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, $f_s$, all have simple physical interpretations. First, we have the particle number density,

$$n_s({\bf r},t) = \int f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$$ (2.2)

and the mean flow velocity,

$${\bf V}_s({\bf r}, t) =\frac{1}{n_s} \int {\bf v}\,f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$$ (2.3)

Next, we have the pressure tensor,

$${\bf p}_s({\bf r}, t) = \int m_s\,{\bf u}_s{\bf u}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v},$$ (2.4)

and the heat flux,

$${\bf q}_s({\bf r}, t) = \int \frac{1}{2}\,m_s\,u_s^{2}\,{\bf u}_s\, f_s({\bf r}, {\bf v},t)\,d^3{\bf v}.$$ (2.5)

Here,

$${\bf u}_s = {\bf v} - {\bf V}_s.$$ (2.6)

The trace of the pressure tensor measures the ordinary (or scalar) pressure,

$$p_s= \frac{1}{3}\,{\rm Tr}\,({\bf p}_s).$$ (2.7)

The (kinetic) temperature is defined as

$$T_s = \frac{p_s}{n_s}.$$ (2.8)

The low-order velocity-space moments of the collision operator also have simple interpretations. The friction force density takes the form

$${\bf F}_{s}= \int m_s\,{\bf v}\,C_{s}(f_e,f_i)\,d^3{\bf v},$$ (2.9)

whereas the collisional heating rate density (in the species-$s$ rest frame) is written

$$W_s \equiv \int \frac{1}{2}\,m_s\,u_s^{2}\,C_{s}(f_e,f_i)\,d^3{\bf v}.$$ (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-$s$ [18]:

$$\frac{d_sn_s}{dt} + n_s\,\nabla\cdot{\bf V}_s =0,$$ (2.11)

$$m_s \,n_s\,\frac{d_s {\bf V}_s}{dt} - e_s \,n_s\, ({\bf E} + {\bf V}_s\times {\bf B})+ \nabla p_s + \nabla\cdot \mbox{\boldmath$\pi$} _s = {\bf F}_s,$$ (2.12)

$$\frac{3}{2}\frac{d_s p_s}{dt} + \frac{5}{2}\,p_s\,\nabla\cdot{\bf V}_s + \mbox{\boldmath$\pi$} _s:\nabla{\bf V}_s + \nabla \cdot{\bf q}_s = W_s.$$ (2.13)

Here,

$$\frac{d_s}{dt} \equiv \frac{\partial}{\partial t} + {\bf V}_s\cdot \nabla,$$ (2.14)

is the well-known convective derivative, and we have written

$${\bf p}_s = p_s\,{\bf I} + \mbox{\boldmath$\pi$} _s,$$ (2.15)

where ${\bf I}$ is the unit (identity) tensor, and $\pi$$_s$ is the viscosity tensor. Obviously, Equation (2.11) is a particle conservation equation for species-$s$, Equation (2.12) is a momentum conservation equation, and Equation (2.13) is an energy conservation equation [18].