Periodic motion of two and four vortices in a cylindrical pipe

The motion of two and four rectilinear vortices inside a cylindrical pipe is studied under the restriction that the total circulation be zero. In the two-vortex case, it is shown that the motion is always periodic and an expression for the period is derived. In the four-vortex case, the motion is determined not to be periodic in general. However, a class of solutions where the motion is periodic is found. Several sample calculations of the vortex motion are included.