Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring

We explore the flow field around a thin axisymmetric vortex ring steadily translating in an ideal fluid, with vorticity proportional to distance from the axis of symmetry, originally treated by Dyson (Philos. Trans. R. Soc. 184 (1893) 1041). By making a higher-order extension of the method of matched asymptotic expansions in a small parameter ε, the ratio of the core radius to the ring radius R₀, an explicit form of the streamfunction is derived, to O(ε³), both inside and outside the core. The pressure and velocity fields are written out in full to the same order. It is shown that the circle of minimum pressure coincides, to O(ε²R₀), with the circle of stagnation points viewed from the frame moving with the core.