A Brief History of MHD
Magnetohydrodynamics, MHD for short, is the study of electrically-conducting fluids. Strictly-speaking, the study dates back as far as to when people first began playing with liquid mercury, although it was Chandrasekhar’s investigations into astrophysics in the early-20th century which really began the theoretical study of the discipline.
Deriving MHD
There are many flavors of MHD. At the lowest level, every one of them begins by taking moments of a kinetic equation with a suitable closure.
Two-Fluid MHD
Single-Fluid MHD
Ideal MHD
Resistive MHD
Hall MHD
5N-Moment Multi-Fluid
This is the penultimate kind of MHD model. 5 variables are evolved that describe the flow of mass, momentum, and energy in each plasma species,
\[\vec{Q} = \colvec{\rho \\ u \\ v \\ w \\ e}\]Obviously, some assumptions are simplifying things here.
13N-Moment Multi-Fluid
This is the most general kind of MHD model. 13 variables are evolved that describe the flow of mass, momentum, energy (based on an anisotropic pressure tensor), and heat flux in each plasma species,
\[\vec{Q} = \colvec{\rho \\ u \\ v \\ w \\ p_{xx} \\ p_{xy} \\ p_{xz} \\ p_{yy} \\ p_{yz} \\ p_{zz} \\ q_{x} \\ q_{y} \\ q_{z}}\]