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On the Mutual Threading of Vortex Rings

W. M. Hicks

1922
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Proceedings of the Royal Society A
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irregularity is due to the term 3W sin# . Sj which rapidly tends to zero. Hence, for t large, the velocity of the liquid tangential to the sphere at a . point of co-latitude 0, has components sin 0c os (2a)£ cos 0), f W sin sin cos which combine into f W sin 0i n a direction making an angle a the negative direction from the meridian (0 increasing) through the point where « = 2 m tc os 0. Hence, ultimately, the velocity of the liquid tangential to the sphere at any point is constant in magnitude
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... nstant in magnitude and equal to f W sin 0, but changes its direction with constant angular speed 2m cos 0, which is a maximum at the poles and zero at the equator. A point to notice, however, is the fact that (dv'/d0)r=a contains the factor 2 m t ,and therefore ultimately increases without limit. This means that we reach a stage beyond which the components of vorticity can no longer be regarded as small. A similar deduction is obtained from dw/dr over the equatorial plane, which contains the factor ( )*. A physical restriction to the solution of the problem is thus introduced. Although the fact that two circular vortices are capable of threading one another in permanent succession has long been known, I do not know that any attempt has been made at a discussion of the general conditions. The following pages contain a contribution to such a discussion for the case of thin circular filaments. The nature of a ring is given when its circulation and volume are known. Its configuration at any time is given when its aperture or the radius of the cross-section of the filament is known. In the case of two rings, the nature of their combination is defined if their ttwo apertures at the instant when they are co-planar are each known. This configuration will be called their standard position. W ith two such rings, the mean area of the two apertures, supposed weighted with their circula tions, remains constant throughout the motion. They can therefore also be uefined by the radius of this mean area and the ratio of the two apertures in VOL. o il.-A. r

doi:10.1098/rspa.1922.0075
fatcat:66rr7jm3azeinkihvezjzjgb5i