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It is conventional to rewrite our fluid equations in terms of the pressure tensor,
,
and the heat flux density, . Substituting from Eqs. (190)(191), and performing a little
tensor
algebra, Eqs. (209)(211) reduce to:
Here,

(217) 
is the wellknown convective derivative, and

(218) 
In the above, and refer to Cartesian components, and repeated
indices are summed (according to the Einstein summation convention).
The convective derivative, of course, measures time variation in
the local rest frame of the species fluid.
Strictly
speaking, we should include an subscript with each convective derivative,
since this operator is clearly different for different plasma species.
There is one additional refinement to our fluid equations which is worth
carrying out. We introduce the generalized viscosity tensor,
,
by writing

(219) 
where is the unit (identity) tensor. We expect the scalar
pressure term to dominate if the plasma is relatively close to thermal
equilibrium. We also expect, by analogy with conventional fluid theory, the
second term to describe viscous stresses. Indeed, this is generally the case
in plasmas,
although the generalized viscosity tensor can also include terms which are
quite unrelated to conventional viscosity. Equations (214)(216) can, thus, be rewritten:
According to Eq. (220), the species density is constant along a fluid
trajectory unless the species flow is nonsolenoidal. For this reason,
the condition

(223) 
is said to describe incompressible species flow. According to
Eq. (221), the species flow accelerates
along a fluid trajectory under the influence of the
scalar pressure gradient, the viscous stresses, the Lorentz force, and the
frictional force due to collisions with other species. Finally,
according to Eq. (222), the species energy density (i.e., )
changes along a fluid trajectory
because of the work done in compressing the fluid, viscous heating,
heat flow, and the local energy gain due to collisions with other species.
Note that the electrical contribution to plasma heating, which was explicit
in Eq. (211), has now become entirely implicit.
Next: Entropy Production
Up: Plasma Fluid Theory
Previous: Moments of the Kinetic
Richard Fitzpatrick
20110331