Prandtl–Batchelor Flows on a Disk

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 vorticity is constant an inner region separated from boundary layer. In this paper, by constructing higher order approximate solutions Navier-Stokes equations establishing validity layer expansion, we give rigorous proof existence Prandtl-Batchelor on disk wall velocity slightly different rigid-rotation. The leading term flow solution rigid rotation) satisfying Batchelor-Wood formula.