Hexagonal structures in 2D Navier-Stokes flows

TL;DR

This paper investigates the spontaneous emergence of hexagonal patterns in two-dimensional viscous fluid flows governed by Navier-Stokes equations, highlighting geometric structures and energy distribution in such flows.

Contribution

It provides a mathematical analysis of hexagonal structure formation in 2D Navier-Stokes flows and discusses the isotropic energy distribution in localized flows.

Findings

Hexagonal patterns can spontaneously form in 2D viscous flows.

Energy density in such flows is isotropic at large distances.

Fluid particles are never at rest at large distances in localized flows.

Abstract

Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn's Hexagon, the huge cloud pattern at the level of Saturn's north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray's solutions of the Navier-Stokes equations. Our analysis also makes evidence of the isotropic character of the energy density of the fluid for sufficently localized 2D flows in the far field: it implies, in particular, that fluid particles of such flows are nowhere at rest at large distances.