On the planar Taylor-Couette system and related exterior problems

TL;DR

This paper investigates the mathematical properties of the planar Taylor-Couette system, revealing conditions for uniqueness and multiplicity of solutions, and connecting finite system behavior to the Stokes paradox as the outer radius grows.

Contribution

It establishes the uniqueness of classical Taylor-Couette flow among smooth solutions and introduces a broader class of incomplete solutions with multiple solutions, analyzing their behavior as the outer radius increases.

Findings

Classical Taylor-Couette flow is unique among smooth rotationally symmetric solutions.

Infinitely many solutions exist in a broader class of incomplete solutions.

Solutions connect to the Stokes paradox as the outer radius tends to infinity.

Abstract

We consider the planar Taylor-Couette system for the steady motion of a viscous incompressible fluid in the region between two concentric disks, the inner one being at rest and the outer one rotating with constant angular speed. We study the uniqueness and multiplicity of solutions to the forced system in different classes. For any angular velocity we prove that the classical Taylor-Couette flow is the unique smooth solution displaying rotational symmetry. Instead, we show that infinitely many solutions arise, even for arbitrarily small angular velocities, in a larger, class of \textit{incomplete} solutions that we introduce. By prescribing the transversal flux, unique solvability of the Taylor-Couette system is recovered among rotationally invariant incomplete solutions. Finally, we study the behavior of these solutions as the radius of the outer disk goes to infinity, connecting our results with the celebrated Stokes paradox.