# 雷诺数

## 定义

### 管内流场

$\mathrm{Re} = {{\rho {\bold \mathrm V} D} \over {\mu}}= {{{\bold \mathrm V} D} \over {\nu}} = {{{\bold \mathrm Q} D} \over {\nu}A}$

• ${\bold \mathrm V}$是平均流速（国际单位：m/s）
• ${D}$管直径（一般為特徵長度）(m)
• ${\mu}$流体动力黏度（Pa·s或N·s/m²）
• ${\nu}$ 运动黏度$\nu = \mu /$ρ）(m²/s)
• ${\rho}$流体密度（kg/m³）
• ${Q}$体积流量（m³/s）
• ${A}$横截面积（m²）

### 搅拌槽

$\mathrm{Re} = {{\rho N D^2} \over {\mu}}.$

## 流动相似性

$\mathrm{Re}_m = \mathrm{Re} \;$
$\mathrm{Eu}_m = \mathrm{Eu} \; \quad\quad \mbox{i.e.} \quad {p_m \over \varrho_m {v_m}^{2}} = {p\over \varrho v^{2}} \; ,$

## 雷诺数的推导

$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}.$

$\frac{D}{\rho V^2}$

$\mathbf{v'} = \frac{\mathbf{v}}{V},\ p' = p\frac{1}{\rho V^2}, \ \mathbf{f'} = \mathbf{f}\frac{D}{\rho V^2}, \ \frac{\partial}{\partial t'} = \frac{D}{V} \frac{\partial}{\partial t}, \ \nabla' = D \nabla$

$\frac{\partial \mathbf{v'}}{\partial t'} + \mathbf{v'} \cdot \nabla' \mathbf{v'} = -\nabla' p' + \frac{\mu}{\rho D V} \nabla'^2 \mathbf{v'} + \mathbf{f'}$

$\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} = -\nabla p + \frac{1}{\mathit{Re}} \nabla^2 \mathbf{v} + \mathbf{f}.$

## 參考文獻

1. ^ R. K. Sinnott Coulson & Richardson's Chemical Engineering, Volume 6: Chemical Engineering Design, 4th ed (Butterworth-Heinemann) ISBN 0-7506-6538-6 page 473