fluid equations

Equations that describe the behavior of a collection of particles in terms of fluid variables, that is quantities which are averaged over the velocity space of all particles.

A system of particles can be described by a distribution function, f(x,v,t). This function describes how many particles there are at a given velocity and position at any time.

The fluid density n (number of particles per unit volume), velocity v and pressure tensor P are defined as follows

n=∫ f(x,v,t) dv'

v= (1/n) ∫ f(x,v,t) v' dv'

P= m ∫ f(x,v,t) (v'-v) (v'-v) dv'

The Fokker-Planck equation is a kinetic equation describing the evolution of the distribution function f including the electromagnetic force. Generalising this to include any force F(so that the equations will describe any fluid and not just plasmas) the equation becomes

δf/δt + v.δf/δx + (F/m).(δf/δv)= (δf/δt)_col

By taking moments of the Fokker-Planck equation (i.e. multiplying it by multiples of the particle velocity v' and integrating it over velocity space), the fluid equations are obtained.

Zeroth moment
Multiply the F-P equation by 1 and integrate over velocity space. Do the math yourself remembering that the integral of the collision term will be zero since collisions will not change the total number of particles. The result is the continuity equation (dealing with the particle flux)

δn/δt + div.(nv)=0

First moment
Multiply the F-P equation by mv' (momentum). Employing a little mathematical dexterity will yield the following

m(δnv/δt) + div.P + m div.(nv v) - nF=R

nm(δv/δt + v(div.v))= -div.P + nF + R

One may take higher moments of the F-P equation ad infinitum. The second moment would be the pressure tensor P and the resulting equation deals with the energy flux. Every new equation introduces a quantity that needs a higher moment to solve it (v in the first equation, P in the second). To achieve closure in the fluid equations text books normally stop at the second moment by, for instance, assuming that the motion is adiabatic.