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A two-dimensional (2D) lattice Boltzmann method (LBM)-cellular automaton (CA) coupled model is developed for the simulation of dendritic growth and bubble formation during solidification. In the model, the dendritic growth is simulated by a CA approach. The driving force of dendritic growth is determined by a local solute equilibrium approach. The LBM based on the Shan-Chen multiphase flow scheme is adopted to simulate the growth and the motion of bubbles in liquid. The interaction mechanism between dendrites and bubbles is embedded in the model. Model validation is carried out by comparing the simulations with the Laplace law, and by simulating the wettability of a bubble on a smooth solid surface. The proposed model is used to study the effect of gas-liquid interaction coefficient on single bubble growth. It is found that the growth velocity and the equilibrium radius of bubble increase with the gas-liquid interaction coefficient. The simulations of the dendritic growth and bubble formation during directional solidification reproduce the physical phenomena, including dendritic competitive growth, the preferential nucleation locations of bubbles, and bubble growth, coalescence, deformation due to the squeeze of neighboring dendrites, as well as bubble motion in the liquid channels. The simulation results are compared reasonably well with the experimental results. In addition, gas pore volume fraction increases with the initial gas content. The simulations of the present LBM-CA model provide an insight into the physical mechanism of bubble nucleation, growth, and motion, as well as the interaction between the dendritic growth and bubble formation during solidification.
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Keywords:
- lattice Boltzmann method /
- celullar automaton /
- dendrite /
- bubble
[1] Han Q Y 2006 Scripta Mater. 55 871
[2] Zhao L, Liao H C, Pan Y, Wang L, Wang Q G 2011 Scripta Mater. 65 795
[3] Atwood R C, Lee P D 2003 Acta Mater. 51 5447
[4] Dong S Y, Xiong S M, Liu B C 2004 Mater Sci Technol. 20 23
[5] Raabe D 2004 Modelling Simul. Mater. Sci. Eng. 12 R13
[6] Guo Z L, Zheng C G 2009 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) p9-10, 61-63 (in Chinese) [郭照立, 郑楚光 2009 格子Boltzmann方法的原理和应用(第一版) (北京: 科学出版出版社)第9—10, 61—63页]
[7] Zeng J B, Li R J, Liao Q, Jiang F M 2011 Acta Phys.Sin. 60 066401 (in Chinese) [曾建邦, 李隆键, 廖全, 蒋方明 2011 60 066401]
[8] Wang W X, Shi J, Qiu B, Li H B 2010 Acta Phys. Sin. 59 8371 (in Chinese) [王文霞, 施娟, 邱冰, 李华兵 2010 59 8371]
[9] Wang J F, Liu Y, Xu Y S 2009 Biomed Microdevices 11 351
[10] Guo Z L, Asinari, Pietro, Zheng C G 2009 Phys. Rev. E 79 026702
[11] Sun D K, Zhu M F, Pan S Y, Raabe D 2009 Acta Mater. 6 1755
[12] Yin H, Felicelli SD, Wang L 2011 Acta Mater. 59 3124
[13] Selzer M, Jainta M, Nestler B 2009 Phys. Status Solid B 246 1197
[14] Zhou F M, Sun D K, Zhu M F 2010 Acta Phys. Sin. 59 3394 (in Chinese) [周丰茂, 孙东科, 朱鸣芳 2010 59 3394]
[15] Li Q, MaY C, Liu K, Kang X H, Li D Z 2007 Acta Metall. Sin. 43 217 (in Chinese) [李强, 马颖澈, 刘奎, 康秀红, 李殿中 2007 金属学报 43 217]
[16] Shan B W, Lin X, Wei L, Huang W D 2009 Acta Phys. Sin. 58 1132 (in Chinese) [单博炜, 林鑫, 魏雷, 黄卫东 2009 58 1132]
[17] Huang H B, Thorne D T, Jr, Schaap M G, Sukop M C 2007 Phys. Rev. E 76 066701
[18] Shan X W 2006 Phys. Rev. E 73 047701
[19] Zhu M F, Stefanescu D M 2007 Acta Mater. 55 1741
[20] Sukop M C, Thorne D T, Jr 2005 Lattice Boltzmann Modeling-An Introduction for Geoscientists and Engineers (2nd ed) (New York: Springer) p114
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[1] Han Q Y 2006 Scripta Mater. 55 871
[2] Zhao L, Liao H C, Pan Y, Wang L, Wang Q G 2011 Scripta Mater. 65 795
[3] Atwood R C, Lee P D 2003 Acta Mater. 51 5447
[4] Dong S Y, Xiong S M, Liu B C 2004 Mater Sci Technol. 20 23
[5] Raabe D 2004 Modelling Simul. Mater. Sci. Eng. 12 R13
[6] Guo Z L, Zheng C G 2009 Theory and Applications of Lattice Boltzmann Method (Beijing: Science Press) p9-10, 61-63 (in Chinese) [郭照立, 郑楚光 2009 格子Boltzmann方法的原理和应用(第一版) (北京: 科学出版出版社)第9—10, 61—63页]
[7] Zeng J B, Li R J, Liao Q, Jiang F M 2011 Acta Phys.Sin. 60 066401 (in Chinese) [曾建邦, 李隆键, 廖全, 蒋方明 2011 60 066401]
[8] Wang W X, Shi J, Qiu B, Li H B 2010 Acta Phys. Sin. 59 8371 (in Chinese) [王文霞, 施娟, 邱冰, 李华兵 2010 59 8371]
[9] Wang J F, Liu Y, Xu Y S 2009 Biomed Microdevices 11 351
[10] Guo Z L, Asinari, Pietro, Zheng C G 2009 Phys. Rev. E 79 026702
[11] Sun D K, Zhu M F, Pan S Y, Raabe D 2009 Acta Mater. 6 1755
[12] Yin H, Felicelli SD, Wang L 2011 Acta Mater. 59 3124
[13] Selzer M, Jainta M, Nestler B 2009 Phys. Status Solid B 246 1197
[14] Zhou F M, Sun D K, Zhu M F 2010 Acta Phys. Sin. 59 3394 (in Chinese) [周丰茂, 孙东科, 朱鸣芳 2010 59 3394]
[15] Li Q, MaY C, Liu K, Kang X H, Li D Z 2007 Acta Metall. Sin. 43 217 (in Chinese) [李强, 马颖澈, 刘奎, 康秀红, 李殿中 2007 金属学报 43 217]
[16] Shan B W, Lin X, Wei L, Huang W D 2009 Acta Phys. Sin. 58 1132 (in Chinese) [单博炜, 林鑫, 魏雷, 黄卫东 2009 58 1132]
[17] Huang H B, Thorne D T, Jr, Schaap M G, Sukop M C 2007 Phys. Rev. E 76 066701
[18] Shan X W 2006 Phys. Rev. E 73 047701
[19] Zhu M F, Stefanescu D M 2007 Acta Mater. 55 1741
[20] Sukop M C, Thorne D T, Jr 2005 Lattice Boltzmann Modeling-An Introduction for Geoscientists and Engineers (2nd ed) (New York: Springer) p114
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