Fluid Theory
Plasma fluid equations are obtained by taking low-order velocity-space moments of the kinetic equation [18].
The low-order moments of the distribution function,
, all have simple physical interpretations.
First, we have the particle number density,
![$\displaystyle n_s({\bf r},t) = \int f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$](img456.png) |
(2.2) |
and the mean flow velocity,
![$\displaystyle {\bf V}_s({\bf r}, t) =\frac{1}{n_s} \int
{\bf v}\,f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$](img457.png) |
(2.3) |
Next, we have the pressure tensor,
![$\displaystyle {\bf p}_s({\bf r}, t) = \int
m_s\,{\bf u}_s{\bf u}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$](img458.png) |
(2.4) |
and the heat flux,
![$\displaystyle {\bf q}_s({\bf r}, t) = \int
\frac{1}{2}\,m_s\,u_s^{2}\,{\bf u}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$](img459.png) |
(2.5) |
Here,
![$\displaystyle {\bf u}_s = {\bf v} - {\bf V}_s.$](img460.png) |
(2.6) |
The trace of the pressure tensor measures the ordinary (or scalar) pressure,
![$\displaystyle p_s= \frac{1}{3}\,{\rm Tr}\,({\bf p}_s).$](img461.png) |
(2.7) |
The (kinetic) temperature is defined as
![$\displaystyle T_s = \frac{p_s}{n_s}.$](img462.png) |
(2.8) |
The low-order velocity-space moments of the collision operator also have simple interpretations.
The friction force density takes the form
![$\displaystyle {\bf F}_{s}= \int m_s\,{\bf v}\,C_{s}(f_e,f_i)\,d^3{\bf v},$](img463.png) |
(2.9) |
whereas the collisional heating rate density (in the species-
rest frame) is written
![$\displaystyle W_s \equiv \int \frac{1}{2}\,m_s\,u_s^{2}\,C_{s}(f_e,f_i)\,d^3{\bf v}.$](img464.png) |
(2.10) |
The zeroth, first, and contracted second velocity-space moments of the kinetic equation, (2.1),
yield the following set of fluid equations for species-
[18]:
Here,
![$\displaystyle \frac{d_s}{dt} \equiv \frac{\partial}{\partial t} + {\bf V}_s\cdot \nabla,$](img474.png) |
(2.14) |
is the well-known convective derivative, and
we have written
where
is the unit (identity) tensor, and
![$\pi$](img478.png)
is the viscosity tensor. Obviously, Equation (2.11) is a particle conservation
equation for species-
, Equation (2.12) is a momentum conservation equation, and Equation (2.13) is an energy conservation equation [18].