Steady motion of conducting fluids in pipes under transverse magnetic fields

TLDR

The study investigates steady flow of an electrically conducting, viscous fluid in channels under a transverse magnetic field with non‑conducting walls. The authors derive and exactly solve the governing equations for velocity, induced currents, and magnetic field in rectangular channels, extend the solution to any symmetric shape, and compare predicted pressure gradients with experiments in square and circular pipes. Strong magnetic fields cause the velocity profile to collapse into a uniform core with novel boundary layers, and the resulting pressure gradients agree with experimental data for square and circular pipes.

Abstract

ABSTRACT This paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.