A model of interacting Navier-Stokes singularities

TL;DR

This paper introduces a Hamiltonian model of interacting Navier-Stokes singularities called pin cons, which generalizes previous models and explores their dynamics, including dipole interactions and transient collapse phenomena.

Contribution

The paper presents a novel Hamiltonian model of Navier-Stokes singularities, extending the vorton model to include interactions and dynamics in viscous flows.

Findings

Dipole of pin cons exhibits transient collapse with radius scaling as (tc - t)^0.63.

Long-term dynamics show the dipole components repel and move apart.

Model captures the influence of singularities on surrounding flow and stress.

Abstract

We introduce a model of interacting singularities of Navier-Stokes, named pin\,cons. They follow a Hamiltonian dynamics, obtained by the condition that the velocity field around these singularities obeys locally Navier-Stokes equations. This model can be seen of a generalization of the vorton model of Novikov, that was derived for the Euler equations. When immersed in a regular field, the pin\,cons are further transported and sheared by the regular field, while applying a stress onto the regular field, that becomes dominant at a scale that is smaller than the Kolmogorov length. We apply this model to compute the motion of a dipole of pin\,cons. When the initial relative orientation of the dipole is inside the interval (0, pi/2), a dipole made of pin\,con of same intensity exhibits a transient collapse stage, following a scaling with dipole radius tending to 0 like (tc - t) power 0.63. For long time, the dynamics of the dipole is however repulsive, with both components running away from each other to infinity.