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 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, Df and Dm. We find that Df + ½Dm, on a perfectly conducting strip, is equal to the viscous drag on an insulating strip for which Dm is zero. Precise values of these drags are given.