On the shape of the cavity created by a thin magnetized airfoil

Abstract

The shape of the cavity created in a conducting fluid by a magnetic field within a thin airfoil is studied by means of an asymptotic expansion in powers of the airfoil width. In the two dimensional case, for a supersonic flow the interface between the cavity and the external flow starts at the tip of the airfoil and to order zero satisfies a simple formula given by the balance between the inside and outside pressures. This may be generalized to three dimensional conical axisymmetric obstacles. For subsonic flows the appropriate formula is more complex, involving the Hilbert transform of the total pressure. In this case cavities do not form unless the mean of the perturbation is zero, and when they do they start far before the airfoil.