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Formal derivation and stability analysis of boundary layer models in MHD

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Abstract

We provide a systematic derivation of boundary layer models in magnetohydrodynamics (MHD), through an asymptotic analysis of the incompressible MHD system. We recover classical linear models, related to the famous Hartmann and Shercliff layers, as well as nonlinear ones, that we call magnetic Prandtl models. We perform their linear stability analysis, emphasizing the stabilizing effect of the magnetic field.

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References

  1. Alexandre, R., Wang, Y.-G., Xu, C.-J., Yang, T.: Well-posedness of the Prandtl equation in Sobolev spaces. J. Am. Math. Soc. 28(3), 745–784 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, D., Wang, Y., Zhang, Z.: Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow. Preprint arXiv:1609.08785

  3. Cowley, S.J., Hocking, L.M., Tutty, O.R.: The stability of solutions of the classical unsteady boundary-layer equation. Phys. Fluids 28(2), 441–443 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  5. Desjardins, B., Dormy, E., Grenier, E.: Boundary layer instability at the top of the Earth’s outer core. J. Comput. Appl. Math. 166(1), 123–131 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23(2), 591–609 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gerard-Varet, D., Masmoudi, N.: Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Sci. Éc. Norm. Supér. (4) 48(6), 1273–1325 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gérard-Varet, D., Nguyen, T.: Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77(1–2), 71–88 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Gerbeau, J.-F., Le Bris, C., Lelièvre, T.: Mathematical methods for the magnetohydrodynamics of liquid metals. In: Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2006)

  10. Gilbert, A.: Dynamo theory. In: Handbook of Mathematical Fluid Dynamics, vol. II, pp. 355–441. North-Holland, Amsterdam

  11. Hartmann, J.: Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 15(6), 1–28 (1937)

    Google Scholar 

  12. Hugues, D.W., Tobias, S.M.: On the instability of magnetohydrodynamic shear flows. Proc. R. Soc. Lond. A 457, 1365–1384 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hunt, J.C.R.: Magnetohydrodynamic flow in rectangular ducts. J. Fluid Mech. 21(4), 577–590 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, W.-X., Yang, T.: Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points. Preprint arXiv:1609.08430, September 2016

  15. Lingwood, R., Alboussière, T.: On the stability of the Hartmann Layer. Phys. Fluids 11, 2058 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, C.-J., Xie, F., Yang, T.: Local-in-time well-posedness theory for MHD boundary layer in Sobolev spaces without monotonicity. Preprint arXiv:1611.05815, November (2016)

  17. Liu, C.-J., Wang, Y.-G., Yang, T.: On the ill-posedness of the Prandtl equations in three dimensional space. Arch. Rat. Mech. Anal. 220(1), 83–108 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Masmoudi, N., Wong, T.K.: Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Commun. Pure Appl. Math. 68(10), 1683–1741 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nunez, M.: MHD shear flows with non-constant transverse magnetic field. Phys. Lett. A 376, 1624–1630 (2012)

    Article  MATH  Google Scholar 

  20. Prandtl, L.: Uber Flussigkeitsbewegung bei sehr kleiner Reibung. Verh. III. Intern. Math. Kongr., Heidelberg, 1904, S. 484-491, Teubner, Leipzig (1905)

  21. Renardy, M.: Well-posedness of the hydrostatic MHD equations. J. Math. Fluid Mech. 14, 355–361 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rousset, F.: Large mixed Ekman–Hartmann boundary layers in magnetohydrodynamics. Nonlinearity 17(2), 503–518 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shercliff, J.A.: Steady motion of conducting fluids in pipes under transverse magnetic fields. Math. Proc. Camb. Philos. Soc. 49(1), 136–144 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wesson, J.: Tokamacs. Oxford University Press, Oxford (2011)

    Google Scholar 

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Authors and Affiliations

  1. Université Paris Diderot, Paris, France

    D. Gérard-Varet

  2. Université Paris Diderot, Sorbonne Paris Cité, Paris, France

    M. Prestipino

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  1. D. Gérard-Varet
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  2. M. Prestipino
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Correspondence to D. Gérard-Varet.

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Gérard-Varet, D., Prestipino, M. Formal derivation and stability analysis of boundary layer models in MHD. Z. Angew. Math. Phys. 68, 76 (2017). https://doi.org/10.1007/s00033-017-0820-x

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  • DOI: https://doi.org/10.1007/s00033-017-0820-x

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