# 位流

## 特性與應用

### 描述與特性

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

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

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

### 不可壓縮流

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

${\displaystyle \nabla ^{2}\varphi =0}$

### 可壓縮流

#### 穩定流

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

${\displaystyle M_{x}={\frac {1}{a}}{\frac {\partial \Phi }{\partial x}}}$ ${\displaystyle M_{y}={\frac {1}{a}}{\frac {\partial \Phi }{\partial y}}}$ ${\displaystyle M_{z}={\frac {1}{a}}{\frac {\partial \Phi }{\partial z}}}$

${\displaystyle \nabla \Phi =V_{\infty }x+\nabla \varphi .}$

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

#### 聲波

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

## 參考

1. Batchelor (1973) pp. 99–101.
2. Batchelor (1973) pp. 378–380.
3. ^ Kirby, B.J. Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices.. Cambridge University Press. 2010 [2011-02-25]. ISBN 978-0521119030. （原始内容存档于2019-04-28）.
4. 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.