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Notes: Fully developed, viscous liquid-metal velocity profiles and induced magnetic field contours were studied for Hartmann numbers of M=2 and 10 and for different load currents for a particular rectangular channel configuration (two-dimensional Couette flow). The rectangular channel was assumed to have a homogeneous external (axial) magnetic field parallel to the moving, perfectly conducting top wall and the stationary, perfectly conducting bottom wall. The two stationary side walls were also perfect conductors. The small gap between the moving wall and each side wall was an insulating, free surface. The method of weighted residuals was used to obtain truncated series solutions for the variables of interest. The heavy load currents across the channel were obtained by simulating an external potential to the conducting moving wall. The load currents in each case were opposed by the induced electric field. Since there is no pressure gradient, the flow along the channel is driven by the viscous effects of the moving wall and the Lorentz body force and is retarded by the stationary walls. In the case where no load current is applied across the channel, the current circulates in the channel.The circulation is driven by the generator that is due to the axial variation of velocity in an axial magnetic field. The numerical results presented show that the radial gap and the free surface region represent electrical resistances in parallel between the perfectly conducting stationary wall and the perfectly conducting moving wall. The numerical results also show that the resistance of the radial gap increases as M2 while that of the free surface increases by M or M1/2. Thus, as M increases, the division of current shifts to the free surface region and the current density in the radial gap decreases as M−1. The theoretical magnetohydrodynamic model presented here was developed to provide numerical parameters to help in the design of liquid-metal current collectors. Numerical results were computed for one-dimensional Couette flow with no pressure gradient in an external, homogeneous axial magnetic field. One-dimensional Couette flow has no end effects, and thus the numerical results were compared with corresponding numerical results for the two-dimensional Couette flow case to determine end effects.

Notes: The liquid-metal flow between a smooth moving wall and a two-dimensional, periodic, static wall with a uniform magnetic field that is perpendicular to both walls is treated. This problem models the flow in a liquid-metal sliding electrical contact with extremely large electric currents. The periodic static wall models the surface of a metal-fiber brush. The flow for a two-dimensional static wall is very different from that for a smooth static wall and involves large electrically driven spatial oscillations of the velocity, with swirling motion around the hills and valleys of the periodic surface. Solutions are presented (1) for large magnetic-field strength and arbitrary dimensionless periodic wall height, and (2) for arbitrary magnetic-field strength and small periodic wall height. Comparison of the two solutions indicates that viscous effects reduce the large velocity oscillations for typical electrical contacts.

Notes: In a previous paper the authors initiated studies of fully developed laminar liquid-metal flows, currents, and power losses in a rectangular channel with a moving perfectly conducting wall and with a skewed homogeneous external magnetic field for high Hartmann numbers, high interaction parameters, low magnetic Reynolds numbers, and different aspect ratios. The channel had insulating side walls that were skewed to the external magnetic field, while the perfectly conducting moving top wall with an external potential and the stationary perfectly conducting bottom wall at zero potential acted as electrodes. These electrodes were also skewed to the external magnetic field. A mathematical solution was obtained for high Hartmann numbers by dividing the flow into three core regions, two free shear layers, and six Hartmann layers along the channel walls. The free shear layers were treated rigorously and in detail with fundamental magnetohydrodynamic theory. The previous work, however, left the solution for the velocity profiles in terms of a complex integral equation which was not solved. In the present work the integral equation is solved numericallyby the method of quadratures to give the velocity profiles, viscous dissipation and Joulean losses in the free shear layers. In addition, expressions for the viscous dissipation in the six Hartmann layers are presented. The best approximation to the viscous dissipation in the channel is the sum of the O(M3/2) contributions from the two free shear layers, the O(M3/2) contributions from the two Hartmann layers separating the free shear layers from the insulators, and the O(M) contributions from three of the Hartmann layers separating core regions from the walls. The best approximation to the Joulean power losses in the channel is the sum of the O(M2) contribution from the central core region which carries an O(1) current between the electrodes and the O(M3/2) contributions from the free shear layers. The expressions for the viscous dissipation and Joulean losses in each region involve the products of universal constants, electrical potentials and geometric factors. The theoretical magnetohydrodynamic model presented here was developed to provide data to help in the design of liquid-metal current collectors.

Notes: Fully developed, laminar liquid-metal flows, currents, and power losses in a rectangular channel in a uniform, skewed high external magnetic field were studied for high Hartmann numbers, high interaction numbers, low magnetic Reynolds numbers, and different aspect ratios. The channel has insulating side walls that are skewed to the external magnetic field. Both the perfectly conducting moving top wall with an external potential and the stationary perfectly conducting bottom wall at zero potential act as electrodes and are also skewed to the external magnetic field. A solution is obtained for high Hartmann numbers by dividing the flow into three core regions, connected by two free-shear regions, and Hartmann layers along all the channel walls. Mathematical solutions are presented in each region in terms of singular perturbation expansions in negative powers of the Hartmann number. The free-shear layers are treated rigorously and in detail with fundamental magnetohydrodynamic theory. Numerical calculations are presented for the total current carried by the core region between top and bottom electrodes, Joulean and viscous power losses, and channel resistance at different skewed external magnetic field angles. With the high external magnetic field, the current through the central core region between the electrodes must be parallel to the external magnetic field lines. The two side core regions carry no current to the zeroth order. The two free-shear layers carry less current than the central core region. The theoretical magnetohydrodynamic model derived here was developed to provide data to help in the design of liquid-metal current collectors.

Notes: In homopolar motors and generators, large dc electric currents pass through the sliding electrical contacts between rotating copper disks (rotors) and static copper surfaces shrouding the rotor tips (stators). A liquid metal in the small radial gap between the rotor tip and concentric stator surface can provide a low-resistance, low-drag electrical contact. Since there is a strong magnetic field in the region of the electrical contacts, there are large electromagnetic body forces on the liquid metal. The primary, azimuthal motion consists of simple Couette flow, plus an electromagnetically driven flow with large extremes of the azimuthal velocity near the rotor corners. The secondary flow involves the radial and axial velocity components, is driven by the centrifugal force associated with the primary flow, and is opposed by the electromagnetic body force, so that the circulation varies inversely as the square of the magnetic-field strength. Three flow regimes are identified as the angular velocity Ω of the rotor is increased. For small Ω, the primary flow is decoupled from the secondary flow. As Ω increases, the secondary flow begins to convect the azimuthal-velocity peaks radially outward, which in turn changes the centrifugal force driving the secondary flow. At some critical value of Ω, the flow becomes periodic through the coupling of the primary and secondary flows. The azimuthal-velocity peaks begin to move radially in and out with an accompanying oscillation in the secondary-flow strength.

Notes: This paper treats the decay of the buoyant convection in a circular cylinder after a brief spike of large acceleration. There is a steady, uniform, axial magnetic field which is sufficiently strong that nonlinear inertial effects and convective heat transfer are negligible. Since the governing equations are linear, the solution for a spike at any angle to the cylinder's axis is given by the superposition of two solutions for an axial spike and a transverse spike. The flows are axisymmetric and three-dimensional for the axial and transverse spikes, respectively. With the transverse spike, the axial temperature gradient drives a single cell of circulation with transverse vorticity, and the radial temperature gradient drives two opposite cells of circulation with axial vorticity on opposite sides of the diameter parallel to the acceleration. Although the axial temperature gradient is larger than the radial one in the Bridgman crystal-growth process, the damping of transverse vorticity with an axial magnetic field is much stronger than the magnetic damping of the axial vorticity, so that the two-cell axial-vorticity circulation dominates. For a typical Bridgman process with a 0.2 T magnetic field and with silicon, the buoyant convections driven by the axial and transverse spikes of acceleration decay to one percent of their magnitudes immediately after the spikes in 3 and 14 s, respectively. © 1997 American Institute of Physics.

Notes: This paper treats the buoyant convection of a liquid metal in a circular cylinder with a uniform, steady, axial magnetic field and with the random residual accelerations encountered on earth-orbiting vehicles. The objective is to model the magnetic damping of the melt motion during semiconductor crystal growth by the Bridgman process in space. For a typical process with a magnetic flux density of 0.2 T, the convective heat transfer and the nonlinear inertial effects are negligible, so that the governing equations are linear. Therefore, for residual accelerations or "g-jitters'' whose directions are random functions of time, the buoyant convection is given by a superposition of the convections for two unidirectional accelerations: an axial acceleration which is parallel to the cylinder's axis and a transverse acceleration which is perpendicular to this axis. Similarly, the response to accelerations whose amplitudes are random functions of time is given by a Fourier-transform superposition of the buoyant convections for sinusoidally periodic accelerations for all frequencies. Since the temperature gradient in the Bridgman process is primarily axial, the magnitude of the three-dimensional buoyant convection for the transverse acceleration is much larger than the magnitude of the axisymmetric convection for the axial acceleration. At a low frequency, only the magnetic damping limits the magnitude of the buoyant convection. As the frequency is increased, linear inertial effects augment the magnetic damping, so that the magnitude of the convection decreases, and its phase shifts to a quarter-period lag after the acceleration. © 1996 American Institute of Physics.

Notes: A novel type of Kelvin–Helmholtz instability model is developed from hydrodynamic theory. The classical Kelvin–Helmholtz instability involves a horizontal interface between two fluids with different parallel, uniform, horizontal velocities. If the upper fluid is a gas with a much smaller density than the lower fluid which is a liquid, then the phase velocity of the critical disturbance equals the liquid's velocity, so that the liquid sees a standing interfacial wave. The inertial force driving the interfacial instability involves only the gas, no matter how small its density is. In a much more realistic flow model, the liquid velocity at the free surface is not uniform, but varies across the free surface. The disturbance phase velocity can only equal the liquid velocity at one point, while liquid on either side of this point moves faster or slower than the wave. The inertial forces in the liquid then dominate and the gas plays a negligible role. The concept is developed from a Couette flow hydrodynamic model where the fluid flows between two parallel vertical walls with a free surface. The importance of a nonuniform liquid velocity is demonstrated. This modified theory will be applied in future work to study the ejection instability at the interface of the liquid metal and inert cover gas in sliding electrical contacts.

Notes: Liquid metals in the small radial gaps between the rotors and stators of homopolar machines represent low-resistance electric current collectors. Design predictions for these liquid-metal current collectors require a thorough knowledge of liquid-metal flows in a narrow gap between a fixed and a moving surface, with a strong applied magnetic field and a free surface beyond each end of the gap. The radial and axial velocities in the secondary flow are reduced by a strong axial or radial magnetic field. For a sufficiently strong field, the azimuthal momentum transport by the secondary flow can be neglected. This assumption reduces the problem for the primary azimuthal velocity to a fully developed magnetohydrodynamic duct flow problem with a moving wall and two free surfaces. Asymptotic solutions for large Hartmann numbers are presented for skewed magnetic fields with both radial and axial components. Collectors without any electrical insulation or with insulation on the stator sides, or rotor sides, or both are considered. Solutions for a purely axial magnetic field and arbitrary Hartmann numbers are also presented.

Notes: Some designs of liquid-metal current collectors in homopolar motors and generators are essentially rotating liquid-metal fluids in cylindrical channels with free surfaces and will, at critical rotational speeds, become unstable. An investigation at David Taylor Research Center is being performed to understand the role of gravity in modifying this ejection instability. Some gravitational effects can be theoretically treated by perturbation techniques on the axisymmetric base flow of the liquid metal. This leads to a modification of previously calculated critical-current-collector ejection values neglecting gravity effects. The purpose of this paper is to document the derivation of the mathematical model which determines the perturbation of the liquid-metal base flow due to gravitational effects. Since gravity is a small force compared with the centrifugal effects, the base flow solutions can be expanded in inverse powers of the Froude number and modified liquid-flow profiles can be determined as a function of the azimuthal angle. This model will be used in later work to theoretically study the effects of gravity on the ejection point of the current collector.