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A Local Instability Mechanism of the Navier–Stokes Flow with Swirl on the No-Slip Flat Boundary

  • Leandro LichtenfelzEmail author
  • Tsuyoshi Yoneda
Article

Abstract

Using numerical simulations of the axisymmetric Navier–Stokes equations with swirl on a no-slip flat boundary, Hsu et al. (J Fluid Mech 794:444–459, 2016) observed the creation of a high-vorticity region on the boundary near the axis of symmetry. In this paper, using a differential geometric approach, we prove that such flows indeed have a destabilizing effect, which is formulated in terms of a lower bound on the \(L^\infty \)-norm of derivatives of the velocity field on the boundary.

Notes

Acknowledgements

The authors would like to thank Doctor Pen-Yuan Hsu for helpful comments. TY was partially supported by Grant-in-Aid for Young Scientists A (17H04825) and Scientific Research B (18H01135), Japan Society for the Promotion of Science (JSPS).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest in the present work.

References

  1. 1.
    Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Chae, D., Lee, J.: On the regularity of the axisymmetric solutions of the Navier–Stokes equations. Math. Z. 239, 645–671 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chan, C.-H., Czubak, M., Yoneda, T.: An ODE for boundary layer separation on a sphere and a hyperbolic space. Physica D 282, 34–38 (2014)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, C.-C., Strain, R.-M., Tsai, T.-P., Yau, H.-T.: Lower bounds on the blow-up rate of the axisymmetric Navier–Stokes equations. II. Commun. Partial Differ. Equ. 34, 203–232 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hsu, P.-Y., Notsu, H., Yoneda, T.: A local analysis of the axi-symmetric Navier–Stokes flow near a saddle point and no-slip flat boundary. J. Fluid Mech. 794, 444–459 (2016)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Koch, G., Nadirashvili, N., Seregin, G.-A., Šverák, V.: Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203, 83–105 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ladyzhenskaya, O.-A.: On the unique global solvability to the Cauchy problem for the Navier–Stokes equations in the presence of the axial symmetry. Zap. Nauc. Sem. LOMI 7, 155–177 (1969)MathSciNetGoogle Scholar
  8. 8.
    Ma, T., Wang, S.: Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Partial differential equations and applications. Discrete Contin. Dyn. Syst. 10, 459–472 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ukhovskii, M.-R., Iudovich, V.-I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. 32, 52–61 (1968)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoMeguroJapan

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