# Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion

## TL;DR

This paper extends a dynamical system model of Navier-Stokes singularities to include vortex reconnection, analyzing maximum vorticity behavior and suggesting a physical singularity at high vortex Reynolds numbers.

## Contribution

It introduces a model incorporating vortex reconnection into the singularity analysis of Navier-Stokes equations, providing new insights into vorticity growth at high Reynolds numbers.

## Key findings

- Maximum vorticity grows exponentially with Reynolds number.
- A physical singularity may occur for Reynolds numbers above 4000.
- Reconnection dynamics influence the approach to singularity.

## Abstract

In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier-Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute the maximum vorticity $\omega_{max}$ as a function of vortex Reynolds number $R_\Gamma$ in the range $2000\le R_\Gamma \le 3400$, and deduce a compatible behaviour $\omega_{max}\sim \omega_{0}\exp{\left[1 + 220 \left(\log\left[R_{\Gamma}/2000\right]\right)^{2}\right]}$ as $R_\Gamma\rightarrow \infty$. This may be described as a physical (although not strictly mathematical) singularity, for all $R_\Gamma \gtrsim 4000$.