On Prandtl-Batchelor theory of a cylindrical eddy: Existence and uniqueness

Abstract

For a steady laminar two-dimensional flow, Prandtl and Batchelor proposed a property in the case of a region of nested closed streamlines. This Prandtl-Batchelor(PB) theory claims the constancy of the vorticity in the limit of infinite Reynolds number R ( or vanishing viscosity $\nu$ ) within such a region. To establish this result rigorously, as a first step we here show that a boundary layer corresponding to the PB theory exists and is unique for the circular eddy under relatively small perturbations of the Euler limit wall velocity.