# 磁流体力学

## 磁流体力学方程组

### 电磁场方程

$\nabla\cdot\boldsymbol{E}=\frac{\rho}{\varepsilon_0}$
$\nabla\times\boldsymbol{E}=-\frac{\partial\boldsymbol{B}}{\partial t}$
$\nabla\cdot\boldsymbol{B}=0$
$\nabla\times\boldsymbol{B}=\mu_0\boldsymbol{J}$

$\boldsymbol{J}=\sigma(\boldsymbol{E}+\boldsymbol{v}\times\boldsymbol{B})$

### 流体力学方程

$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\boldsymbol{v})=0$

$\rho\frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t} = \nabla\cdot\boldsymbol{P}+\boldsymbol{J}\times\boldsymbol{B}$

$\boldsymbol{P}=2\eta\boldsymbol{S}-\bigg(p+\frac23\eta\nabla\cdot\boldsymbol{v} - \eta'\nabla\cdot\boldsymbol{v}\bigg)\boldsymbol{I}$

$\rho\frac{\mathrm{d}}{\mathrm{d}t}\bigg(\varepsilon+\frac{\boldsymbol{v}^2}{2}\bigg) = \nabla\cdot(\boldsymbol{P}\cdot\boldsymbol{v})+\boldsymbol{E}\cdot\boldsymbol{J} - \nabla\cdot\boldsymbol{q}$

### 状态方程

$p=p(\rho,T)$

$\frac{\mathrm{d}}{\mathrm{d}t}(p\rho^{-\gamma})=0$$p\rho^{-\gamma}=\mathrm{const}$

### 理想磁流体力学方程组

$\nabla\times\boldsymbol{E}=-\frac{\partial\boldsymbol{B}}{\partial t}$
$\nabla\times\boldsymbol{B}=\mu_0\boldsymbol{J}$
$\boldsymbol{E}+\boldsymbol{v}\times\boldsymbol{B}=0$
$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\boldsymbol{v})=0$
$\rho\frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t} = \nabla\cdot\boldsymbol{P}+\boldsymbol{J}\times\boldsymbol{B}$
$p\rho^{-\gamma}=\mathrm{const}$

## 磁张力与磁压力

$\boldsymbol{f}=\boldsymbol{J}\times\boldsymbol{B} = \frac{1}{\mu_0}(\boldsymbol{B}\cdot\nabla)\boldsymbol{B} - \nabla(\frac{B^2}{2\mu_0})$

## 磁扩散与磁冻结

$\frac{\partial\boldsymbol{B}}{\partial t}=\nabla\times(\boldsymbol{v}\times\boldsymbol{B})+\eta\nabla^2\boldsymbol{B}$

$\frac{\partial\boldsymbol{B}}{\partial t}=\eta\nabla^2\boldsymbol{B}$

$\frac{\partial\boldsymbol{B}}{\partial t}=\nabla\times(\boldsymbol{v}\times\boldsymbol{B})$