Towards a finite-time singularity of the Navier-Stokes equations

Abstract

A model is developed describing the approach to a finite-time singularity of the Navier-Stokes equations for two interacting vortices. The model is derived from a combination of the Biot-Savart law and an equation describing the evolution of the vortex core cross-sections. A nonlinear dynamical system is derived for the minimum separation parameter $s(\tau)$, the curvature $\kappa(\tau)$, and the radial scale $\delta(\tau)$ of the vortex cross-section at the points of closest approach, where $\tau$ is dimensionless time. The scaling properties of this system are investigated.