Asymptotic expansions of the Biot-Savart law for a slender vortex with core variation

Abstract

The Method of Matched Asymptotic Expansion of Singular Integrals (MAESI) is used to expand the Biot-Savart law in terms of different parameters. This method is first used to find, in terms of the small distance r to a line vortex, the first orders of the known expansion of the potential flow induced by this line vortex. This method is also used to easily compare two equations of motion of a slender vortex filament: the one obtained in an ad-hoc way by a cut-off line-integral technique and the other derived from the Navier-Stokes equations by Callegari and Ting. Finally, this method is used to give the inner expansion of the flow induced by a slender vortex in terms of its slenderness ε. This is the first inner expansion up to order one in terms of ε of the Biot-Savart law for a slender vortex. An application of this inner expansion is then given to find the induced velocity of a family of non-circular vortex rings with axisymmetric axial-core variation. In order to understand the time-evolution of these initial conditions to the Navier-Stokes equations, a short time scale is introduced. A quasi-hyperbolic system that describes the leading-order dynamics of the axisymmetric axial core variation on a curved slender vortex filament is finally extracted from the Navier-Stokes equations.