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Interaction of a Vortex Induced by a Rotating Cylinder with a Plane

  • Daozhi Han (a1), Yifeng Hou (a1) and Roger Temam (a1)

In this article, we study theoretically and numerically the interaction of a vortex induced by a rotating cylinder with a perpendicular plane. We show the existence of weak solutions to the swirling vortex models by using the Hopf extension method, and by an elegant contradiction argument, respectively. We demonstrate numerically that the model could produce phenomena of swirling vortex including boundary layer pumping and two-celled vortex that are observed in potential line vortex interacting with a plane and in a tornado.

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*Corresponding author. Email addresses: (D. Han), (Y. Hou) (R. Temam)
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[1] R. Alexandre , Y.-G. Wang , C.-J. Xu , and T. Yang . Well-posedness of the Prandtl equation in Sobolev spaces. J. Amer. Math. Soc., 28(3):745784, 2015.

[2] Charles J. Amick . Existence of solutions to the nonhomogeneous steady Navier-Stokes equations. Indiana Univ. Math. J., 33(6):817830, 1984.

[3] R. Brown , I. Mitrea , M. Mitrea , and M. Wright . Mixed boundary value problems for the Stokes system. Trans. Amer. Math. Soc., 362(3):12111230, 2010.

[6] E. B. Fabes , C. E. Kenig , and G. C. Verchota . The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J., 57(3):769793, 1988.

[7] David Gérard-Varet and Emmanuel Dormy . On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc., 23(2):591609, 2010.

[8] M. A. Gol'dshtik . Viscous-flow paradoxes. In Annual review of fluid mechanics, Vol. 22, pages 441472. Annual Reviews, Palo Alto, CA, 1990.

[9] M.A. Gol'dshtik . A paradoxical solution of the Navier-Stokes equations. Journal of Applied Mathematics and Mechanics, 24(4):913929, 1960.

[11] Makram Hamouda , Chang-Yeol Jung , and Roger Temam . Boundary layers for the 3D primitive equations in a cube: the supercritical modes. Nonlinear Anal., 132:288317, 2016.

[12] F. Hecht . New development in FreeFem++. J. Numer. Math., 20(3-4):251265, 2012.

[14] E. Hopf . Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr., 4:213231, 1951.

[15] Mikhail V. Korobkov , Konstantin Pileckas , and Remigio Russo . On the flux problem in the theory of steady Navier-Stokes equations with nonhomogeneous boundary conditions. Arch. Ration. Mech. Anal., 207(1):185213, 2013.

[18] Loredana Lanzani , Luca Capogna , and Russell M. Brown . The mixed problem in L p for some two-dimensional Lipschitz domains. Mathematische Annalen, 342(1):91124, 2008.

[19] Vladimir Maz'ya and Tatyana Shaposhnikova . Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains. Bull. Math. Sci., 1(1):3377, 2011.

[20] Marius Mitrea and Michael Taylor . Navier-Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann., 321(4):955987, 2001.

[23] Marco Sammartino and Russel E. Caflisch . Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm. Math. Phys., 192(2):463491, 1998.

[24] J. Serrin . The swirling vortex. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 271(1214):325360, 1972.

[25] Zhong Wei Shen . A note on the Dirichlet problem for the Stokes system in Lipschitz domains. Proc. Amer. Math. Soc., 123(3):801811, 1995.

[26] Shagi-Di Shih and R. Bruce Kellogg . Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18(5):14671511, 1987.

[27] J. L. Taylor , K. A. Ott , and R. M. Brown . The mixed problem in Lipschitz domains with general decompositions of the boundary. Trans. Amer. Math. Soc., 365(6):28952930, 2013.

[29] R. Jeffrey Trapp . A clarification of vortex breakdown and tornadogenesis. Monthly Weather Review, 128(3):888895, 3 2000.

[30] Zhouping Xin and Liqun Zhang . On the global existence of solutions to the Prandtl's system. Adv. Math., 181(1):88133, 2004.

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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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