Motion of a Thin Vortex Ring in a Viscous Fluid: Higher-Order Asymptotics

Abstract

The motion of an axisymmetric vortex ring of small cross-section in a viscous incompressible fluid is investigated using the method of matched asymptotic expansions. A general formula for the ring speed is obtained up to third order in epsilon = delta/R-0 (= (nu/Gamma)(1/2)), the ratio of core to curvature radii, which takes account of the influence of the self-induced strain. Here Gamma is the circulation and nu is the kinematic viscosity of fluid. It is pointed out that the dipole distributed along the centerline of the ring plays a vital role in its movement. Its strength needs be specified at the initial instant in order to remove the indeterminacy of the theory. A new asymptotic development of the Biot-Savart law enables us to calculate the non-local induction velocity at O(epsilon(3)) from the dipole. In a special case, we recover Dyson's inviscid formula (1893). It is demonstrated that the viscosity acts, at O(epsilon(3)), to expand the radius of the loop consisting of the stagnation points in the core, when viewed from a certain comoving frame.