氣泡之生成與脫離在工業上有很重要的應用，但目前尚無解三維
(非軸對稱)靜態氣泡之文獻公開發表。由於沒有動量方程式可用，計
算靜態氣泡必須解Young-Laplace equation。本文發展一套計算三維氣
泡形狀的數值方法，將三維氣泡形狀利用球座標系統表示，並在氣泡
表面網格點上進行切平面座標轉換，得到局部表面曲率及形狀的關
係，在已知形狀條件下求得曲率，也可由已知曲率的條件下反求形狀。
本研究的主要成果有三：
1. 首度以雙三次函數bi-cubic function 來近似氣泡表面局部區域之形
狀，算則簡單，精準度高。
2. 由氣泡表面之曲率值反求氣泡形狀之首篇文獻。
3. 由Young-Laplace equation 取得之曲率值求解三維氣泡形狀之首篇
文獻。
相信本論文所研發之技術，有助於氣泡動力學之發展。

參考文獻
[1] M. van Sint Annaland, N.G. Deen, and J.A.M. Kuipers, 2005,
“Numerical simulation of gas bubbles behaviour using a
three-dimensional volume of fluid method,” Chemical Engineering
Science, 60, pp. 2999-3011.
[2] M. van Sint Annaland, W. Dijkhuizen, N. G. Deen, and J. A. M.
Kuipers, 2006, “Numerical Simulation of Behavior of Gas Bubbles
Using a 3-D Front-Tracking Method,” AIChE J., 52, pp. 99-110.
[3] E.M. Stokely and S.Y. Wu, 1992, “Surface Parameterization and
Curvature Measurement of Arbitrary 3-D Objects: Five Practical
Methods,” IEEE Trans. Part. Anal. Machine Intell., 14, pp. 833-840.
[4] Alan M., MCIvor , Robert J. Valkenburg, 1997, “A comparison of
local surface geometry estimation methods,” Machine Vision and
Applications, 10, pp. 17-26.
[5] D. Liao, H. Gregersen, T. Hausken, O.H. Gilja, M. Mundt, and G.
Kassab, 2004, “Analysis of surface geometry of the human stomach
using real-time 3-D ultrasonography in vivo,” Neurogastroenterol
Motil, 16, pp. 315-324.
[6] Donghua Liao, Jingbo Zhao, and Hans Gregersen1, 2005, “Regional
surface geometry of the rat stomach based on three-dimensional
curvature analysis,” Phys. Med. Biol., 50, pp. 231-246.
[7] S. L. Lee, and H. D. Lee, 2007, “Evolution of Liquid Meniscus
Shape in a Capillary Tube,” Journal of Fluids Engineering,
ASME 129, pp. 957-965.
[8] S. L. Lee, and C. F. Yang, 2010, “Bubble formation on the orifice of
a submerged vertical nozzle,” National Tsing Hua University, P.M.E.
PhD Thesis.
- 18 -
[9] Michael S. Longuet-Higgins, Bryan R. Kerman, and Knud Lunde,
1991, “The release of air bubbles from an underwater nozzle,” J.
Fluid Mech., 230, pp. 365-390.
[10] Hasan N. Oguz a1, and Andrea Prosperetti, 1993, “Dynamics of
bubble growth and detachment from a needle,” J. Fluid Mech., 257,
pp. 111-145.
[11] S. T. Thoroddsen, T. G. Etoh, and K. Takehara, 2007, “Experiments
on bubble pinch-off,” Phys. Fluids, 19, 042101.
[12] J. M. Gordilloa, 2008, “Axisymmetric bubble collapse in a quiescent
liquid pool. I. Theory and numerical simulations,” Phys. Fluids, 20,
112103.
[13] R. Bolanos-Jimenez, A. Sevilla, C. Martinez-Bazan, and J. M.
Gordillo, 2008, “Axisymmetric bubble collapse in a quiescent liquid
pool. II. Experimental study,” Phys. Fluids, 20, 112104.
[14] R. Bolanos-Jimenez, A. Sevilla, C. Martinez-Bazan, D. van der Meer,
and J. M. Gordillo, 2009, “The effect of liquid viscosity on bubble
pinch-off,” Phys. Fluids, 21, 072103.
[15] P. C. Duineveld, 1995, “The Rise Velocity and Shape of Bubbles in
Pure Water at High Reynolds Number,” J. Fluid Mech., 292, pp.
325-332.
[16] Mingming Wu, and Morteza Gharib, 2002, “Experimental Studies on
the Shape and Path of Small Air Bubble Rising in Clean Water,”
Phys. Fluids, 14, pp. L49-L52.