On the motion of vortex rings in an incompressible media

The paper deals with the motion of an axisymmetric vortex ring in an incompressible media whose velocity $\vec{v}$ and density $ρ$ satisfy the equations $div\,\vec{v}=0$, $\vec{v}\cdot grad\, ρ=0$. The second equation allows us to consider the case when the density varies across the ring. It is shown that the media’s density can vary only in the vicinity of the flow possessing vorticity and must be constant if the flow is potential. Thus, the ring’s velocity and the shape of its atmosphere depend not only on the size of the vortex core and circulation but also on the spatial distribution of the density across the ring.