Couette-Taylor instabilities in the small gap regime: the very counter-rotating case

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.