HMHD equations in conservative form
\[
\begin{aligned}
\frac{\partial \rho}{\partial t} + \nabla\!\cdot(\rho\,\mathbf{u}) &= 0
\\[6pt]
\frac{\partial (\rho \mathbf{u})}{\partial t}
+ \nabla\!\cdot\!\left(\;\rho\,\mathbf{u}\,\mathbf{u}
+ p_t \, \mathbf{I} + \frac{B^2}{2\mu_0}\,\mathbf{I}
- \frac{\mathbf{B}\mathbf{B}}{\mu_0}\right)
&= \rho\,\mathbf{g}
\\[6pt]
\frac{\partial \varepsilon}{\partial t}
+ \nabla\!\cdot\!\left[
\!\left(\varepsilon - \frac{B^2}{2\mu_0} + p_t \right)\mathbf{u}
+ \frac{ \mathbf{E}\times\mathbf{B} }{\mu_0}
\right] &= \rho\,\mathbf{g}\!\cdot\!\mathbf{u}
\\[6pt]
\frac{\partial \mathbf{B}}{\partial t} + \nabla\!\cdot\!(\mathbf{u}_e \mathbf{B} -
\mathbf{B}\mathbf{u}_e) &= -\nabla\times(\eta\,\mathbf{J})
\end{aligned}
\]
Notation: \(\mathbf{I}\) is the 3\(\times\)3 identity tensor;
\(\varepsilon = \frac{p_t}{\gamma-1} + \frac{\rho\,u^2}{2} + \frac{B^2}{2\mu_0}\) is the total energy density (without gravitational potential energy); \(\mathbf{J}=\nabla\times\mathbf{B}/\mu_0\); and \(\mathbf{u}_e=\mathbf{u}-\mathbf{J}/(e n_e)\). The induction equation is written in Hall form, so the ideal magnetic advection uses the electron velocity. For constant \(\eta\), and using \(\nabla\cdot\mathbf{B}=0\), \(-\nabla\times(\eta\mathbf{J})=(\eta/\mu_0)\nabla^2\mathbf{B}\).
The two-fluid MHD (ion fluid + electron fluid) equations with \(p_t\) and E expanded:
\[
\begin{aligned}
\frac{\partial \rho}{\partial t} + \nabla\!\cdot(\rho\,\mathbf{u}) &= 0
\\[6pt]
\frac{\partial (\rho \mathbf{u})}{\partial t}
+ \nabla\!\cdot\!\left[\;\mathbf{u}\,\rho\,\mathbf{u}
+ \mathbf{I}\,(p_i + p_e)
+ \mathbf{I}\,\frac{B^2}{2\mu_0}
- \frac{\mathbf{B}\mathbf{B}}{\mu_0}\right]
&= \rho\,\mathbf{g}
\\[6pt]
\frac{\partial \varepsilon_i}{\partial t}
+ \nabla\!\cdot\!\left[
\!\left(\varepsilon_i + p_i - \frac{B^2}{2\mu_0}\right)\mathbf{u}
+ \!\frac{ \gamma \, p_e}{\gamma-1} \mathbf{u}_e
+ \frac{\left(\mathbf{I}\,B^2
- \mathbf{B}\mathbf{B}\right)}{\mu_0}\!\cdot\!\mathbf{u}_e
- \frac{\mathbf{B}}{\mu_0} \times\left( \eta\,\mathbf{J}
- \frac{\nabla p_e}{e\,n_e} \right)
\right] &= \rho\,\mathbf{g}\!\cdot\!\mathbf{u}
\\[6pt]
\frac{\partial p_e}{\partial t} + \nabla\!\cdot(p_e\,\mathbf{u}_e)
&= (1-\gamma)(\,p_e\,\nabla\!\cdot\mathbf{u}_e-\eta J^2)
\\[6pt]
\frac{\partial \mathbf{B}}{\partial t}
+ \nabla\cdot
\left( \mathbf{u}_e\mathbf{B} - \mathbf{B}\mathbf{u}_e \right)
= -\nabla\times(\eta\mathbf{J})
+ \nabla\times \left( \frac{\nabla p_e}{e n_e} \right)
\end{aligned}
\]
In this form, the input pressure is the ion thermal pressure \(p_i\), which is advanced with the ion total energy equation, while the electron pressure \(p_e\) is advanced separately. The electric field is \(\mathbf{E} = \eta\,\mathbf{J} -\mathbf{u}\times\mathbf{B} + \frac{\mathbf{J}\times\mathbf{B}}{e\,n_e} - \frac{\nabla p_e}{e\,n_e}\). This equation is named the generalized Ohm's law, but it enters the dynamic equation coz it comes from the electron momentum equation after neglecting electron mass. We note that by keeping \(p_e\), we are still assuming non-zero electron mass, but as single particles, not as a bulk fluid.
Sometimes we may also play with these equations
\[
\begin{aligned}
R_m &= \frac{uL}{\eta}; R = \frac{uL}{\nu}
\\[6pt]
\end{aligned}
\]