Motion of a Curved Vortex Filament with Decaying Vortical Core and Axial Velocity

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The motion and decay of a vortex filament with large axial and circumferential velocity components in a three-dimensional stream are studied. Solutions are constructed to the Navier–Stokes equations by use of matched asymptotic expansions; the small parameter used is a measure of the ratio of the viscous effects to the vortex strength. The outer flow, which corresponds to the classical Biot–Savart type analysis is matched to the solution in an inner viscous region. The radius of the viscous core is assumed to be much smaller than the radius of curvature. The present viscous analysis yields the classical inviscid theory as a limiting case for the leading term in the outer region and thus can be used to correct various deficiencies in the latter. We show in particular, that the inner solution yields a finite velocity at all points in the filament and we determine how the components of both vorticity and velocity diffuse due to the viscous forces. The matching conditions guarantee the continuity of velocity components and define the velocity of the filament which depends not only on the geometry of the vortex filament but also on the circumferential and axial velocity variations in the inner core.