Axial core-variations of axisymmetric shape on a curved slender vortex filament with a similar, Rankine, or bubble core

Abstract

The dynamics of axial core-variations of axisymmetric shape on a vortex filament is derived from the Navier–Stokes equations in the slenderness limit. The core of the vortex is of similar, Rankine, or bubble type with a centerline of any shape. In this limit, a two-time-scale asymptotic approach is used to study the dynamics of the axial core-variations and of the centerline. The short-time dynamical equations of the axial core-variations are given and are inviscid at leading and first orders. The induced short-time and normal-time dynamics of the centerline is obtained. The full two-time-scale dynamics of the axial variations and of the filament motion is discussed qualitatively. The normal-time dynamics of vortex filaments without axial core-variations is given in a short form. Within the two-time-scale framework, the dynamics of axial core-variations around this one-time base flow is then studied in the small amplitude limit. The normal-time equations of a vortex bubble are given. The bubble has no axial variations, a centerline of any shape and can have a nonpotential core. The equation for the ultra-short-time dynamics of axial variations on this bubble is given.