Taylor–Couette system in the narrow-gap limit, revisited, with a corrigendum to the paper by Nagata (2023, Philosophical Transaction A.)

We study transition from circular Couette flow to Taylor–Couette flow in the narrow-gap limit. We find that subcritical bifurcations of a nonlinear axisymmetric flow occur at the linear critical point when the angular velocity ratio for the cylinders, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>μ</mml:mi> </mml:math> , is less than approximately −0.5. We also demonstrate that two nonlinear axisymmetric flow branches with the same wavenumber as that for the critical mode exist for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Introducing the modified Taylor number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>τ</mml:mi> </mml:math> , analyses for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> are made possible, so that this study supplements the recent analysis of Nagata M. 2023 Taylor-couette flow in the narrow-gap limit. Philos. Trans. R. Soc. A 381, 20220134. (doi: 10.1098/rsta.2022.0134 ), in which the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:mo form="prefix" stretchy="false">−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> was excluded, because the critical value of the conventional Taylor number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>a</mml:mi> </mml:mrow> </mml:math> used therein approaches zero as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo form="prefix" stretchy="false">−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . The present analysis takes into account all the contributions from the inertia term, including a component which was neglected in [1]. The effect of the inclusion of this previously neglected component is surveyed at the end.