# 伯努利定律

## 物理量及定律

### 原表达形式

${\displaystyle {\frac {1}{2}}\rho v^{2}+\rho gh+p={\mbox{constant}}}$

${\displaystyle v=\;}$ 流体速度
${\displaystyle g=\;}$ 重力加速度（地球表面的值约为 9.8 m/s2
${\displaystyle h=\;}$ 流体处于的深度（从某参考点计）
${\displaystyle p=\;}$ 流体所受的压力强度
${\displaystyle \rho =\;}$ 流体质量密度
${\displaystyle {\mbox{constant}}=\;}$ 常数

### 定理假设

• 不可压缩流（Incompressible flow）：密度为常数，在流体为气体适用于马赫数${\displaystyle M}$小于0.3的情况。
• 无摩擦流（Frictionsless flow）：摩擦效应可忽略，忽略黏滞性效应。
• 流体沿着流线流动（Flow along a streamline）：流体元素（element）沿着流线而流动，流线间彼此是不相交的。

## 推论过程

${\displaystyle F_{1}s_{1}-F_{2}s_{2}=p_{1}A_{1}v_{1}\Delta t-p_{2}A_{2}v_{2}\Delta t.\;}$

${\displaystyle mgh_{1}-mgh_{2}=\rho gA_{1}v_{1}\Delta th_{1}-\rho gA_{2}v_{2}\Delta th_{2}.\;}$

${\displaystyle {\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}={\frac {1}{2}}\rho A_{2}v_{2}\Delta tv_{2}^{2}-{\frac {1}{2}}\rho A_{1}v_{1}\Delta tv_{1}^{2}}$

${\displaystyle p_{1}A_{1}v_{1}\Delta t-p_{2}A_{2}v_{2}\Delta t+\rho gA_{1}v_{1}\Delta th_{1}-\rho gA_{2}v_{2}\Delta th_{2}={\frac {1}{2}}\rho A_{2}v_{2}\Delta tv_{2}^{2}-{\frac {1}{2}}\rho A_{1}v_{1}\Delta tv_{1}^{2}}$
${\displaystyle {\frac {\rho A_{1}v_{1}\Delta tv_{1}^{2}}{2}}+\rho gA_{1}v_{1}\Delta th_{1}+p_{1}A_{1}v_{1}\Delta t={\frac {\rho A_{2}v_{2}\Delta tv_{2}^{2}}{2}}+\rho gA_{2}v_{2}\Delta th_{2}+p_{2}A_{2}v_{2}\Delta t.}$

${\displaystyle A_{1}v_{1}=A_{2}v_{2}={\mbox{constant}}}$

${\displaystyle {\mbox{constant}}=\Delta V\;}$

${\displaystyle {\frac {1}{2}}\rho v^{2}+\rho gh+p={\mbox{constant}}}$

${\displaystyle {\frac {v^{2}}{2g}}+h+{\frac {p}{\rho g}}={\mbox{constant}}}$

## 垂直流线方向的加速度定律

${\displaystyle \textstyle \sum \delta F_{n}\displaystyle ={\frac {\delta mV^{2}}{\Re }}={\frac {\rho \delta \mathbb {V} V^{2}}{\Re }}{\bar {V}}}$，其中${\displaystyle \mathbb {V} =\delta s\delta n\delta y}$为微小流体质点体积，${\displaystyle \rho }$为流体密度。

${\displaystyle \textstyle \delta W_{n}=-\delta W\cos \theta =-\gamma \delta \mathbb {V} \cos \theta }$，其中${\displaystyle \textstyle \gamma =\rho g}$

${\displaystyle \delta F_{pn}}$为质点于垂直方向上所受净压

{\displaystyle {\begin{aligned}\sum \delta F_{pn}&=(p-\delta p_{n})\delta s\delta y-(p+\delta p_{n})\delta s\delta y=-2\delta p_{n}\delta s\delta y\\&=-{\frac {\partial p}{\partial n}}\delta n\delta s\delta y=-{\frac {\partial p}{\partial n}}\delta \mathbb {V} \\\end{aligned}}}

${\displaystyle \textstyle \sum \delta F_{n}\displaystyle =\delta W_{n}+\delta F_{pn}=(-\gamma \cos \theta -{\frac {\partial p}{\partial n}})\delta \mathbb {V} }$

${\displaystyle (-\gamma {dz \over dn}-{\frac {\partial p}{\partial n}})={\rho V^{2} \over \Re }}$

${\displaystyle {\partial p \over \partial n}={dp \over dn}}$

${\displaystyle \int {dp \over \rho }+\int {V^{2} \over \Re }dn+gz={\mbox{constant along the streamline}}}$

${\displaystyle p+\rho \int {V^{2} \over \Re }dn+\gamma z={\mbox{constant along the streamline}}}$

1.跨过流线的运动方程 ${\displaystyle (-\gamma {dz \over dn}-{\frac {\partial p}{\partial n}})={\rho V^{2} \over \Re }}$

${\displaystyle p+\rho \int {V^{2} \over \Re }dn+\gamma z={\mbox{constant along the streamline}}}$

2.沿着流线的运动方程 同上述做法[3]，可得出沿着流线方向之运动方程

${\displaystyle (-\gamma {dz \over ds}-{\frac {\partial p}{\partial s}})={\rho \over 2}{dV^{2} \over ds}}$

${\displaystyle p+{\tfrac {1}{2}}\rho V^{2}+\gamma z={\mbox{constant along the streamline}}}$

## 可压缩流体的伯努利定律

### 可压缩流体之流体力学

${\displaystyle {\frac {v^{2}}{2}}+\int _{p_{1}}^{p}{\frac {d{\tilde {p}}}{\rho ({\tilde {p}})}}\ +\Psi ={\text{constant}}}$   （流线型下的守恒）

${\displaystyle p=\;}$ 压力
${\displaystyle \rho =\;}$ 密度
${\displaystyle v=\;}$ 流速
${\displaystyle \Psi =\;}$ 保守力场下的位势，通常指重力位势

${\displaystyle {\frac {v^{2}}{2}}+gz+\left({\frac {\gamma }{\gamma -1}}\right){\frac {p}{\rho }}={\text{constant}}}$   （流线型下的守恒）

${\displaystyle \gamma =\;}$ 绝热指数
${\displaystyle g=\;}$ 重力加速度
${\displaystyle z=\;}$ 离参考平面的高度

${\displaystyle {\frac {v^{2}}{2}}+\left({\frac {\gamma }{\gamma -1}}\right){\frac {p}{\rho }}=\left({\frac {\gamma }{\gamma -1}}\right){\frac {p_{0}}{\rho _{0}}}}$

${\displaystyle p_{0}=\;}$ 总压力
${\displaystyle \rho _{0}=\;}$ 总密度

### 可压缩流动的热力学

${\displaystyle {v^{2} \over 2}+\Psi +w={\text{constant}}}$

${\displaystyle v=\;}$ 流速
${\displaystyle \Psi =\;}$ 重力位势
${\displaystyle w=\;}$ 单位质量的（通常写作${\displaystyle h}$，但注意并非表示高度）

${\displaystyle \Psi }$变化可以忽略，一个非常有用的形式的方程是:

${\displaystyle {v^{2} \over 2}+w=w_{0}}$

## 参考资料

1. ^ Bernoulli's Law -- from Eric Weisstein's World of Physics. [2017-11-22]. （原始内容存档于2017-06-27）.
2. ^ 伯努利定理的误解与错误 物理双月刊
3. BRUCE R. MUNSO； DONALD F. YOUNG；THEODORE H. OKIISHI；WADE W. HUEBSCH. Fundamentals of Fluid Mechanics. John Wiley & Sons Inc. 2013-01-22: page 96. ISBN 1118318676.
4. ^ Dennis Zill. Advanced Engineering Mathematics. Jones & Bartlett. 2012: 第22页. ISBN 9781449689803.
5. ^ 张慧贞 物理双月刊 37卷3期 教科书对于演示实例的理解及误解