Prandtl-Batchelor theorem for flows with quasi-periodic time dependence

TL;DR

This paper extends the classical Prandtl-Batchelor theorem to quasi-periodic flows, showing that in the zero-viscosity limit, vorticity remains constant within closed contours in such flows.

Contribution

It generalizes the Prandtl-Batchelor theorem to quasi-periodic flows using ergodic and geometric analysis, revealing vorticity behavior in viscous flows approaching inviscid limits.

Findings

Vorticity is constant within closed contours in quasi-periodic flows at zero viscosity.

Quasi-periodic viscous flows cannot converge to non-uniform vorticity inviscid flows.

Vorticity contours form closed curves with constant vorticity in the zero-viscosity limit.

Abstract

The classical Prandtl-Batchelor theorem (Prandtl 1904; Batchelor 1956) states that in the regions of steady 2D flow where viscous forces are small and streamlines are closed, the vorticity is constant. In this paper, we extend this theorem to recirculating flows with quasi-periodic time dependence using ergodic and geometric analysis of Lagrangian dynamics. In particular, we show that 2D quasi-periodic viscous flows, in the limit of zero viscosity, cannot converge to recirculating inviscid flows with non-uniform vorticity distribution. A corollary of this result is that if the vorticity contours form a family of closed curves in a quasi-periodic viscous flow, then at the limit of zero viscosity, vorticity is constant in the area enclosed by those curves at all times.