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This study focuses on the dynamic behavior of rising bubble in water with taking surface tension and viscous effect into consideration. The Young-Laplace equation is adopted to obtain the viscous component of the normal stress, and the tangential component is evaluated by viscous correction pressure, which is based on viscous dissipation energy equivalence principle. Firstly, both axi-symmetric and three-dimensional model is established by employing boundary integral method. The validation of our model is confirmed by comparing the analytical results with the experimental results, and they are in good agreement with each other. Secondly, the shape and balanced velocity of millimetre-sized bubble are studied, and the influences of initial condition, surface tension and viscosity on the dynamic behaviour of the bubble are also discussed. Finally a numerical technic is put forward to handle the coalescence of two rising bubbles, which can show some detailed information about the coalescence process.
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Keywords:
- rising bubble /
- viscous effect /
- balanced velocity /
- coalescence
[1] Moore D W 1963 J. Fluid Mech. 16 161
[2] Moore D W 1965 J. Fluid Mech. 23 749
[3] Duineveld P C 1995 J. Fluid Mech. 292 325
[4] Raymond F, Rosant J M 2000 Chem. Eng. Sci. 55 943
[5] Zenit R, Magnaudet J 2008 Phys. Fluids 20 061702
[6] Wu M, Gharib M 2002 Phys. Fluids 14 49
[7] Amirnia S, de Bruyn J R, Bergougnou M A, Margaritis A 2013 Chem. Eng. Sci. 94 60
[8] Duineveld P C 1998 In Fascination of Fluid Dynamics (Berlin: Springer Netherlands) pp409-439
[9] Sanada T, Sato A, Shirota M, Watanabe M 2009 Chem. Eng. Sci. 64 2659
[10] Grenier N, Antuono M, Colagrossi A, Touze L 2009 J. Comput. Phys. 228 8380
[11] Hua J, Lou J 2007 J. Comput. Phys. 222 769
[12] Hua J, Stene J F, Lin P 2008 J. Comput. Phys. 227 3358
[13] Chen R H, Tian W X, Su G H, Qiu S Z, Ishiwatari T, Oki Y 2011 Chem. Eng. Sci. 66 5055
[14] Wang H, Zhang Z Y, Yang Y M, Hu Y, Zhang H S 2008 Chin. Phys. B 17 3847
[15] Wang H, Zhang Z Y, Yang Y M, Zhang H S 2010 Chin. Phys. B 19 026801
[16] Wang Q X, Yeo K S, Khoo B C, Lam K Y 1996 Theoret. Comput. Fluid Dyn. 8 73
[17] Wang Q X, Yeo K S, Khoo B C, Lam K Y 1996 Comput. Fluids 25 607
[18] Klaseboer E, Hung K C, Wang C, Wang C W, Khoo B C, Boyce P, Debono S, Charlier H 2005 J. Fluid Mech. 537 387
[19] Rungsiyaphornrat S, Klaseboer E, Khoo B C, Yeo K S 2003 Comput. Fluids 32 1049
[20] Li Z R, Sun L, Zong Z, Jing D 2012 Acta Mech. 223 2331
[21] Lundgren T S, Mansour N N 1991 J. Fluid Mech. 224 177
[22] Lind S J, Phillips T N 2010 J. Non-Newton. Fluid 165 852
[23] Li S, Zhang A M 2014 Acta Phys. Sin. 63 054705(in Chinese)[李帅, 张阿漫 2014 63 054705]
[24] Klaseboer E, Manica R, Khoo B C, Derek Y C C 2011 Eng. Anal. Bound. Elem. 35 489
[25] Zhang A M, Ni B Y 2014 Comput. Fluids 92 22
[26] Newman J N 1977 Marine Hydrodynamics (1st Ed.) (London: MIT Press) p131
[27] Yang W J, Yeh H C 1966 AIChE J. 12 927
[28] Best J P 1993 J. Fluid Mech. 251 79
[29] Lamb H 1932 Hydrodynamics (Cambridge: Cambridge University Press) p473
[30] Joseph D D, Wang J 2004 J. Fluid Mech. 505 365
[31] Zhang A M, Yao X L 2008 Chin. Phys. B 17 927
[32] Wu G X 1991 App. Ocean Res. 13 317
[33] Best J P 1994 Bubble Dynamics and Interface Phenomena (Berlin: Springer) p405
[34] Lee M, Klaseboer E, Khoo B C 2007 J. Fluid Mech. 570 407
[35] Malysa K, Krasowska M, Krzan M 2005 Adv. Colloid Interfac. 114 205
[36] Bhaga D, Weber M E 1981 J. Fluid Mech. 105 61
[37] Suñol F, González-Cinca R 2010 Colloid. Surface. A 365 36
[38] Haberman W L, Morton R K 1953 An Experimental Investigation of the Drag and Shape of Air Bubbles Rising in Various Liquids (Washington: David Taylor Model Basin Washington DC) p18
[39] Clift R, Grace J R, Weber M E 1978 Bubbles, Drops and Particles (New York: Academic Press) p346
[40] Batchelor G K 1967 An introduction to fluid dynamics (Cambridge: Cambridge University Press) p368
[41] Loth E 2008 Int. J Multiphas. Flow 34 523
[42] Khan A R, Richardson J F 1987 Chem. Eng. Commun. 62 135
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[1] Moore D W 1963 J. Fluid Mech. 16 161
[2] Moore D W 1965 J. Fluid Mech. 23 749
[3] Duineveld P C 1995 J. Fluid Mech. 292 325
[4] Raymond F, Rosant J M 2000 Chem. Eng. Sci. 55 943
[5] Zenit R, Magnaudet J 2008 Phys. Fluids 20 061702
[6] Wu M, Gharib M 2002 Phys. Fluids 14 49
[7] Amirnia S, de Bruyn J R, Bergougnou M A, Margaritis A 2013 Chem. Eng. Sci. 94 60
[8] Duineveld P C 1998 In Fascination of Fluid Dynamics (Berlin: Springer Netherlands) pp409-439
[9] Sanada T, Sato A, Shirota M, Watanabe M 2009 Chem. Eng. Sci. 64 2659
[10] Grenier N, Antuono M, Colagrossi A, Touze L 2009 J. Comput. Phys. 228 8380
[11] Hua J, Lou J 2007 J. Comput. Phys. 222 769
[12] Hua J, Stene J F, Lin P 2008 J. Comput. Phys. 227 3358
[13] Chen R H, Tian W X, Su G H, Qiu S Z, Ishiwatari T, Oki Y 2011 Chem. Eng. Sci. 66 5055
[14] Wang H, Zhang Z Y, Yang Y M, Hu Y, Zhang H S 2008 Chin. Phys. B 17 3847
[15] Wang H, Zhang Z Y, Yang Y M, Zhang H S 2010 Chin. Phys. B 19 026801
[16] Wang Q X, Yeo K S, Khoo B C, Lam K Y 1996 Theoret. Comput. Fluid Dyn. 8 73
[17] Wang Q X, Yeo K S, Khoo B C, Lam K Y 1996 Comput. Fluids 25 607
[18] Klaseboer E, Hung K C, Wang C, Wang C W, Khoo B C, Boyce P, Debono S, Charlier H 2005 J. Fluid Mech. 537 387
[19] Rungsiyaphornrat S, Klaseboer E, Khoo B C, Yeo K S 2003 Comput. Fluids 32 1049
[20] Li Z R, Sun L, Zong Z, Jing D 2012 Acta Mech. 223 2331
[21] Lundgren T S, Mansour N N 1991 J. Fluid Mech. 224 177
[22] Lind S J, Phillips T N 2010 J. Non-Newton. Fluid 165 852
[23] Li S, Zhang A M 2014 Acta Phys. Sin. 63 054705(in Chinese)[李帅, 张阿漫 2014 63 054705]
[24] Klaseboer E, Manica R, Khoo B C, Derek Y C C 2011 Eng. Anal. Bound. Elem. 35 489
[25] Zhang A M, Ni B Y 2014 Comput. Fluids 92 22
[26] Newman J N 1977 Marine Hydrodynamics (1st Ed.) (London: MIT Press) p131
[27] Yang W J, Yeh H C 1966 AIChE J. 12 927
[28] Best J P 1993 J. Fluid Mech. 251 79
[29] Lamb H 1932 Hydrodynamics (Cambridge: Cambridge University Press) p473
[30] Joseph D D, Wang J 2004 J. Fluid Mech. 505 365
[31] Zhang A M, Yao X L 2008 Chin. Phys. B 17 927
[32] Wu G X 1991 App. Ocean Res. 13 317
[33] Best J P 1994 Bubble Dynamics and Interface Phenomena (Berlin: Springer) p405
[34] Lee M, Klaseboer E, Khoo B C 2007 J. Fluid Mech. 570 407
[35] Malysa K, Krasowska M, Krzan M 2005 Adv. Colloid Interfac. 114 205
[36] Bhaga D, Weber M E 1981 J. Fluid Mech. 105 61
[37] Suñol F, González-Cinca R 2010 Colloid. Surface. A 365 36
[38] Haberman W L, Morton R K 1953 An Experimental Investigation of the Drag and Shape of Air Bubbles Rising in Various Liquids (Washington: David Taylor Model Basin Washington DC) p18
[39] Clift R, Grace J R, Weber M E 1978 Bubbles, Drops and Particles (New York: Academic Press) p346
[40] Batchelor G K 1967 An introduction to fluid dynamics (Cambridge: Cambridge University Press) p368
[41] Loth E 2008 Int. J Multiphas. Flow 34 523
[42] Khan A R, Richardson J F 1987 Chem. Eng. Commun. 62 135
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