Magnetogasdynamics and Plasma Dynamics pp 139-154 | Cite as

# Two- and Three-Dimensional Flows in Magnetogasdynamics

## Abstract

In the analysis of one-dimensional flow (chapter IX), we actually considered only the average values of the velocity, pressure, etc., over certain sections. In order to study the details of the flow patterns, we have to find the velocity, pressure, etc., at every point in space. Hence we have to study the three dimensional flow and consider all three components of the velocity and magnetic field together with the pressure, density and temperature of the plasma as functions of the three spatial coordinates *x*, *y*, and *z* and time *t*. In general the three dimensional flow problem is too complicated to be analyzed. We have to make some reasonable simplifications in order to discuss some flow problems which are good approximations to the actual conditions. It is well known that the effects of viscosity µ and heat conductivity ϰ are negligibly small for high Reynolds number flow except in the boundary region near the solid wall or some other transition regions where the velocity and the temperature gradients are large. Hence in a majority of flow problems, the plasma may be considered as an inviscid and non-heat-conducting fluid. We shall make such an assumption in our treatments in §§ 2 to 5. For a plasma, besides viscosity and heat conductivity, there is a third diffusion property which may be characterized by the magnetic viscosity *v* _{ H } (or by electrical conductivity σ). If we assume *v* _{ H } = 0 in addition to µ = 0 and ϰ = 0, we have an ideal plasma (cf. chapter IV, § 5). The properties of an ideal plasma have some similarities to those of an inviscid and non-heat-conducting gas. Particularly the system of the fundamental equations for the three dimensional and unsteady flows of an ideal plasma is hyperbolic just as that for a similar flow of ordinary gasdynamics of an inviscid fluid. We may use the method of characteristics to solve such problems. In § 2, we shall discuss the three dimensional unsteady flows of an ideal plasma.

## Keywords

Boundary Layer Flow Magnetic Reynolds Number Blunt Body Thin Body Boundary Layer Region## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Bush, W. B.: Magnetohydrodynamic-Hypersonic Flow Past a Blunt Body. Jour. Aero-Space Sci, vol. 25, No. 11, Nov. 1958, pp. 685–690.MathSciNetMATHGoogle Scholar
- 2.Friedrichs, K. O.: Nonlinear Wave Motion in Magneto-Hydrodynamics. Los Almos Laboratory Report, 1954.Google Scholar
- 3.Friedrichs, K. O., and H. Kranzer: Nonlinear Wave Motion. Notes on Magnetohydrodynamics VIII, Institute of Math. Sei., New York Univ., 1958.Google Scholar
- 4.Kuimovsgn, A. G.: On the Flow of a Conducting Fluid Around a Magnetized Body. Doklady, AN USSR, vol. 117, No. 2, 1957, pp. 199–202.Google Scholar
- 5.Mcgune, J. E., and E. L. ResleilJr.: Compressibility Effects in Magneto-aerodynamic Flows past thin Bodies. Jour. Aero-Space Sci., vol. 27, No. 7, July 1960, pp. 493–503.Google Scholar
- 6.Pai, S. I.: Viscous flow theory, I — Laminar flow. D. Van Nostrand Co., Inc., Princeton, N. J., 1956.Google Scholar
- 7.Pai, S. I.: Linearized Theory of Airfoils in Fluids of Low Electric Conductivity. Tech. Inf. Series R 60 SD 311, Space Sciences Lab. General Electric Co., Dec. 1959.Google Scholar
- 8.Sears, W. R., and E. L. Resler Jr.: Sub- and Super-Alfvenic Flows Past Bodies. Preprint, Second Intern. Congress of the Aero. Sci., Zurich, Switzerland, Sept. 1960.Google Scholar