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 є = δ/R ₀ (= (v/Г)^1/2), the ratio of core to curvature radii, which takes account of the influence of the self-induced strain. Here Г is the circulation and v 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(є ³)from the dipole. In a special case, we recover Dyson’s inviscid formula (1893). It is demonstrated that the viscosity acts, at O(є ³), to expand the radius of the loop consisting of the stagnation points in the core, when viewed from a certain comoving frame.