Theoretical Model of Gravitational Perturbation of Current Collector Axisymmetric Flow Field

摘要

Some designs of liquid metal 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 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 report 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. A rederivation of the hydrodynamic instability threshold of a liquid metal current collector is presented. Keywords: Current collector; Magnetohydrodynamics; Liquid metal flows; Fluid mechanics; Gravity perturbation; Hydrodynamics. (jhd)