Examples of steady vortex rings of small cross-section in an ideal fluid

The existence theory for steady vortex rings of small cross-section is used to derive asymptotic formulae that describe the shape and overall properties of such rings. A certain two-parameter family of rings is studied in detail, to a first approximation; for members of this family, the ratio ω/ r (of vorticity to cylindrical radius) falls from a positive maximum at a central point of the core cross-section to a value at the core boundary that can be substantially smaller or even negative. The case of uniform ω/ r is considered to a higher order of approximation, and the formulae given for this case appear to be useful for quite substantial cross-sections.