# Magnetogasdynamics and Electromagnetogasdynamics

• Shih-I Pai
Chapter

## 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.

## Keywords

Magnetic Field Mach Number External Electric Field Fundamental Equation Compressible Fluid
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Alfvén, H.: On the Existence of Electromagnetic-hydrodynamic Waves. Arkiv F. mat. astr. o. fysik, Bd. 29B, No. 2, 1942.Google Scholar
2. 2.
Chandrasekhar, S.: An Introduction to the Study of Stellar Structure. Dover Publications, Inc., 1957.Google Scholar
3. 3.
Friedrichs, K. O.: Nonlinear Wave Motion in Magneto-hydrodynamics. Los Alamos Laboratory Report, 1954.Google Scholar
4. 4.
Hartmann, J.: Hg-dynamics. I. Kgl. Danske Vidensk. Selskab Math.-fys. Medd. XV: 6, 1937.Google Scholar
5. 5.
De Hoffmann, F., and E. Teller: Magnetogasdynamic Shock. Phys. Rev., vol. 80, No. 4, Nov. 15, 1950, pp. 692–703.Google Scholar
6. 6.
Lundquist, S.: Studies in Magneto-hydrodynamics. Arkiv F. Fysik, Bd. 5, No. 15, 1952, pp. 297–347.
7. 7.
Pai, S. I.: Energy Equation of Magneto-gasdynamics. Phys. Rev., vol. 105, No. 5, March 1, 1957, pp. 1424–1426.Google Scholar
8. 8.
Pai, S. I.: Introduction to the Theory of Compressible Flow. Van Nostrand Co., Inc., Princeton, N. J., 1958.Google Scholar
9. 9.
Pai, S. I.: Viscous Flow Theory, I-Laminar Flow. Van Nostrand Co., Inc., Princeton, N. J., 1956.Google Scholar
10. 10.
Stratton, J. A.: Electromagnetic Theory. McGraw-Hill Book Company, Inc., New York, 1941.