| 类似流体力学，等离子体物理的方程同样是非线性偏微分方程，等离子体可以认为就是带电的流体。不同于流体，等离子体的温度非常高，所以尽量不要让等离子体接触容器壁。此外，等离子体高温导致即使压强很高，密度也非常低。等离子体的运动除了需要类似流体力学的方程外，还需要电磁学方程。 磁流体（MHD）方程 最简单的等离子体方程就是磁流体方程，只需要在可压缩流体N-S方程中加入安培力项\({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over f}}=\frac{{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over F}}}{V}=\frac{I{\mathord{ \buildrel{\tiny \rightharpoonup} \over l}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}}{{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over S}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over l}}}=\frac{({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over J}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over S}}){\mathord{ \buildrel{\tiny \rightharpoonup} \over l}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}}{{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over S}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over l}}}={\mathord{ \buildrel{\scriptsize \rightharpoonup} \over J}}\times {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}\)，其中\({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over J}}//{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over l}}\)，得到方程：$$\rho\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p+\eta\nabla^2{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}+\left(\frac{1}{3}\eta+\eta_V\right){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}})+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over J}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over f}}$$ 如果等离子体带电，例如静电约束，还要加入电场项\(\rho_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}\)：$$\rho\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p+\eta\nabla^2{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}+\left(\frac{1}{3}\eta+\eta_V\right){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}})+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over J}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}+\rho_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over f}}$$ 很多时候可以忽略黏性和其他力，近似为：$$\rho\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over J}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}$$$$\rho\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over J}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}+\rho_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}$$ 此外，还需要可压缩流体的连续性方程：$$\frac{\partial\rho}{\partial t}+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot\rho {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}=0$$ 以及能量方程，增加焦耳项\(\frac{\delta Q_V}{{\rm d} t}=\frac{1}{V}\frac{I^2R{\rm d}t}{{\rm d} t}=\frac{J^2S^2}{Sl}\rho_R\frac{l}{S}=\rho_R J^2\)：$$\rho\left(\frac{\partial u}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}})u\right)=k\nabla^2 T-p({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}})+\rho_R J^2+\mathit{\Phi}$$ 此外，还有广义欧姆定律\({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over J}}=\frac{{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\times {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}}{\rho_L}\)以及麦克斯韦方程组。 NSP方程 纳维-斯托克斯-泊松（NSP）方程，就是在N-S方程加入静电项，\({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over f}}=\rho_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}=\rho_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\varphi\)，可以认为是磁流体方程的特例，主要用于描述带电流体而不是等离子体。同时需要麦克斯韦方程组的第一个方程。$$\rho\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p+\eta\nabla^2{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}+\left(\frac{1}{3}\eta+\eta_V\right){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}})+\rho_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\varphi+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over f}}$$$$\nabla^2\varphi=\frac{\rho_e}{\varepsilon}$$ 双流体MHD方程 等离子体的不同粒子会在电磁场的作用下发生一定的分离，例如霍尔效应等，很复杂，需要考虑两种成分。连续性方程需要分别列为：$$\frac{\partial n_e}{\partial t}+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot n_e {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}=0$$$$\frac{\partial n_i}{\partial t}+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot n_i {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}=0$$ \(n_e\)和\(n_i\)分别是电子和阳离子的数量密度。对于类N-S方程则变为：$$m_en_e\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_e}}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_e}}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_e}}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p_e+\eta_e\nabla^2{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}+\left(\frac{1}{3}\eta_e+\eta_{e,V}\right){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}})+en_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_e}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}+en_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}-\nu_em_en_e({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_e}}-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_i}})+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over f_e}}$$$$m_in_i\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_i}}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_i}}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_i}}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p_i+\eta_i\nabla^2{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}}+\left(\frac{1}{3}\eta_i+\eta_{i,V}\right){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v}})+Zen_i{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_i}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}+Zen_i{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}-\nu_im_in_i({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_i}}-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_e}})+{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over f_i}}$$ 增加了碰撞项，其中多出了碰撞频率。具有\(\nu_e n_e m_e=\nu_i n_i m_i\)的关系。可以近似为：$$m_en_e\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_e}}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_e}}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_i}}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p_e+en_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_e}}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}+en_e{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}-\nu_em_en_e({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_e}}-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_i}})$$$$m_in_i\left(\frac{\partial {\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_i}}}}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_i}}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}){\mathord{ \buildrel{\scriptsize \rightharpoonup} \over {v_i}}}\right)=-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}p_i+Zen_i{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_i}}\times{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over B}}+Zen_i{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over E}}-\nu_im_in_i({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_i}}-{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_e}})$$ 然后是能量方程，欧姆项基本只作用于电子，加入了热传递项，因为阳离子质量远大于电子，电子速度可以认为是入射角与反射角相等，速度大小几乎不变，得到动量改变大小为\(2m_ev_e\sin\theta=m_i\Delta v_i\)，平方得到\(\Delta E=4\sin^2\theta\frac{m_e}{m_i}E_e\)，积分平均\(\frac{\int^{\frac{\pi}{2}}_{0}\sin^2\theta{\rm d}\theta}{\frac{\pi}{2}}=\frac{1}{2}\)，得到\(\Delta E=\frac{2m_e}{m_i}E_e\)，从而得到\(\frac{2m_e}{m_i}n_e\nu_e\frac{3}{2}k_B\Delta T=3\frac{m_e}{m_i}n_e\nu_ek_B(T_e-T_i)\)，得到方程：$$m_en_e\left(\frac{\partial u_e}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_e}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}})u_e\right)=k_e\nabla^2 T_e-p_e({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_e}})+\rho_R J^2-3\frac{m_e}{m_i}n_e\nu_ek_B(T_e-T_i)+\mathit{\Phi}_e$$$$m_in_i\left(\frac{\partial u_i}{\partial t}+({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_i}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}})u_i\right)=k_i\nabla^2 T_i-p_i({\mathord{ \buildrel{\scriptsize \rightharpoonup} \over \nabla}}\cdot{\mathord{ \buildrel{\scriptsize \rightharpoonup} \over v_i}})+3\frac{m_e}{m_i}n_e\nu_ek_B(T_e-T_i)+\mathit{\Phi}_i$$ 总结 这里只介绍了公式，计算在后面。 |
