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Planar Limits of Three-Dimensional Incompressible Flows with Helical Symmetry

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Abstract

Helical symmetry is invariance under a one-dimensional group of rigid motions generated by a simultaneous rotation around a fixed axis and translation along the same axis. The key parameter in helical symmetry is the step or pitch, the magnitude of the translation after rotating one full turn around the symmetry axis. In this article we study the limits of three-dimensional helical viscous and inviscid incompressible flows in an infinite circular pipe, with respectively no-slip and no-penetration boundary conditions, as the step approaches infinity. We show that, as the step becomes large, the three-dimensional helical flow approaches a planar flow, which is governed by the so-called two-and-half Navier–Stokes and Euler equations, respectively.

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Acknowledgments

The authors thank the anonymous referee for a careful reading of the manuscript. M. L. F. and H. N. L. wish to thank the University of California at Riverside, where part of this work was conducted, for their hospitality. M. L. F. is partially supported by Brazil CNPq Grant 303089/2010-5, and CNPq fellowship 200434/2011-0. H. N. L. is partially supported by Brazil CNPq Grant 306331/2010-1, CAPES fellowship 6649/10-6, and FAPERJ Grant E-26/103.197/2012. A. M. would like to thank the Institute of Mathematics at the Federal University in Rio de Janeiro for their hospitality and support. A. M.’s work was partially supported by the US National Science Foundation Grants DMS-1009713, DMS-1009714, and DMS-1312727. D. N.’s work is partially supported by the Chinese National Youth Grant No. 11001184 and the Beijing Natural Science Foundation grants No. 1142004. The work of E. S. T. was supported in part by the Minerva Stiftung/Foundation, and the US National Science Foundation Grants DMS-1009950, DMS-1109640 and DMS-1109645.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão, Caixa Postal 68530, 21941-909, Rio de Janeiro, RJ, Brazil

    Milton C. Lopes Filho & Helena J. Nussenzveig Lopes

  2. Department of Mathematics, Penn State University, University Park, PA, 16801, USA

    Anna L. Mazzucato

  3. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Dongjuan Niu

  4. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Edriss S. Titi

  5. Department of Mathematics and Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, 92697-3875, USA

    Edriss S. Titi

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  1. Milton C. Lopes Filho
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  2. Anna L. Mazzucato
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  3. Dongjuan Niu
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  4. Helena J. Nussenzveig Lopes
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  5. Edriss S. Titi
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Correspondence to Anna L. Mazzucato.

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Lopes Filho, M.C., Mazzucato, A.L., Niu, D. et al. Planar Limits of Three-Dimensional Incompressible Flows with Helical Symmetry. J Dyn Diff Equat 26, 843–869 (2014). https://doi.org/10.1007/s10884-014-9411-0

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  • DOI: https://doi.org/10.1007/s10884-014-9411-0

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