Asymptotic expansions of the Biot-Savart law for a slender vortex with core variation (Q5944605)

The authors use the method of matched asymptotic expansions of singular integrals in order to expand Biot-Savart law in terms of different parameters. First, the authors obtain, in terms of small distance \(r\) to a line vortex, the first order of the known expansion of potential flow induced by this line vortex. Also, they compare two equations of motion of a slender vortex filament: the first is obtained in an ad-hoc way by a cut-off line-integral techique, and the latter is obtained from Navier-Stokes equations. Finally, the authors derive the inner expansion of the flow induced by a slender vortex in terms of its slenderness. This inner expansion is applied to the calculation of induced velocity of a family of non-circular vortex rings with axisymmetric axial-core variation.