## 预先知识

### 微分符号的欧拉表示法

$$$\partial_t \mathbf u \equiv \frac{\partial \mathbf u}{\partial t}$$$

### 求和的下标表示法

$$$a_i b_i \equiv \sum_{i=1}^{3}{a_i b_i} = a_1 b_1+a_2 b_2+a_3 b_3$$$

$$$\frac{\partial u_i}{\partial x_i} \equiv \sum_{i=1}^{3} \frac{\partial u_i}{\partial x_i} = \frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}+\frac{\partial u_3}{\partial x_3}$$$

$$$\frac{\partial (u_i u_j)}{\partial x_j} \equiv \sum_{j=1}^{3} \frac{\partial (u_i u_j)}{\partial x_j} = \frac{\partial (u_i u_1)}{\partial x_1}+\frac{\partial (u_i u_2)}{\partial x_2}+\frac{\partial (u_i u_3)}{\partial x_3}, \qquad i=1,2,3$$$

$$$\frac{\partial (u_i u_j)}{\partial x_j} = u_j \frac{\partial u_i}{\partial x_j} \equiv u_1 \frac{\partial u_i}{\partial x_1}+u_2\frac{\partial u_i}{\partial x_2}+u_3\frac{\partial u_i}{\partial x_3} , \qquad i=1,2,3$$$

### 张量运算的矢量表示法

$$$\nabla \cdot \mathbf u \equiv \frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}+\frac{\partial u_3}{\partial x_3}$$$

\nabla \cdot (\mathbf u \otimes \mathbf u) \equiv \left[ \begin{aligned} \sum_{j=1}^{3} \frac{\partial (u_1 u_j)}{\partial x_j} \\ \sum_{j=1}^{3} \frac{\partial (u_2 u_j)}{\partial x_j} \\ \sum_{j=1}^{3} \frac{\partial (u_3 u_j)}{\partial x_j} \end{aligned}\right] = \left[ \begin{aligned}\frac{\partial (u_1 u_1)}{\partial x_1}+\frac{\partial (u_1 u_2)}{\partial x_2}+\frac{\partial (u_1 u_3)}{\partial x_3} \\ \frac{\partial (u_2 u_1)}{\partial x_1}+\frac{\partial (u_2 u_2)}{\partial x_2}+\frac{\partial (u_2 u_3)}{\partial x_3} \\ \frac{\partial (u_3 u_1)}{\partial x_1}+\frac{\partial (u_3 u_2)}{\partial x_2}+\frac{\partial (u_3 u_3)}{\partial x_3} \end{aligned} \right]

\nabla \cdot (\mathbf u \mathbf u) = (\mathbf u \cdot \nabla)\mathbf u \equiv \left[ \begin{aligned}u_1\frac{\partial u_1}{\partial x_1}+u_2\frac{\partial u_1}{\partial x_2}+u_3\frac{\partial u_1}{\partial x_3} \\ u_1\frac{\partial u_2}{\partial x_1}+u_2\frac{\partial u_2}{\partial x_2}+u_3\frac{\partial u_2}{\partial x_3} \\ u_1\frac{\partial u_3}{\partial x_1}+u_2\frac{\partial u_3}{\partial x_2}+u_3\frac{\partial u_3}{\partial x_3} \end{aligned} \right]

$$$(\mathbf u \cdot \nabla) \equiv \left( u_1\frac{\partial}{\partial x_1}+u_2\frac{\partial}{\partial x_2}+u_3\frac{\partial}{\partial x_3} \right)$$$

### 张量运算符的缩写表示

$$$\operatorname{grad}\phi \equiv \nabla \phi$$$

$$$\operatorname{div}\mathbf u \equiv \nabla \cdot \mathbf u$$$

$$$\operatorname{div}\operatorname{grad}\phi \equiv \nabla \cdot (\nabla \phi) = \nabla^2\phi = \Delta \phi$$$

$$$\operatorname{curl}\mathbf u \equiv \nabla \times \mathbf u$$$

## 不可压缩 Navier-Stokes 方程

### 下标表示

$$$\frac{\partial u_i}{\partial x_i} = 0$$$

$$$\frac{\partial u_i}{\partial t} + \frac{\partial (u_i u_j)}{\partial x_j} = - \frac{1}{\rho} \frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}\left[\nu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)\right]$$$

$$$\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = - \frac{1}{\rho} \frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}\left[\nu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)\right]$$$

### 矢量表示

$$$\nabla \cdot \mathbf u = 0$$$

$$$\frac{\partial \mathbf u}{\partial t} + \nabla \cdot (\mathbf u \mathbf u) = -\frac{1}{\rho}\nabla p + \nabla \cdot \left\{\nu \left[ \nabla \mathbf u + (\nabla \mathbf u)^T \right] \right\}$$$

$$$\frac{\partial \mathbf u}{\partial t} + (\mathbf u \cdot \nabla) \mathbf u = -\frac{1}{\rho}\nabla p + \nabla \cdot \left\{\nu \left[ \nabla \mathbf u + (\nabla \mathbf u)^T \right] \right\}$$$

### 缩写表示

$$$\operatorname{div} \mathbf u = 0$$$

$$$\frac{\partial \mathbf u}{\partial t} + \operatorname{div} (\mathbf u \mathbf u) = -\frac{1}{\rho} \operatorname{grad} p + \operatorname{div} \left\{\nu\left[\operatorname{grad}\mathbf u + (\operatorname{grad}\mathbf u)^T\right]\right\}$$$

$$$\frac{\partial \mathbf u}{\partial t} + (\mathbf u \cdot \operatorname{grad} )\mathbf u = -\frac{1}{\rho} \operatorname{grad} p + \operatorname{div} \left\{\nu\left[\operatorname{grad}\mathbf u + (\operatorname{grad}\mathbf u)^T\right]\right\}$$$

### 简化形式

$$$\nabla \cdot \left\{\nu \left[ \nabla \mathbf u + (\nabla \mathbf u)^T \right] \right\} = \nu \nabla \cdot \left[ \nabla \mathbf u + (\nabla \mathbf u)^T\right]$$$

$$$\nabla \cdot (\nabla \mathbf u)^T = \nabla (\nabla \cdot \mathbf u) = 0$$$

$$$\nu \nabla \cdot \left[ \nabla \mathbf u + (\nabla \mathbf u)^T\right] = \nu \nabla^2 \mathbf u$$$

$$$\frac{\partial \mathbf u}{\partial t} + (\mathbf u \cdot \nabla) \mathbf u = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf u$$$

### 其他

Navier-Stokes 方程更一般的形式为柯西动量方程，其形式如下：

$$$\frac{D \mathbf{u}}{D t} = \frac{1}{\rho} \nabla \cdot \boldsymbol{\sigma}$$$

$$$\frac{D \mathbf u}{D t} = \frac{\partial \mathbf u}{\partial t} + (\mathbf u \cdot \nabla) \mathbf u$$$

$\boldsymbol \sigma$ 为柯西张量，包含压力和剪切应力，可写作

$\boldsymbol \sigma = -p \mathbf I + \boldsymbol \tau$

$$$\boldsymbol \tau = 2 \mu \mathbf S = \mu \left[\nabla \mathbf u + (\nabla \mathbf u)^T\right]$$$

\begin{aligned} \frac{\partial \mathbf u}{\partial t} + (\mathbf u \cdot \nabla) \mathbf u &= \frac{1}{\rho} \nabla \cdot \left\{-p \mathbf I + \mu \left[\nabla \mathbf u + (\nabla \mathbf u)^T\right]\right\} \\ & = - \frac{1}{\rho}\nabla p + \nabla \cdot \left\{\nu \left[\nabla \mathbf u + (\nabla \mathbf u)^T\right]\right\} \end{aligned}