Гидродинамика

А NOTE ON THE MOTION Ob VORTEX RINGS In this note Prof. Joukowsky discusses the problem of motion of a circular vortex ring carrying the fluid contained within in a motion of translation, and having on its surface a layer of vorticeg. The method of attack used by Prof. Joukow ­ sky in solving the problem is much simpler than t ha t given b y Basset in his treatise. There is first determined t h e velocity o f a point N situated at a small distance p from a point О of a vortex filament of vanishingly small thicknessi the strength of t h e filament being m and the shape that of a ring of radius a. T h e velocity is mede up of three component velocities (formulae 4 and 5), viz- J) the velocity m /, 8a to = -?r Ig 2IIA \ P in the direction at right angles to the plane of the vortex, 2) t h e velocity of vortical rotation about the tangent to the ring a t th © point O, and 3) the velocity of motion of rotation about t h e same tangent, this velocity having a finite variable ma g n i t u d e where ? —projection of p on the radius of the ring. Assuming the vortex rings to be disposed on the surface of a thoroid, of which the radius of cross-sections is b, Prof. J o u s kowsky determines the velocity imparted by all these r i n g - to a point N of the fluid contained within the thoroid. T h e resultant velocity of the point N due to all motions of the s e cond group is found to be zero, that due to motions of the f i r s t group is M \. 8a