Couette Taylor instabilities for counter-rotating cylinders in the small-gap regime

We study the Couette Taylor instabilities for an incompressible viscous fluid between two coaxial cylinders of nearly equal radii, allowing counter-rotation with the ratio of rotation rate $\mu \in [-1,1]$. Working in a rotating frame and in a small-gap and small-viscosity regime, we derive the corresponding limiting Navier Stokes system and analyze the linear stability of the Couette flow. In particular, we numerically compute the critical Taylor number for general perturbations and identify a transition near $\mu_c \approx -0.8$: for $\mu > \mu_c$ the most unstable mode is axisymmetric, whereas for $\mu < \mu_c$ the most unstable mode is non-axisymmetric. Near criticality, slowly varying traveling waves are governed by a time-independent Ginzburg Landau equation. The nonlinear coefficient changes sign near $\hat{\mu}_c \approx -0.65$, yielding a supercritical regime for $\mu > \hat{\mu}_c$ and a subcritical regime for $\mu_c < \mu < \hat{\mu}_c$. In the subcritical range, we classify small-amplitude steady states, including Taylor vortex flows, wavy vortices, a two-parameter family of quasi-periodic flows, and a localized traveling perturbation of the Couette flow.