[物理学与PDEs]第3章第2节 磁流体力学方程组 2.3 磁流体力学方程组

1.  磁流体力学方程组 $$\beex \bea \cfrac{\p {\bf H}}{\p t} &-\rot({\bf u}\times{\bf H})=\cfrac{1}{\sigma\mu_0}\lap{\bf H},\\ \Div&{\bf H}=0,\\ \cfrac{\p \rho}{\p t}&+\Div(\rho {\bf u})=0,\\ \cfrac{\p (\rho{\bf u})}{\p t}&+\Div(\rho{\bf u}\times{\bf u}-{\bf P}) -\mu_0\rot{\bf H}\times{\bf H}=\rho {\bf F},\\ \cfrac{\p}{\p t}&\sex{\rho e+\cfrac{1}{2}\rho u^2+\cfrac{1}{2}\mu_0 H^2} +\Div\sez{\sex{\rho e+\cfrac{1}{2}\rho u^2}{\bf u}-{\bf P} {\bf u}}\\ &+\Div\sez{\cfrac{1}{\sigma}\rot{\bf H}\times{\bf H}-\mu_0({\bf u}\times{\bf H})\times{\bf H}} =\Div(\kappa \n T)+\rho {\bf F}\cdot{\bf u}. \eea \eeex$$

2.  由于电导率 $\sigma$ 的存在, 磁场具有扩散性.