A Cartesian representation of the Taylor–Couette system in the vanishing limit of the gap between coaxial cylinders is presented, where the ratio, μ , of the angular velocities, ω i and ω o , of the inner and the outer cylinders, respectively, affects its axisymmetric flow structures. Our numerical stability study finds remarkable agreement with previous studies for the critical Taylor number, T c ( μ ) , for the onset of axisymmetric instability. The Taylor number T can be expressed as T = Ω ( R − Ω ) , where Ω (the rotation number) and R (the Reynolds number) in the Cartesian system are related to the average and the difference of ω i and ω o . The instability sets in the region ( Ω , R ) → ( 0 , ∞ ) , while the product of Ω and R is kept finite. Furthermore, we developed a numerical code to calculate nonlinear axisymmetric flows. It is found that the mean flow distortion of the axisymmetric flow is antisymmetric across the gap when μ = 1 , while a symmetric part of the mean flow distortion appears additionally when μ ≠ 1 . Our analysis also shows that for a finite R all flows with μ ≠   1 approach the R axis, so that the plane Couette flow system is recovered in the vanishing gap limit. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper (part 2)’.