The magneto-hydrodynamic boundary layer in the two-dimensional steady flow past a semi-infinite flat plate. I. Uniform conditions at infinity

Abstract

The problem investigated is the flow of a viscous, electrically conducting liquid past a fixed, semi-infinite, unmagnetized but conducting flat plate. The liquid flowUand also the magnetic field H0at a distance from the plate are both assumed to be uniform and parallel to the plate. It is assumed that the Reynolds numberRand magnetic Reynolds numberRmare large enough for momentum and magnetic boundary layers to have developed. The standard boundary-layer techniques as used in the Blasius solution then apply and the problem reduces to the solution of two simultaneous non-linear ordinary differential equations. These are examined by the use of an iteration method suggested in the non ­ magnetic problem by Weyl and a solution of reasonable accuracy has been obtained for the drag coefficient. This confirms a similar result obtained in a different way by Carrier & Greenspan. The principal result of the paper is that the boundary layer thickens and drag coefficient diminishes steadily as the parameterS = µH20/4πρU2increases. WhenSattains the finite value of unity the drag coefficient obtained here actually vanishes with the flow having been reduced to rest by the action of the magnetic field. This result might be inferred qualitatively since a finite amount of work has to be done in conveying liquid particles across the lines of magnetic force.