Theory of unsteady flow about thin cylinders in fluids of high electrical conductivity

Abstract

A theory is developed for the incompressible flow of a fluid with high electrical conductivity about thin cylinders (airfoils) in non-uniform motion. A uniform magnetic field is applied parallel to the free stream and solutions are obtained subject to the restriction of small perturbations. The effects of viscosity are included, for the most part, only through the use of the Kutta condition, where applicable, for lifting airfoils. The validity and range of applicability of the infinite-conductivity assumption are determined on the basis of an order-of-magnitude analysis; the general character of the flow is discussed at length.The flow-field for infinite conductivity is changed from the non-magnetic case only through the new transport speed of vorticity; the forces on the airfoil are changed due to surface currents. For the case of the Alfvén speed less than the free-stream speed, the airfoil lift and pitching moment are given in integral form for general unsteady-airfoil motion and are given in closed form for harmonic ocsillations. The forces at moderate frequencies may be larger than in the corresponding non-magnetic case. The response to a unit-step change in the downwash is studied and the asymptotic form of the lift is obtained for small and large time.For the case of the Alfvén speed greater than the free-stream speed, vorticity and current are shed from both the leading and trailing edges. Therefore the extension of the usual Kutta condition is not obvious. It is shown that if finite viscosity and / or conductivity tend to remove the trailing-edge singularity, the flow is unstable and no steady flow can be obtained.