Fluid Equations

It is conventional to rewrite our fluid equations in terms of the pressure tensor, ${\bf p}_s$ , and the heat flux density, ${\bf q}_s$ . Substituting from Equations (4.17) and (4.18), and performing a little tensor algebra, Equations (4.36)-(4.38) reduce to:

$$\frac{dn_s}{dt} + n_s\,\nabla\cdot{\bf V}_s =0,$$ (4.41)

$$m_s\, n_s\,\frac{d {\bf V}_s}{dt} + \nabla\cdot {\bf p}_s - e_s \,n_s\, ({\bf E} + {\bf V}_s\times {\bf B}) = {\bf F}_s,$$ (4.42)

$$\frac{3}{2}\frac{d p_s}{dt} + \frac{3}{2}\,p_s\,\nabla\cdot{\bf V}_s + {\bf p}_s:\nabla{\bf V}_s + \nabla\cdot{\bf q}_s = W_s.$$ (4.43)

Here,

$$\frac{d}{dt} \equiv \frac{\partial}{\partial t} + {\bf V_s}\cdot \nabla$$ (4.44)

is the well-known convective derivative, and

$${\bf p} : \nabla {\bf V}_s \equiv (p_s)_{\alpha\beta} \,\frac{\partial (V_s)_{\beta}}{\partial r_\alpha}.$$ (4.45)

In the previous expression, $\alpha$ and $\beta$ refer to Cartesian components, and repeated indices are summed (in accordance with the Einstein summation convention) (Riley 1974). The convective derivative, of course, measures time variation in the local rest frame of the species-$s$ fluid. Strictly speaking, we should include an $s$ subscript with each convective derivative, because this operator is clearly different for different plasma species.

There is one additional refinement to our fluid equations that is worth carrying out. We introduce the generalized viscosity tensor, $\pi$ $_s$ , by writing

$${\bf p}_s = p_s\,{\bf I} + \mbox{\boldmath$\pi$} _s,$$ (4.46)

where ${\bf I}$ 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 that are quite unrelated to conventional viscosity. Equations (4.41)-(4.43) can, thus, be rewritten:

$$\frac{dn_s}{dt} + n_s\,\nabla\cdot{\bf V}_s =0,$$ (4.47)

$$m_s \,n_s\,\frac{d {\bf V}_s}{dt} + \nabla p_s + \nabla\cdot \mbox{\boldmath$\pi$} _s - e_s \,n_s\, ({\bf E} + {\bf V}_s\times {\bf B}) = {\bf F}_s,$$ (4.48)

$$\frac{3}{2}\frac{d p_s}{dt} + \frac{5}{2}\,p_s\,\nabla\cdot{\bf V}_s + \mbox{\boldmath$\pi$} _s:\nabla{\bf V}_s + \nabla \cdot{\bf q}_s = W_s.$$ (4.49)

According to Equation (4.47), the species-$s$ density is constant along a fluid trajectory unless the species-$s$ flow is non-solenoidal. For this reason, the condition

$$\nabla\cdot {\bf V}_s = 0$$ (4.50)

is said to describe incompressible species-$s$ flow. According to Equation (4.48), the species-$s$ 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 Equation (4.49), the species-$s$ energy density (that is, $p_s$ ) 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. The electrical contribution to plasma heating, which was explicit in Equation (4.38), has now become entirely implicit.