Steady longitudinal motion of a cylinder in a conducting fluid

Abstract

The steady motion of an infinitely long solid cylinder parallel to its length in a conducting fluid in the presence of a uniform magnetic field is discussed. Due to Alfvén waves originating at the cylinder we find two opposite ‘wakes’ parallel to the applied magnetic field. A formula which relates the total drag on the cylinder to the electric potential difference δΦ between the two undisturbed regions outside these two wakes is derived $D|\;|\delta \Phi| = 2\surd {\rho}v \sigma$ where ρν is the viscosity and σ is the conductivity of the fluid. The reduction to a classical boundary-value problem is made for the case of an insulating cylinder. Exact solutions are obtained for the case of a perfectly conducting or an insulating flat strip of semi-infinite width. These give a clear picture of the field, especially in the transition region near the edge of the strip. The case of a strip of finite width is also discussed with special reference to the viscous and the magnetic drags, D f and D m . We find that D f + ½ D m , on a perfectly conducting strip, is equal to the viscous drag on an insulating strip for which D m is zero. Precise values of these drags are given.