Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity

A large-Reynolds-number asymptotic solution of the Navier-Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched asymptotic expansions is extended to a higher order in a small parameter epsilon = (nu/Gamma)(1/2), where nu is the kinematic viscosity of fluid and Gamma is the circulation. Alternatively, epsilon is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics. We establish a general formula for the translation speed of the ring valid up to third order in epsilon. This is a natural extension of Fraenkel-Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R-0, that viscosity acts, at third order, to expand the circles of stagnation points of radii R-s(t) and (R) over tilde(s)(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R-p(t) as R-s approximate to R-0 + [2 log(4R(0)/root nu t) + 1.4743424] nu t/R-0, (R) over tilde(s) approximate to R-0 + 2.5902739 nu t/R-0, and R-p approximate to R-0 + 4.5902739 nu t/R-0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet & Widmayer (1974) and Weigand & Gharib (1997). The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.