在本研究中,吾人使用速度-渦度法(velocity-vorticity method)來得到一些著名問題之解析解,如柏傑斯漩渦(Burgers vortex),庫蒂流(Couette flow),海根-蒲休葉流(Hagen-Poiseuille flow),以及史托克斯第二問題(Stokes* second problem)等等.當然,控制方程式即是奈維爾-史托克斯方程式(Navier-Stokes equations),而其中的對流項(convective terms)也許可以拿掉,或者對時間的導數可以忽略,或是壓力梯度已經完全給定.
藉由渦度傳輸方程式(vorticity transport equations),不管壓力已知還是未知,速度和渦度場可以被決定.如此一來,不但可以顯示出速度-渦度法的效用而且能較方便地以數值方法來決定其他更複雜的流場.
同時,應該要強調的是吾人所得到的解析解與傳統的方法, 例如主要變數法(primitive variable method),所得到的是一樣的.
至於將來,則是希望利用速度-渦度法找出二維和三維更複雜流場的解析解,如方形穴室流(square cavity flows),或圓形穴室流(circular cavity flows),等.

In this study, we make use of the velocity-vorticity method to obtain the exact solutions of some well-known problems, for instance the Burgers vortex, Couette flow, Hagen-Poiseuille flow, and Stokes* second problem, etc. Of course, the governing equations are, namely, the Navier-Stokes equations, for which the convective terms may be dropped out, or the time derivative can be neglected, or the pressure gradient is given exactly.
By virtue of the vorticity transport equations, the velocity and vorticity fields can be determined regardless of whether the pressure is known or not. It not only shows the effect of the velocity-vorticity method but makes it convenient to determine other more complicated flow fields by numerical methods.
Meanwhile, it should be emphasized that the solutions we obtained are exactly the same as those solved by the conventional methods, such as the primitive variable method.
In the future, we are in expectation of finding exact solutions for more complex two dimensional and three dimensional flow fields by virtue of the velocity-vorticity method, such as square and circular cavity flows.

CHAPTER 1. INTRODUCTION**************.1
1.1. Prelude**.******************.**.1
1.2. Motives and Purposes*.****************2
1.3. Literature Review ****************..*.*2
CHAPTER 2. THEORETICAL DERIVATION AND
GOVERNING EQUATIONS****************5
2.1. The Navier-Stokes Equations**************..5
2.2. Derivation of Vorticity Transport Equations********..11
CHAPTER 3. SOME EXACT SOLUTIONS FOR
NAVIER-STOKES EQUATIONS**********..***..19
3.1. The Burgers Vortex**.****************19
3.2. Couette Flow due to Pressure Gradient
between a Fixed and a Moving Plate*************..31
3.3. Hagen-Poiseuille Flow*****************.39
3.4. Pulsating Flow between Parallel Surfaces****.*****..46
3.5. Steady Flow of a Viscous Fluid under
Gravity Sliding down an Inclined Plane ************.52
3.6. Stokes* Second Problem******.**********..61
CHAPTER 4. CONCLUSIONS***************..68
REFERENCES*************..********...70

劉英宏 ,"利用速度渦度法解析三維黏性不可壓縮流場", 國立台灣
大學土木工程學研究所碩士論文,1998.
Archeson, D. J. (1990). Elementary Fluid Dynamics , Oxford University
Press.
Aris, R. (1989). Vector, Tensors, and the Basic Equations of Fluid
Mechanics, Dover Publications, Inc..
Batchelor, G. K. (1967) , An Introdution to Fluid Dynamics, Cambridge
University Press.
Chorin, T. P. (1994) , Vorticity and Turbulence , Springer-Verlag.
Courant, C., and John, F. (1965), Introduction to Calculus and Analysis
Volume 1, Springer-Verlag, New York.
Currie, I. G. (1993) , Fundamental Mechanics of Fluids, M
Goda, K. (1979) , A Multistep Technique with Implicit Difference
Schemes for Calculating Two- or Three -Dimensional Cavity
Flows, J. Comput. Phys. , 30, pp. 76-95.
Holmes, P. , Lumley, J. C. and Berkooz, G. (1996), Turbulence, Coherent
Structures, Dynamical Systems and Symmetry, Cambridge University
Press.
Ku, H. C. , Hirsh, R. S. , and Taylor, T. D. (1986) , A Pseudospectral
Method for Solution of the Three - Dimensional Incompressible
Navier-Stokes Equations , J. Comput. Phy. , 70, pp. 439-462.
Landau, L. D., and Lifshitz, E. M. (1959). Fluid Mechanics, J. B. Sykes
and W. H. Reid, translators, Pergamon Press, Oxford, England.
Liggett, J. A. (1994).Fluid Mechanics , McGraw-Hill, Inc. .
Liu, Y. H. and Young, D. L. (1998). Velocity-Vorticity Method for the
Three-Dimensional Incompressible Viscous Flows, Proceedings of the
22nd National Conference on Theoretical and Applied Mechanics,
Tainan, Taiwan.
Logan, J. D. (1994). An Introduction to Nonlinear Partial Differential
Equations. John Wiley & Sons, Inc..
Munson, B. R., Young, D. F., and Okiishi, T. H. (1994). Fundamental of
Fluid Mechanics, John Wiley & Sons, Inc..
O*Neil, P. V. (1992). Advanced Engineering Mathematics, PWS-KENT
Publishing Company.
Smirnov, M. M. (1964).Second-Order Partial Differential
Equations .translated by Scripta Technica Ltd. , King*s
College,London.
Sokolnikoff, I. S. (1964). Tensor Analysis . John Wiley & Sons, Inc..
Trim, D. W. (1990). Applied Partial Differential Equations . PWS-KENT
Publishing Company.