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arXiv · 2410.07759

Prandtl-Batchelor flows with a point vortex on a disk

Zhi Chen·Mingwen Fei·Zhiwu Lin·Jianfeng Zhao
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Abstract

We study steady two-dimensional incompressible Navier-Stokes flow in the unit disk in a high Reynolds number regime, driven by a localized central forcing and a nearly rigid rotating boundary. The forcing models a compact source of circulation that sustains a vortex core in a viscous flow. For sufficiently small viscosity and boundary perturbation, we construct a very weak steady solution on the whole disk and identify its inviscid limit: a point vortex embedded in a constant vorticity background and joined to the boundary motion by a thin boundary layer. The background vorticity is selected by the Batchelor-Wood formula. The singular core introduces non-coercive interactions in the linearized problem that are absent for regular Prandtl-Batchelor flows. To overcome this difficulty, we map the punctured disk to a semi-infinite cylinder by a logarithmic radial transformation and develop a Fourier-mode analysis that isolates the delicate first Fourier mode, yielding uniform stability and vorticity estimates in the presence of the point-vortex singularity.

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Zhi Chen, Mingwen Fei, Zhiwu Lin, Jianfeng Zhao. 2024-10-10. Prandtl-Batchelor flows with a point vortex on a disk. https://arxiv.org/abs/2410.07759

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