Magnetogasdynamics and Electromagnetogasdynamics

Abstract

In this chapter, we consider the plasma as a single fluid. The fundamental equations are given in chapter II, § 8. They are as follows:

$$p={{R}_{p}}\rho T$$

(4.1)

$$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial {{x}^{i}}}(\rho {{u}^{i}})=0$$

(4.2)

$$\rho \frac{D{{u}^{i}}}{Dt}=-\frac{\partial pt}{\partial {{x}^{i}}}+\frac{\partial {{\tau }^{ij}}}{\partial {{x}^{j}}}+{{F}_{{{e}^{i}}}}+{{F}_{{{g}^{i}}}}$$

(4.3)

$$\frac{\partial \rho \overline{em}}{\partial t}+\frac{\partial \rho \overline{em}{{u}^{j}}}{\partial {{x}^{j}}}=-\frac{\partial {{u}^{j}}pt}{\partial {{x}^{j}}}+\frac{\partial {{u}^{i}}{{\tau }^{ij}}}{\partial {{x}^{j}}}+{{E}^{j}}{{J}^{j}}+\frac{\partial {{Q}^{j}}}{\partial {{x}^{j}}}$$

(4.4)

$$\nabla \times \overrightarrow{H}=\overrightarrow{J}+\frac{\partial \varepsilon \overrightarrow{E}}{\partial t}$$

(4.5)

$$\nabla \times \overrightarrow{E}=-\frac{\partial {{\mu }_{e}}\overrightarrow{H}}{\partial t}$$

(4.6)

$$\frac{\partial {{\rho }_{e}}}{\partial t}+\frac{\partial {{J}^{j}}}{\partial {{x}^{j}}}=0$$

(4.7)

$${{J}^{i}}={{i}^{i}}+{{\rho }_{e}}{{u}^{i}}=\sigma [{{E}^{i}}+{{\mu }_{e}}(\overrightarrow{u}\times {{\overrightarrow{H}}^{i}})]+{{\rho }_{e}}{{u}^{i}}$$

(4.8)

$$\nabla \cdot \overrightarrow{H}=0$$

(4.9)

$$\nabla \cdot \overrightarrow{E}=\frac{1}{\varepsilon }\rho e$$

(4.10)

where the unknowns to be investigated are u ⁱ, p, ρ, T,E ⁱ, H ⁱ, ρ _e and J ⁱ. Equations (4.9) and (4.10) are not independent equations which may be derived from equations (4.5), (4.6) and (4.7) (cf. chapter III, § 5); but they are important relations, hence we list them here for reference.