Finite-time singularity in the vortex dynamics of a string

Abstract

We analyze the dynamics of a perfectly flexible string with a constant length and a vanishing inner friction. The local angular velocity of line elements in this seemingly simple mechanical system is shown to have many mathematical and physical properties in common with vorticity in the three-dimensional incompressible Euler equation. It is demonstrated that initially smooth vorticity fields lose their regularity within finite time in a self-similar process, and that the peak vorticity grows as ${\ensuremath{\omega}}_{\mathrm{max}}\ensuremath{\sim}(T\ensuremath{-}{t)}^{\ensuremath{-}1}.$