Some General Properties of Induced Velocities

Abstract

In preparation for subsequent investigations, we must study the properties of the velocity field generated (induced) by vortex sheets. In this chapter we prove some auxiliary theorems that will be used later. We establish the basic properties of the velocity perturbations caused by a vortex sheet at points close to the surface, and we also find the order of the rate at which these disturbances decay at large distances from the vortex sheet. By a vortex sheet we mean some surface Σ, on which there is a continuously distributed vortex layer such that the axis of each elementary vortex of the layer is in the plane tangent to the surface. Our analysis is valid, not only for steady but also for unsteady motion. In the latter case the time t will be assumed to be fixed.