Prandtl-Batchelor flows on an annulus

TL;DR

This paper extends Prandtl-Batchelor theory to forced steady Navier-Stokes flows on an annulus, proving the existence of flows with nested closed streamlines in the high Reynolds number limit.

Contribution

It rigorously proves the existence of Prandtl-Batchelor flows on an annulus with boundary conditions close to rigid rotation, using boundary layer expansion techniques.

Findings

In the inviscid limit, solutions with nested streamlines tend to a rotating shear flow.

Constructed higher order approximate solutions validate boundary layer expansion.

Proved existence of Prandtl-Batchelor flows under specific boundary conditions.

Abstract

For steady two-dimensional Navier-Stokes flows with a single eddy (i.e. nested closed streamlines) in a simply connected domain, Prandtl (1905) and Batchelor (1956) found that in the inviscid limit, the vorticity is constant inside the eddy. In this paper, we consider the generalized Prandtl-Batchelor theory for the forced steady Navier-Stokes equations on an annulus. First, we observe that in the limit of infinite Reynolds number, if forced steady Navier-Stokes solutions has nested closed streamlines on an annulus, then the inviscid limit is a rotating shear flow uniquely determined by the external force and boundary conditions. We call solutions of steady Navier-Stokes equations with the above property Prandtl-Batchelor flows. Then, by constructing higher order approximate solutions of the forced steady Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on an annulus with the wall velocities slightly different from the rigid-rotations along the same direction.