Couette Taylor instabilities in the small-gap regime

TL;DR

This paper rigorously analyzes the onset of Taylor vortices in a viscous fluid between nearly identical rotating cylinders, revealing a critical Taylor number and a variety of steady flow patterns near instability.

Contribution

It provides a rigorous proof of the critical Taylor number and characterizes the bifurcation structure, including exotic steady flow solutions, in the small-gap limit.

Findings

Existence of a critical Taylor number $T_c$ for instability.

Near $T_c$, solutions are governed by a Ginzburg-Landau PDE.

A family of steady solutions including wavy vortices and exotic patterns emerges beyond $T_c$.

Abstract

The Couette-Taylor instability occurs in a viscous fluid confined between two coaxial rotating cylinders. When the Taylor number surpasses a critical value, the stable Couette flow destabilizes, giving way to steady Taylor vortices. As the Taylor number increases further, these vortices themselves become unstable, transitioning into wavy Taylor vortices. In this article, we focus on the small-gap limit, where the ratio of the cylinder radii approaches unity and the rotation rates of the cylinders are nearly identical. We provide a rigorous proof of the existence of a critical Taylor number $T_c$, at which the Couette flow loses stability. For Taylor numbers just above $T_c$, under fixed axial periodicity, the solutions to the limiting Navier-Stokes system are governed by a Ginzburg-Landau-type partial differential equation. Beyond the classical Taylor vortex flow, we demonstrate that a two-parameter family of solutions emerges at criticality for $T>T_c$. This family includes not only wavy vortices but also a variety of other exotic flow patterns, all of which remain steady in the frame rotating at the average angular velocity of the cylinders.