Analytic continuation of the Birkhoff–Rott equation in complex-time domain

Abstract

A vortex sheet is a surface across which the velocity field of incompressible and inviscid flows has a jump discontinuity. Mathematical and numerical studies reveal that a two-dimensional vortex sheet, which is governed by the Birkhoff–Rott equation, acquires a singularity in finite time without forming rolling-up spiral. On the other hand, numerical computation of a regularized Birkhoff–Rott equation shows that the vortex sheet evolves into a rolling-up doubly branched spiral. Because of the finite-time singularity, it is impossible to regard the rolling-up spiral as a solution of the Birkhoff–Rott equation as long as time is real. However, it may be possible to analytically continue the equation to the spiral along a path to get around the singularity in complex-time plane. In the present article, we consider singularities in complex-time plane for the regularized Birkhoff–Rott equation by numerical means. Distribution of the complex singularities and their limiting behaviour indicate that it is absolutely impossible to perform analytic continuation in complex-time domain to the spiral solution. Furthermore, we propose a simple model of a doubly branched spiral and investigate it mathematically. The model is successful in approximating the rolling-up motion of the vortex sheet. Comparing the vortex-sheet motion with the model indicate that the doubly branched spiral with infinite windings at the centre could be a solution of the Birkhoff–Rott equation beyond the singularity time.