Lohse D, Brenner M P 1996 Phys. Fluids 8 2808
Google Scholar
[2] Yasui K 1997 Phys. Rev. E 56 6750
Google Scholar
[3] Toegel R, Lohse D 2003 J. Chem. Phys. 118 1863
Google Scholar
[4] Shen Y, Yasui K, Sun Z C, Mei B, You M Y, Zhu T 2016 Ultrason. Sonochem. 29 394
Google Scholar
[5] 胡静, 林书玉, 王成会, 李锦 2013 62 134303
Google Scholar
Hu J, Lin S Y, Wang C H, Li J 2013 Acta Phys. Sin. 62 134303
Google Scholar
[6] 张鹏利, 林书玉, 朱华泽, 张涛 2019 68 134301
Google Scholar
Zhang P L, Lin S Y, Zhu H Z, Zhang T 2019 Acta Phys.Sin. 68 134301
Google Scholar
[7] 清河美, 那仁满都拉 2020 69 184301
Google Scholar
Qinghim, Naranmandula 2020 Acta Phys. Sin. 69 184301
Google Scholar
[8] Mettin R, Akhatov I, Parlitz U, Ohl C D, Lauterborn W 1997 Phys. Rev. E 56 2924
Google Scholar
[9] Doinikov A A 2001 Phys. Rev. E 64 026301
Google Scholar
[10] Sadighi-Bonabi R, Rezaee N, Ebrahimi H, Mirheydari M 2010 Phys. Rev. E 82 016316
Google Scholar
[11] Zhang Y N, Zhang Y N, Li S C 2016 Ultrason. Sonochem. 29 129
Google Scholar
[12] Pandey V 2019 Phys. Rev. E 99 042209
Google Scholar
[13] Luo J, Niu Z P 2019 Sci. Rep. 9 1352
Google Scholar
[14] Zhang L L, Chen W Z, Zhang Y Y, Wu Y R, Wang X, Zhao G Y 2020 Chin. Phys. B 29 034303
Google Scholar
[15] 李想, 陈勇, 封皓, 綦磊 2020 69 184703
Google Scholar
Li X, Chen Y, Feng H, Qi L 2020 Acta Phys. Sin. 69 184703
Google Scholar
[16] Shen Y, Zhang L L, Wu Y R, Chen W Z 2021 Ultrason. Sonochem. 73 105535
Google Scholar
[17] Qin D, Zou Q Q, Lei S, Wang W, Li Z Y 2021 Ultrason. Sonochem. 78 105712
Google Scholar
[18] Zhang X M, Li F, Wang C H, Mo R Y, Hu J, Guo J Z, Lin S Y 2022 Ultrason. 126 106809
Google Scholar
[19] 王德鑫, 那仁满都拉 2018 67 037802
Google Scholar
Wang D X, Naranmandula 2018 Acta Phys.Sin. 67 037802
Google Scholar
[20] 王寻, 黎奥, 周敏, 梁金福, 张泽坤, 吴伟 2022 应用声学 41 735
Wang X, Li A, Zhou M, Liang J F, Zhang Z K, Wu W 2022 J. Appl. Acoust. 41 735
[21] Yasui K, Iida Y, Tuziuti T, Kozuka T, Towata A 2008 Phys. Rev. E 77 016609
Google Scholar
[22] Shen Z Z 2020 Chin. Phys. B 29 014304
Google Scholar
[23] 徐珂, 许龙, 周光平 2021 70 194301
Google Scholar
Xu K, Xu L, Zhou G P 2021 Acta Phys. Sin. 70 194301
Google Scholar
[24] Métayer O L, Saurel R 2016 Phys. Fluids 28 046102
Google Scholar
[25] Toegel R, Gompf B, Pecha R, Lohse D 2000 Phys. Rev. Lett. 85 3165
Google Scholar
[26] Longwell P A, Olin J B, Sage B H 1958 Ind. Eng. Chem. Chem. Eng. Data Series 3 175
Google Scholar
[27] Hirschfelder J A, Curtiss C F, Bird R B 1954 Molecular Theory of Gases and Liquids (New York: Wiley) pp293–302
[28] Dehane A, Merouani S, Hamdaoui O, Alghyamah A 2021 Ultrason. Sonochem. 73 105511
Google Scholar
[29] Yasui K, Tuziuti T, Kanematsu W 2016 Phys. Rev. E 94 013106
Google Scholar
[30] Jishnu C R, Salih A M 2018 Fluid Phase Equilib. 483 182
Google Scholar
[31] Yasui K 1995 J. Acoust. Soc. Am. 98 2772
Google Scholar
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图 1 空化气泡内气体最大总分子数
$ N_\mathrm{\rm{T}, max} $ 的变化 (a)当$R_{20}=5 \;\text{μm}$ ,$ A=1.32 P_{0} $ ,$ f_{0}=20\;\mathrm{kHz} $ 时,$ N_\mathrm{\rm{T}, max} $ 随气泡1初始半径的变化; (b)当$ R_{10}=R_{20}=5 \;\text{μm} $ ,$ f_{0}=20\;\mathrm{kHz} $ 时,$ N_\mathrm{\rm{T}, max} $ 随驱动声压的变化; (c)当$ R_{10}=R_{20}=5 \;\text{μm} $ ,$ A=1.32 P_{0} $ 时,$ N_\mathrm{\rm{T}, max} $ 随驱动频率的变化Figure 1. Change of the maximum total molecular number
$ N_\mathrm{\rm{T}, max} $ of gas in cavitation bubble: (a)$ N_\mathrm{\rm{T}, max} $ changes with the initial radius of bubble 1 at$ R_{20}=5\; \text{μm} $ ,$ A=1.32 P_{0} $ ,$ f_{0}=20\;\mathrm{kHz} $ ; (b)$ N_\mathrm{\rm{T}, max} $ changes with driving sound pressure at$ R_{10}=R_{20}=5 \;\text{μm} $ ,$ f_{0}=20\;\mathrm{kHz} $ ; (c)$ N_\mathrm{\rm{T}, max} $ changes with driving frequency at$ R_{10}=R_{20}=5 \;\text{μm} $ ,$ A=1.32 P_{0} $ .
图 2 空化气泡归一化最大半径和次Bjerknes力系数随气泡1初始半径的变化 (a)归一化最大半径; (b)次Bjerknes力系数
Figure 2. Maximum normalized radius of cavitation the bubbles and the secondary Bjerknes force coefficient change with the initial radius of bubble 1: (a) Maximum normalized radius; (b) the secondary Bjerknes force coefficient
图 3 空化气泡归一化最大半径和次Bjerknes力系数随驱动声压的变化 (a)归一化最大半径; (b)次Bjerknes力系数
Figure 3. Maximum normalized radius of cavitation the bubbles and the secondary Bjerknes force coefficient change with driving sound pressure: (a) Maximum normalized radius; (b) the secondary Bjerknes force coefficient
图 4 空化气泡归一化最大半径和次Bjerknes力系数随驱动频率的变化 (a) 归一化最大半径; (b) 次Bjerknes力系数
Figure 4. Maximum normalized radius of cavitation the bubbles and the secondary Bjerknes force coefficient change with driving frequency: (a) Maximum normalized radius; (b) the secondary Bjerknes force coefficient
表 1 模型1的相关物理参数
Table 1. Related physical parameters of model 1
参数 值 单位 液体密度 $ \rho_{0} $ 998 $ \mathrm{kg{\cdot} m^{-3}} $ 黏性系数 μ 0.001 $ \mathrm{Pa{\cdot} s} $ 表面张力 σ 0.0725 $ \mathrm{N{\cdot} m^{-1}} $ 环境温度 $ T_{\mathrm{0}} $ 300 $ \mathrm{K} $ 液体热容比 $ \gamma_\mathrm{l} $ 1.19 $ - $ 液体热容 $ C_\mathrm{V, l} $ 3610 $ \mathrm{J{\cdot} kg^{-1} {\cdot} K^{-1}} $ 饱和蒸汽压 $ P_\mathrm{v} $ $ 2.3381\times 10^{3} $ $ \mathrm{Pa} $ 液体分子共体积 $ b_\mathrm{l} $ $ 6.7145\times 10^{-4} $ $ \mathrm{m^3{\cdot} kg^{-1}} $ 液体压力常数 $ B_\mathrm{l} $ $ 6.2239\times 10^8 $ $ \mathrm{Pa} $ 气体压力常数 $ B_\mathrm{g} $ 0 $ \mathrm{Pa} $ 气体摩尔质量 $ M_{\mathrm{H_{2}O}} $ 0.018 $ \mathrm{kg{\cdot} mol^{-1}} $ 气体摩尔质量 $ M_{\mathrm{Ar}} $ 0.04 $ \mathrm{kg{\cdot} mol^{-1}} $ 阿伏伽德罗常数 $ N_{\mathrm{A}} $ $ 6.023\times 10^{23} $ $ \mathrm{mol^{-1}} $ 玻尔斯曼常数 $ k_{\mathrm{B}} $ $ 1.38\times 10^{-23} $ $ \mathrm{J{\cdot} K^{-1}} $ 环境压力 $ P_{\mathrm{0}} $ $ 10^{5} $ $ \mathrm{Pa} $ 气泡间距 $d_{ij}$ $ 10^{-3} $ $ \mathrm{m} $ 
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表 2 模型2的相关物理参数
Table 2. Related physical parameters of model 2
参数 值 单位 液体密度 $ \rho_{0} $ 998 $ \mathrm{kg{\cdot} m^{-3}} $ 黏性系数 μ 0.001 $ \mathrm{Pa{\cdot} s} $ 表面张力 σ 0.0725 $ \mathrm{N{\cdot} m^{-1}} $ 环境压力 $ P_{\mathrm{0}} $ $ 1\times 10^{5} $ $ \mathrm{Pa} $ 气体热容比 $ \gamma_\mathrm{g} $ 1.67 $ - $ 液体中的声速 $ C_\mathrm{l} $ 1500 $ \mathrm{m{\cdot} s^{-1}} $ 气泡间距 $d_{ij}$ $ 1\times 10^{-3} $ $ \mathrm{m} $
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[1] Hilgenfeldt S, Lohse D, Brenner M P 1996 Phys. Fluids 8 2808
Google Scholar
[2] Yasui K 1997 Phys. Rev. E 56 6750
Google Scholar
[3] Toegel R, Lohse D 2003 J. Chem. Phys. 118 1863
Google Scholar
[4] Shen Y, Yasui K, Sun Z C, Mei B, You M Y, Zhu T 2016 Ultrason. Sonochem. 29 394
Google Scholar
[5] 胡静, 林书玉, 王成会, 李锦 2013 62 134303
Google Scholar
Hu J, Lin S Y, Wang C H, Li J 2013 Acta Phys. Sin. 62 134303
Google Scholar
[6] 张鹏利, 林书玉, 朱华泽, 张涛 2019 68 134301
Google Scholar
Zhang P L, Lin S Y, Zhu H Z, Zhang T 2019 Acta Phys.Sin. 68 134301
Google Scholar
[7] 清河美, 那仁满都拉 2020 69 184301
Google Scholar
Qinghim, Naranmandula 2020 Acta Phys. Sin. 69 184301
Google Scholar
[8] Mettin R, Akhatov I, Parlitz U, Ohl C D, Lauterborn W 1997 Phys. Rev. E 56 2924
Google Scholar
[9] Doinikov A A 2001 Phys. Rev. E 64 026301
Google Scholar
[10] Sadighi-Bonabi R, Rezaee N, Ebrahimi H, Mirheydari M 2010 Phys. Rev. E 82 016316
Google Scholar
[11] Zhang Y N, Zhang Y N, Li S C 2016 Ultrason. Sonochem. 29 129
Google Scholar
[12] Pandey V 2019 Phys. Rev. E 99 042209
Google Scholar
[13] Luo J, Niu Z P 2019 Sci. Rep. 9 1352
Google Scholar
[14] Zhang L L, Chen W Z, Zhang Y Y, Wu Y R, Wang X, Zhao G Y 2020 Chin. Phys. B 29 034303
Google Scholar
[15] 李想, 陈勇, 封皓, 綦磊 2020 69 184703
Google Scholar
Li X, Chen Y, Feng H, Qi L 2020 Acta Phys. Sin. 69 184703
Google Scholar
[16] Shen Y, Zhang L L, Wu Y R, Chen W Z 2021 Ultrason. Sonochem. 73 105535
Google Scholar
[17] Qin D, Zou Q Q, Lei S, Wang W, Li Z Y 2021 Ultrason. Sonochem. 78 105712
Google Scholar
[18] Zhang X M, Li F, Wang C H, Mo R Y, Hu J, Guo J Z, Lin S Y 2022 Ultrason. 126 106809
Google Scholar
[19] 王德鑫, 那仁满都拉 2018 67 037802
Google Scholar
Wang D X, Naranmandula 2018 Acta Phys.Sin. 67 037802
Google Scholar
[20] 王寻, 黎奥, 周敏, 梁金福, 张泽坤, 吴伟 2022 应用声学 41 735
Wang X, Li A, Zhou M, Liang J F, Zhang Z K, Wu W 2022 J. Appl. Acoust. 41 735
[21] Yasui K, Iida Y, Tuziuti T, Kozuka T, Towata A 2008 Phys. Rev. E 77 016609
Google Scholar
[22] Shen Z Z 2020 Chin. Phys. B 29 014304
Google Scholar
[23] 徐珂, 许龙, 周光平 2021 70 194301
Google Scholar
Xu K, Xu L, Zhou G P 2021 Acta Phys. Sin. 70 194301
Google Scholar
[24] Métayer O L, Saurel R 2016 Phys. Fluids 28 046102
Google Scholar
[25] Toegel R, Gompf B, Pecha R, Lohse D 2000 Phys. Rev. Lett. 85 3165
Google Scholar
[26] Longwell P A, Olin J B, Sage B H 1958 Ind. Eng. Chem. Chem. Eng. Data Series 3 175
Google Scholar
[27] Hirschfelder J A, Curtiss C F, Bird R B 1954 Molecular Theory of Gases and Liquids (New York: Wiley) pp293–302
[28] Dehane A, Merouani S, Hamdaoui O, Alghyamah A 2021 Ultrason. Sonochem. 73 105511
Google Scholar
[29] Yasui K, Tuziuti T, Kanematsu W 2016 Phys. Rev. E 94 013106
Google Scholar
[30] Jishnu C R, Salih A M 2018 Fluid Phase Equilib. 483 182
Google Scholar
[31] Yasui K 1995 J. Acoust. Soc. Am. 98 2772
Google Scholar
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