位流

NACA 0012翼型周圍的位流流線，攻角11°。

特性與應用

描述與特性

$\mathbf{v} = \nabla \varphi.$

$\nabla \times \nabla \varphi = \mathbf{0},$

$\nabla \times \mathbf{v} = \mathbf{0}.$

不可壓縮流

$\nabla \cdot \mathbf{v} =0$

$\nabla^2 \varphi = 0$

可壓縮流

穩定流

$\left( 1 - M_x^2 \right) \frac{\partial^2 \Phi}{\partial x^2} + \left( 1 - M_y^2 \right) \frac{\partial^2 \Phi}{\partial y^2} + \left( 1 - M_z^2 \right) \frac{\partial^2 \Phi}{\partial z^2} - 2 M_x M_y \frac{\partial^2 \Phi}{\partial x\, \partial y} - 2 M_y M_z \frac{\partial^2 \Phi}{\partial y\, \partial z} - 2 M_z M_x \frac{\partial^2 \Phi}{\partial z\, \partial x} = 0$

$M_x = \frac{1}{a} \frac{\partial \Phi}{\partial x}$ $M_y = \frac{1}{a} \frac{\partial \Phi}{\partial y}$ $M_z = \frac{1}{a} \frac{\partial \Phi}{\partial z}$

$\nabla \Phi = V_\infty x + \nabla \varphi.$

$\left(1-M_\infty^2\right) \frac{\partial^2 \varphi}{\partial x^2} + \frac{\partial^2 \varphi}{\partial y^2} + \frac{\partial^2 \varphi}{\partial z^2} = 0$

聲波

$\frac{\partial^2 \varphi}{\partial t^2} = \overline{a}^2 \Delta \varphi$

參考

1. ^ 1.0 1.1 1.2 1.3 1.4 Batchelor (1973) pp. 99–101.
2. ^ 2.0 2.1 2.2 Batchelor (1973) pp. 378–380.
3. ^ Kirby, B.J. Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices.. Cambridge University Press. 2010. ISBN 978-0521119030.
4. ^ 4.0 4.1 4.2 Anderson, J.D., Modern compressible flow, McGraw-Hill, 2002, ISBN 0072424435, pp. 358–359.
5. ^ Lamb (1994) §287, pp. 492–495.
6. ^ Feynman, R.P.; Leighton, R.B.; Sands, M., The Feynman Lectures on Physics 2, Addison-Wesley, 1964, p. 40-3. Chapter 40 has the title: The flow of dry water.
7. ^ Batchelor (1973) pp. 404–405.