Prandtl-Batchelor flows on a disk

TL;DR

This paper rigorously proves the existence of Prandtl-Batchelor flows on a disk, showing that in the vanishing viscosity limit, the flow exhibits a constant vorticity region separated from the boundary layer, with the leading order being a rigid rotation.

Contribution

It provides a rigorous mathematical proof of Prandtl-Batchelor flows on a disk, confirming the constant vorticity region in the vanishing viscosity limit with higher order approximate solutions.

Findings

Existence of Prandtl-Batchelor flows on a disk established.

Flow's leading order is a constant vorticity solution (rigid rotation).

Validation of Prandtl boundary layer expansion for these flows.

Abstract

For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. In this paper, by constructing higher order approximate solutions of the 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 a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying Batchelor-Wood formula.