The single fluid Magnetohydrodynamics equation

The single-fluid magnetohydrodynamics (MHD) equations describe the behavior of a magnetized plasma as a single fluid. They combine the equations of fluid dynamics with Maxwell's equations to account for the motion of charged particles in the presence of magnetic fields. The single-fluid MHD equations are derived by assuming that the plasma is well-mixed and exhibits a single, average velocity and temperature.

In the fully ionized plasma where we have regarded the plasma composed of two interpenetrating fluid i.e ion fluids and electron fluids. The collision between the oppositely charged particles takes place that causes diffusion. But the problem is in tackling such diffusion equation is that the term Pei depends upon Vi - Ve i.e two unknowns. Hence a linear combination of equation for two different fluids is made and the combined form reduces the two fluid plasma problem into single fluid plasma and the obtained equations are of magnetohydrodynamics

In single fluid plasma equation, Pei just depends on diff(Vi - Ve) as a whole but not on two velocities separately. The ion and electron equation are

Adding 1 and 2, we get

Defining mass density

Mass velocity

Current density

This is the conservation of mass in a fluid or plasma. This consider the plasma as a single fluid with mass density. The electric field appear explicitly because the fluid is neutral.

The two equations may be combined as m(3) - m(4) which on simplification

Equation 5 is called generalized ohm's law which describes the electric properties of conducting fluid. The hall current term J xB and last term are small enough to be neglected.

The set of MHD equations are

This collection of equations is frequently used to describe the equilibrium state of plasma along with Maxwell's equations. When dealing with complex magnetic geometries and MHD energy conservation, the MHD equations have been employed extensively.

This note is taken from Plasma, Msc physics, Nepal.

This note is a part of the Physics Repository.