A model of interacting Navier-Stokes singularities
Abstract: We introduce a model of interacting singularities of Navier-Stokes, named pinçons. They follow a non-equilibrium 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 pinçons 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 pair of pinçons. A pinçons dipole is intrinsically repelling and generically run away from each other in the early stage of their interaction. At late times, the dissipation takes over, and the dipole dies over a viscous time scale. In the presence of a stochastic forcing, the dipole tends to orientate itself so that its components are perpendicular to their separation, and it can then follow during a transient time a near out-of-equilibrium state, with forcing balancing dissipation.
In the general case where the pinçons have arbitrary intensity and orientation, we observe three generic dynamics in the early stage: one collapse with infinite dissipation, and two expansions mode, the dipolar anti-aligned runaway and a anisotropic aligned runaway. The collapse of a pair of pinçons follows several characteristics of the reconnection between two vortex rings, including the scaling of the separation between the two components, following Leray scaling (tc-t)1/2.
This work was done in collaboration with H. Faller, L. Fery and D. Geneste