AI 中文总结
针对小间隙等工况下两旋转圆柱间粘性流体的极强反向旋转情形，推导极限系统并分析弱非线性行为，讨论相关波的存在性、稳定性及空间调制解的分类问题。
AI 中文摘要
本文研究两旋转圆柱间粘性流体的Couette-Taylor不稳定性，研究范围涵盖小间隙、慢归一旋转速率、高雷诺数工况，重点关注极强反向旋转情形μ < μ_c ≈ -0.8，该情形下一次不稳定性为非轴对称。从Navier-Stokes方程出发，推导捕捉主导阶动力学的极限系统，计算临界泰勒数T_c(μ)及临界轴向、周向波数。近临界时，弱非线性行为由两个耦合复Ginzburg-Landau方程描述；从线性化特征函数及伴随问题数值计算该振幅系统的所有系数，包括三次非线性项。简化方程存在螺旋波（沿轴向与周向传播）和带状波（轴向静止、周向传播），并讨论其存在性与稳定性判据。还研究满足三阶动力学系统的更奇特空间调制解，其完整分类仍是开放挑战。
英文摘要
In this paper, we study the Couette-Taylor instability of a viscous fluid between two rotating cylinders in the small-gap, slow rescaled rotation rate, high Reynolds number regime, focusing on the very counter-rotating case $μ< μ_c \approx -0.8$ where the primary instability is non-axisymmetric. Starting from the Navier-Stokes equations, we derive a limit system that captures the leading-order dynamics and compute the critical Taylor number $T_c(μ)$ together with the critical axial and azimuthal wavenumbers. Near criticality, the weakly nonlinear behaviour is governed by a system of two coupled complex Ginzburg-Landau equations. All coefficients of this amplitude system including the cubic nonlinear terms are evaluated numerically from the linearised eigenfunctions and the associated adjoint problem. The reduced equations admit helicoidal waves (travelling in both the axial and azimuthal directions) and ribbon waves (standing axially, travelling azimuthally), and their existence and stability criteria are discussed. We also examine more exotic spatially modulated solutions that satisfy a third-order dynamical system, whose complete classification remains an open challenge.
