A model of interacting Navier-Stokes singularities

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.

24 pages 13 figures

References in corpus (3)

• Irreversible dynamics of vortex reconnections in quantum fluids
• Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion
• A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations