HMHD Equations in conservative form

\[ \begin{aligned} \frac{\partial \rho}{\partial t} + \nabla\!\cdot(\rho\,\mathbf{u}) &= 0 \\[6pt] \frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla\!\cdot\!\left(\;\rho\,\mathbf{u}\,\mathbf{u} + p_t \, \mathbf{I} + \frac{B^2}{2\mu_0}\,\mathbf{I} - \frac{\mathbf{B}\mathbf{B}}{\mu_0}\right) &= \rho\,\mathbf{g} \\[6pt] \frac{\partial \varepsilon}{\partial t} + \nabla\!\cdot\!\left[ \!\left(\varepsilon + p_t \right)\mathbf{u} + \frac{ \mathbf{E}\times\mathbf{B} }{\mu_0} \right] &= 0 \\[6pt] \frac{\partial \mathbf{B}}{\partial t} + \nabla\!\cdot\!(\mathbf{u} \mathbf{B} - \mathbf{B}\mathbf{u}) &= \frac{\eta \nabla^2 B}{\mu_0} \end{aligned} \] }

\[ \begin{aligned} \frac{\partial \rho}{\partial t} + \nabla\!\cdot(\rho\,\mathbf{u}) &= 0 \\[6pt] \frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla\!\cdot\!\left[\;\mathbf{u}\,\rho\,\mathbf{u} + \mathbf{I}\,(p_i + p_e) + \mathbf{I}\,\frac{B^2}{2\mu_0} - \frac{\mathbf{B}\mathbf{B}}{\mu_0}\right] &= \rho\,\mathbf{g} \\[6pt] \frac{\partial \varepsilon_i}{\partial t} + \nabla\!\cdot\!\left[ \!\left(\varepsilon_i + p_i - \frac{B^2}{2\mu_0}\right)\mathbf{u} + \!\frac{ \gamma \, p_e}{\gamma-1} \mathbf{u}_e + \frac{\left(\mathbf{I}\,B^2 - \mathbf{B}\mathbf{B}\right)}{\mu_0}\!\cdot\!\mathbf{u}_e - \mathbf{B}\times\left( \eta\,\mathbf{J} - \frac{\nabla p_e}{e\,n_e} \right) \right] &= 0 \\[6pt] \frac{\partial p_e}{\partial t} + \nabla\!\cdot(p_e\,\mathbf{u}_e) &= (1-\gamma)(\,p_e\,\nabla\!\cdot\mathbf{u}_e-\eta J^2) \\[6pt] \frac{\partial \mathbf{B}}{\partial t} &= -\,\nabla\times\mathbf{E} \end{aligned} \]

\[ \begin{aligned} R_m &= \frac{uL}{\eta}; R = \frac{uL}{\nu} \\[6pt] \end{aligned} \]