Some electrically driven flows in magnetohydrodynamics Part 3. The asymptotic theory for flow between circular electrodes

Abstract

In this, the third, part we present a complete asymptotic analysis of the distribution of velocity and electric potential in an electrically conducting liquid between two circular electrodes of finite diameter, 2b, when a current is passed between them. The electrodes are set opposite to each other in insulating planes, a distance 2aapart, and a magnetic field is applied perpendicular to these planes.The asymptotic solution is obtained under the restriction that the Hartmann numberMsatisfies both the conditionsM[Gt ] 1 andM^½l[Gt ] 1, wherel=b/a. It enables us to calculate the distribution of velocity and electrical potential throughout the flow field, and provides an expansion for the resistanceRbetween the electrodes in descending powers ofM^½l, which is correct provided terms of orderM⁻¹and (lM^½)⁻³are neglected. Comparison of the theoretical results with experiment shows good agreement in the measurements ofRand in the direct probe measurements of electrical potential within the fluid. This is one of the first experiments in which direct probe measurements of an MHD flow, as well as external measurements, have provided such a satisfactory confirmation of the theory. Direct measurements of the velocity, by means of a Pitot tube as in part 2, or by means of a hot-film anemometer undertaken by Malcolm (1968), agree less well with the theory.