Prandtl-Batchelor flows on an annulus

For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines) in a simply connected domain, Prandtl (1905) and Batchelor (1956) found that in the limit of vanishing viscosity, the vorticity is constant in an inner region separated from the boundary layer. In this paper, we consider the generalized Prandtl-Batchelor theory on an annulus that is non-simply-connected. First, we observe that in the vanishing viscosity if steady Navier-Stokes solutions with nested closed streamlines on an annulus converge to steady Euler flows which are rotating shear flows, then the vorticity of Euler flows must be $a\ln r+b$ and the associated velocity must be $(ar+\frac{b}{r}+cr\ln r,0)$ in polar coordinates. We call solutions of steady Navier-Stokes equations with the above property Prandtl-Batchelor flows. Then, 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 an annulus with the wall velocity slightly different from the rigid-rotation. In particular, for the above boundary conditions, we prove that there is a continuous curve (i.e. infinitely many) of solutions to the steady Navier-Stokes equations when the viscosity is sufficiently small.