Vortical ring

A vortical thread having the shape of a torus of small cross-section. The general formulas which determine the velocity of liquid particles from vortices make it possible to represent the velocity potential $\phi(z,r)$ and the Stokes function of the current $\psi(z,r)$ of a flow generated in an unbounded liquid by a vortical ring as integrals containing the Bessel functions of order zero and one:

$$\phi(z,r)=\frac12a\kappa\int\limits_0^\infty e^{-kz}J_0(kr)J_1(ka)dk,$$

$$\phi(z,r)=-\frac12a\kappa r\int\limits_0^\infty e^{-kz}J_1(kr)J_1(ka)dk.$$

In these formulas, which apply if $z>0$, $a$ is the radius of the ring and $\kappa$ is the stress of the vortical ring. The coordinate $z$ is taken from the plane of the moving ring. The ring, when acted upon by the velocities it itself produces in the ring, moves in the direction of the $z$-axis at a constant velocity $c$, which is given by the following approximate formula:

$$c=\frac{\kappa}{4\pi\epsilon}\left[\ln\frac{8\epsilon}{a}-\frac14}\right],$$

where $\epsilon$ is the radius of a cross-section of the vortical ring. For several vortical rings the functions $\phi$ and $\psi$ are represented as sums of the respective functions of each ring.

References

[1]	L.M. Milne-Thomson, "Theoretical hydrodynamics" , Macmillan (1950)