A SHALLOW-LIQUID THEORY IN MAGNETOHYDRODYNAMICS

Technical Report · Mon Jun 01 04:00:00 EDT 1959

OSTI ID:4223369

Fraenkel, L E

The nonlinear and linear shallow-water' theories, which describe long gravity waves on the free surface of an inviscid liquid, are extended to the case of an electrically conducting liquid on a horizontal bottom, in the presence of a vertical magnetic field. The dish holding the liquid, and the medium outside it, are assumed to be non-conducting. The approximate equations are based on a small ratio of depth to wavelength, on the properties of mercury, and on a moderate magnetic field strength, These equations have a magnetohydraulic, character, for in the shallow liquid layer the horizontal fluid velocity and current density are independent of the vertical coordinate. Some explicit solutions of the linear equations are obtained for plane flows and for axi-symmetric flows in which the velocity vector lies in a vertical, meridianal plane. The amplitudes of waves in a dish, and the amplitudes behind wave fronts progressing into undisturbed liquid, are found to be exponentially damped, the mechanical energy associated with a disturbance being dissipated by Joule heating. The approximate nonlinear equations for plane flow are studied by means of characteristic variables, and it appears that, because of the magnetic damping effect, there is less qualitative difference between solutions of the nonlinear and linear approximate equations at large times than is the case when the magnetic field is absent. In particular, the characteristic curves depart only a finite distance form their undistured positions. (auth) process could account for at most a small part of the observed resistivity. (auth)

• Research Organization: California. Inst. of Tech., Pasadena. Guggenheim Aeronautical Lab.
• NSA Number: NSA-13-019418
• OSTI ID: 4223369
• Report Number(s): AFOSR-TN-59-563
• Country of Publication: United States
• Language: English

Related Subjects

CURRENTS
ELECTRIC CONDUCTIVITY
EQUATIONS
FLUID FLOW
FREQUENCY
HEATING
HYDRAULICS
LAYERS
LIQUIDS
MAGNETIC FIELDS
MAGNETOHYDRODYNAMICS
MERCURY
NUMERICALS
PHYSICS AND MATHEMATICS
SURFACES
TURBULENCE
VELOCITY
VISCOSITY