What is the most general form of the momentum equation in MHD?

The momentum equation for an electrically neutral, conducting, nonpermeable fluid has the form (Jackson, 1962):

$$\rho \frac{d \mathbf v}{d t} = - \nabla p + \mathbf J \times \mathbf B + \mathbf F_v + \rho \mathbf g$$

where $\mathbf F_v$ is the viscous force, which in the Navier-Stokes momentum equation is usually written in terms of the stress deviator tensor $\mathbf \tau$:

$$\mathbf F_v = \nabla \cdot \mathbf \tau$$

What is the corresponding form for an electrically non-neutral, permeable fluid?

I am particularly interested in the case of a permeable material. I expect that in this case the magnetic field $\mathbf H$ should appear in the equation instead of the magnetic induction $\mathbf B$.

References from book/articles are much appreciated.

• I've only ever seen the MHD equations using either the magnetic field B (as in my answer here) or the vector potential A (as in here), on in terms of H. – Kyle Kanos Jul 16 '17 at 20:02
• Whoops. The last phrase above should be , not in terms of H. Hope that didn't confuse anyone. – Kyle Kanos Jul 17 '17 at 10:10
• @KyleKanos I am starting to suspect that this form is ok for both permeable and nonpermeable fluids, as long as we consider $J$ to be the sum of both bound (microscopic) and free (macroscopic) currents. – valerio Jul 17 '17 at 10:22
• @valerio92 - I would look up the generalized Ohm's law in its most general form, then use that to replace the electric field. For instance, you are missing some terms in your present form. – honeste_vivere Jul 17 '17 at 15:36
• @honeste_vivere Which terms am I missing? – valerio Jul 17 '17 at 22:43