# 磁流体力学

## 磁流体力学方程组

### 电磁场方程

${\displaystyle \nabla \cdot {\boldsymbol {E}}={\frac {\rho }{\varepsilon _{0}}}}$
${\displaystyle \nabla \times {\boldsymbol {E}}=-{\frac {\partial {\boldsymbol {B}}}{\partial t}}}$
${\displaystyle \nabla \cdot {\boldsymbol {B}}=0}$
${\displaystyle \nabla \times {\boldsymbol {B}}=\mu _{0}{\boldsymbol {J}}}$

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

### 流体力学方程

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

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

${\displaystyle {\boldsymbol {P}}=2\eta {\boldsymbol {S}}-{\bigg (}p+{\frac {2}{3}}\eta \nabla \cdot {\boldsymbol {v}}-\eta '\nabla \cdot {\boldsymbol {v}}{\bigg )}{\boldsymbol {I}}}$

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

### 状态方程

${\displaystyle p=p(\rho ,T)}$

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

### 理想磁流体力学方程组

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

## 磁张力与磁压力

${\displaystyle {\boldsymbol {f}}={\boldsymbol {J}}\times {\boldsymbol {B}}={\frac {1}{\mu _{0}}}({\boldsymbol {B}}\cdot \nabla ){\boldsymbol {B}}-\nabla ({\frac {B^{2}}{2\mu _{0}}})}$

## 磁扩散与磁冻结

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

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

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