Boundary Current Effects in Magnetohydrodynamics with Anisotropic Conductivity

Citation

Smisek, Richard Franklin (1965) Boundary Current Effects in Magnetohydrodynamics with Anisotropic Conductivity. Engineer's thesis, California Institute of Technology. doi:10.7907/ANC6-1M76. https://resolver.caltech.edu/CaltechETD:etd-01262004-143607

Abstract

A theoretical investigation is conducted to determine the effects of currents flowing through a boundary into the magnetohydrodynamic flow of an inviscid, incompressible fluid with anisotropic conductivity. The particular arrangement of an externally applied magnetic field parallel to the velocity field is investigated for two flow geometries; (i) semi-infinite flow over a conducting flat wall, and (ii) channel flow between a conducting lower wall and an insulating upper wall. In both cases the applied boundary currents are assumed to be sinusoidal in shape and flow into the fluid normal to the boundary. A small perturbation analysis is used to linearize the macroscopic steady flow equations of a fully ionized gas. A Cartesian coordinate system is adopted in which the x-axis is in the flow direction and the y-axis is normal to the conducting wall. The problem is considered two dimensional from the standpoint that the perturbed quantities are independent of the z-coordinate although the z-components are, in general, non-zero. The general solution to the linearized equations is obtained for case (i). Because of the complexity of this solution, it is studied in detail only in the limits of small and large magnetic Reynold's number. Solutions for case (ii) are obtained in the limits of small and large magnetic Reynold's number by applying the limiting procedure to the linearized equations before solving them. In the limit of small magnetic Reynold's number for both cases (i) and (ii), the magnetic and velocity field vectors are composed of an irrotational part and a rotational part. The irrotational portion always remains in the x-y plane. However, the rotational portion and, hence, the currents lie in a plane which is rotated about the x-axis; the angle between this plane and the x-y plane being strongly dependent upon the degree of anisotropy in the fluid's electrical conductivity. The currents in the fluid form symmetric loops closing at the conducting boundary. Anisotropic effects on the magnitude of the magnetic and velocity field components and the currents are generally moderate except near the conducting wall. At this wall the x and z current components can become quite large for strong anisotropic conductivity. Both the irrotational and rotational portions of the velocity field vector behave in a manner analogous to ordinary incompressible flow with the applied sinusoidal boundary current in the flat wall replaced by a solid sinusoidal wall. In the limit of large magnetic Reynold's number for both cases (i) and (ii), anisotropic effects are absent to the order of the inverse square root of the magnetic Reynold's number. In addition, the currents and field perturbations are found to be confined to a thin magnetic boundary layer near the conducting wall. The currents lie entirely in the x-y plane and again form loops closing at the conducting boundary, but are steeply inclined toward the x-axis. The x-component of the current flowing in the fluid is found to be. larger than the applied boundary current by a factor of the square root of the magnetic Reynold's number.

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