Magnetohydrodynamic circulation of a liquid of finite conductivity in an annulus

Summary

Unsteady circulation of an incompressible fluid of finite conductivity between two infinite concentric cylinders under the influence of a radial magnetic field is considered). Three cases are treated: (1) an external axial electric field is imposed, while the walls remain stationary; (2) and (3) no external electric field is imposed, but the walls either rotate at constant angular velocities, or oscillate sinusoidally. It is shown that none of the variables are dependent upon axial or angular position, and that the radial magnetic field component is independent of the fluid motion. These fortunate circumstances result in a coupled linear system of equations for the fluid velocity and the tangential magnetic field component. They are uncoupled by assuming the induced electric field to be space — independent at every instant. This is exact for steady flows. Zero and first-order approximate solutions may then be obtained either by neglecting the induced electric field, or by satisfying conservation of charge requirements. It is shown that neglect of the induced electric field is permissible only at small Hartmann numbers. An interesting feature of the solution is that the orthogonal expansion is in terms of Bessel functions whose order depends upon the Hartmann number, and hence may be arbitrarily large. Finally, it is thought that the first flow may find application in a novel viscometer for monitoring physical property variations in remote, hazardous, or highly reactive fluid systems. Some numerical examples are presented in support of the feasibility of such a device.