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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 article, 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 sufficiently localized 2D flows in the far field: it implies, in particular, that fluid particles of such flows are nowhere at rest at large distances.

Hexagonal structures in 2D Navier–Stokes flows