A Cartesian representation of the Taylor-Couette system in the vanishing limit of the gap between coaxial cylinders is presented, where the ratio, [Formula: see text], of the angular velocities, [Formula: see text] and [Formula: see text], 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, [Formula: see text], for the onset of axisymmetric instability. The Taylor number [Formula: see text] can be expressed as [Formula: see text], where [Formula: see text] (the rotation number) and [Formula: see text] (the Reynolds number) in the Cartesian system are related to the average and the difference of [Formula: see text] and [Formula: see text]. The instability sets in the region [Formula: see text], while the product of [Formula: see text] and [Formula: see text] 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 [Formula: see text], while a symmetric part of the mean flow distortion appears additionally when [Formula: see text]. Our analysis also shows that for a finite [Formula: see text] all flows with [Formula: see text] approach the [Formula: see text] 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)'.
Keywords: Taylor–Couette flow; plane Couette flow; rotating plane Couette flow.