Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations

In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations (\textit{ANS}). In order to do so, we first introduce the scaling invariant Besov-Sobolev type spaces, $B^{-1+\frac{2}{p},{1/2}}_{p}$ and $B^{-1+\frac{2}{p},{1/2}}_{p}(T)$, $p\geq2$. Then, we prove the global wellposedness for (\textit{ANS}) provided the initial data are sufficient small compared to the horizontal viscosity in some suitable sense, which is stronger than $B^{-1+\frac{2}{p},{1/2}}_{p}$ norm. In particular, our results imply the global wellposedness of (\textit{ANS}) with high oscillatory initial data.Comment: 39 page

Direct and inverse pumping in flows with homogeneous and non-homogeneous swirl

The conditions in which meridional recirculations appear in swirling flows above a fixed wall are analysed. In the classical Bodew\"adt problem, where the swirl tends towards an aysmptotic value away from the wall, the well-known "tea-cup effect" drives a flow away from the plate at the centre of the vortex. Simple dimensional arguments applied to a single vortex show that if the intensity of the swirl decreases away from the wall, the sense of the recirculation can be inverted, and that the associated flow rate scales with the swirl gradient. Only if the flow is quasi-2D, does the classical tea-cup effect take place. This basic theory is confirmed by numerical simulations of a square array of steady, electrically driven vortices. Experiments in the turbulent regimes of the same configuration reveal that these mechanisms are active in the average flow and in its fluctuating part. The mechanisms singled out in this letter provide an explanation for previously observed phenomena in electrolyte flows. They also put forward a possible mechanism for the generation of helicity in flows close to two-dimensionality, which plays a key role in the transition between 2D and 3D turbulence