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arXiv · 2111.07114

Prandtl-Batchelor flows on an annulus

Mingwen Fei·Chen Gao·Zhiwu Lin·Tao Tao
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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.

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Mingwen Fei, Chen Gao, Zhiwu Lin, Tao Tao. 2021-11-13. Prandtl-Batchelor flows on an annulus. https://arxiv.org/abs/2111.07114

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