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Liouville-type theorems for the Taylor–Couette–Poiseuille flow of the stationary Navier–Stokes equations

Published online by Cambridge University Press:  29 July 2024

Hideo Kozono Open the ORCID record for Hideo Kozono [Opens in a new window] ,
Yutaka Terasawa  and
Yuta Wakasugi
Hideo Kozono*
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Waseda University, Tokyo 169-8555, Japan Mathematical Research Center for Co-creative Society, Tohoku University, Sendai 980-8578, Japan
Yutaka Terasawa
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho Chikusaku Nagoya 464-8602, Japan
Yuta Wakasugi
Affiliation:
Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan
*
†Email address for correspondence: kozono@waseda.jp
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Abstract

We study the stationary Navier–Stokes equations in the region between two rotating concentric cylinders. We first prove that, for a small Reynolds number, if the fluid flow is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily the Taylor–Couette–Poiseuille flow. If, in addition, the associated pressure is bounded or periodic in the $z$ axis, then it coincides with the well-known canonical Taylor–Couette flow. We discuss the relation between uniqueness and stability of such a flow in terms of the Taylor number in the case of narrow gap of two cylinders. The investigation in comparison with two Reynolds numbers based on inner and outer cylinder rotational velocities is also conducted. Next, we give a certain bound of the Reynolds number and the $L^\infty$-norm of the velocity such that the fluid is, indeed, necessarily axisymmetric. As a result, it is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the Taylor–Couette–Poiseuille flow with the exact form of the pressure.

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 989 , 25 June 2024 , A7
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Bang, J., Gui, C., Wang, Y. & Xie, C. 2023 Liouville-type theorems for steady solutions to the Navier–Stokes system in a slab. arXiv:2205.13259v4.Google Scholar
Carrillo, B., Pan, X. & Zhang, Q.-S. 2020 Decay and vanishing of some axially symmetric D-solutions of the Navier–Stokes equations. J. Funct. Anal. 279, 108504.CrossRefGoogle Scholar
Carrillo, B., Pan, X., Zhang, Q.-S. & Zhao, N. 2020 Decay and vanishing of some D-solutions of the Navier–Stokes equations. Arch. Rat. Mech. Anal. 237, 1383–1419.CrossRefGoogle Scholar
Chae, D. 2014 Liouville-type theorems for the forced Euler equations and the Navier–Stokes equations. Commun. Math. Phys. 326, 37–48.CrossRefGoogle Scholar
Chae, D. & Wolf, J. 2016 On Liouville type theorems for the steady Navier–Stokes equations in $\mathbb {R}^3$. J. Differ. Equ. 261, 5541–5560.CrossRefGoogle Scholar
Chamorro, D., Jarrín, O. & Lemarié-Rieusset, P.-G. 2021 Some Liouville theorems for stationary Navier–Stokes equations in Lebesgue and Morrey spaces. Ann. Henri Poincaré Anal. Non Linéaire 38, 689–710.CrossRefGoogle Scholar
Chossat, P. & Iooss, G. 1994 The Couette–Taylor problem. In Applied Mathematical Sciences, vol. 102. Springer-Verlag.CrossRefGoogle Scholar
Galdi, G.P. 2011 An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady-State Problems, 2nd edn. Springer Monographs in Mathematics. Springer-Verlag.CrossRefGoogle Scholar
Guy Raguin, L. & Georgiadis, Y. 2024 Kinematics of the stationary helical vortex mode in Taylor–Couette–Poiseuille flow. J. Fluid Mech. 516, 125–154.CrossRefGoogle Scholar
Kagei, Y. & Nishida, T. 2015 Instability of plane Poiseuille flow in viscous compressible gas. J. Math. Fluid Mech. 17, 129–143.CrossRefGoogle Scholar
Kagei, Y. & Nishida, T. 2019 Traveling waves bifurcating from plane Poiseuille flow of the compressible Navier–Stokes equation. Arch. Rat. Mech. Anal. 231, 1–44.CrossRefGoogle Scholar
Kagei, Y. & Teramoto, Y. 2020 On the spectrum of the linearized operator around compressible Couette flows between two concentric cylinders. J. Math. Fluid Mech. 22 (2), paper 21.CrossRefGoogle Scholar
Kirchgässner, K. & Sorger, P. 1969 Branching analysis for the Taylor problem. Q. J. Mech. Appl. Math. 22, 183–209.CrossRefGoogle Scholar
Koch, G., Nadirashvili, N., Seregin, G. & Sverak, V. 2009 Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203, 83–105.CrossRefGoogle Scholar
Kozono, H., Terasawa, Y. & Wakasugi, Y. 2017 A remark on Liouville-type theorems for the stationary Navier–Stokes equations in three space dimensions. J. Funct. Anal. 272, 804–818.CrossRefGoogle Scholar
Ma, T. & Wang, S. 2009 Boundary-layer and interior separations in the Taylor–Couette–Poiseuille flow. J. Math. Phys. 50, 033101.CrossRefGoogle Scholar
Matsukawa, Y. & Tsukahara, T. 2022 Subcritical transition of Taylor–Couette–Poiseuille flow at high radius ratio. Phys. Fluids 34, 074109.CrossRefGoogle Scholar
Seregin, G. 2018 Remarks on Liouville type theorems for steady-state Navier–Stokes equations. Algebra Anal. 30, 238–248.Google Scholar
Taylor, G.I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. Ser. A 223, 289–343.Google Scholar
Temam, R. 1977 Navier–Stokes Equations Theory and Numerical Analysis. North-Holland.Google Scholar
Tsai, T.-P. 2021 Liouville type theorems for stationary Navier–Stokes equations. SN Part. Differ. Equ. Appl. 2, 10.CrossRefGoogle Scholar