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.
