“Vortex Rings in a Compressible Fluid”

Extract

In a paper recently printed in the Society's Proceedings, I considered the effect of compressibility in the fluid on the motion of straight vortices; the present paper treats of circular vortex rings in a compressible fluid. The circle passing through the centres of the circular cross sections of the vortex filament will be called the “circular axis,” and the perpendicular to the plane of the circular axis through its centre, the “axis” of the vortex. In the notation employed, a denotes the radius of the circular axis, and e that of the cross section of the filament, while ω represents vorticity, and ρ density. It is also convenient to denote the area of the cross section— i.e., πe, by σ. Following Helmholtz, it will be supposed that e/a is always very small, and that the cross section is truly circular. Certain small inconsistencies in the ordinary theory following from this last assumption will be pointed out, though they do not seem seriously to affect the general applicability of the results. The axis of the vortex ring is taken as axis of z, and z, r, θ are the ordinary cylindrical co-ordinates. It is also convenient to denote by r' the distance of a point from the circular axis of a ring, and by ψ the inclination of this distance to the plane of the circular axis. The effects of the vorticity and variation in density may be considered separately.