Asymmetric Vortex Sheet

Abstract

We present a steady analytical solution of the incompressible Navier-Stokes equation for arbitrary viscosity in an arbitrary dimension $d$ of space. It represents a $d-1$ dimensional vortex "sheet" with an asymmetric profile of vorticity as a function of the normal coordinate $z$. This profile is related to the Hermite polynomials $H_\mu(z)$ which are analytically continued to the negative fractional index $\mu = -\frac{d}{d-1}$. In $d=2$ dimensions, the solution degenerates to a constant vorticity flow. In $ d \ge 3$ dimensions, the vorticity is confined to the thin layer around the hyperplane with Gaussian decay on one side of the hyperplane and the power decay on another side. One can adjust the common scale of velocity so that the dissipation will stay finite at vanishing viscosity. In this limit, the width $w$ of the viscous lawyer will shrink to zero as $\nu^{\frac{3}{5}}$ for arbitrary dimension $d>3$. In $d=3$ dimensions, this power law is also accompanied by powers of the logarithm.