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On the Navier boundary condition for viscous fluids in rough domains

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Abstract

In this paper we review some recent results concerning the study of the asymptotic behavior of viscous fluids in rough domains assuming Navier boundary conditions on the rough boundary. Our main interest is to study the relation between both the adherence and the Navier boundary conditions in the case of a boundary with weak rugosities. We show that the roughness acts on the fluid as a friction term. In particular, if the roughness is sufficiently strong, Navier condition implies adherence condition. This generalizes previous results of other authors.

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References

  1. G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23:1482–1518, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Arbogast, J. Douglas, and U. Hornung. Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal., 21:823–836, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Bucur, E. Feireisl, S. Nečasová, and J. Wolf. On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries. J. Differential Equations, 244: 2890–2908, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Bucur, E. Feireisl, and S. Nečasová. Influence of wall roughness on the slip behavior of viscous fluids. Proc. Royal Soc. Edinburgh A, 138,5:957–973, 2008.

    Article  MATH  Google Scholar 

  5. D. Bucur, E. Feireisl, and S. Nečasová. On the asymptotic limit of flows past a ribbed boundary, J. Math. Fluid Mech., 10,4: 554–568, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Bucur, E. Feireisl, and S. Nečasová. Boundary behavior of viscous fluids: Influence of wall roughness and friction-driven boundary conditions. Arch. Rational Mech. Anal., 197:117–138, 2010.

    Article  MATH  Google Scholar 

  7. C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez. Homogenization of elliptic problems with Dirichlet and Neumann conditions imposed on varying subsets. Math. Methods Appl. Sci. 30,14:1611–1625, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Calvo-Jurado, J. Casado-Díaz, M. Luna-Laynez. Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets. ESAIM Control, Optim. and Calc. Var., 15: 49–67, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Casado-Díaz. Two-scale convergence for nonlinear Dirichlet problems in perforated domains. Proc. Roy. Soc. Edimburgh A, 130: 249–276, 2000.

    Article  MATH  Google Scholar 

  10. J. Casado-Díaz, E. Fernández-Cara, and J. Simon. Why viscous fluids adhere to rugose walls: A mathematical explanation. J. Differential Equations, 189,2: 526–537, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Casado-Díaz, M. Luna-Laynez, and F.J. Suárez-Grau. Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall. Math. Mod. Meth. Appl. Sci., 20: 121–156, 2010.

    Article  MATH  Google Scholar 

  12. J. Casado-Díaz, M. Luna-Laynez, and F.J. Suárez-Grau. A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary. C. R. Acad. Sci. Paris, Ser. I, 348: 967–971, 2010.

    Article  MATH  Google Scholar 

  13. J. Casado-Díaz, M. Luna-Laynez, and F.J. Suárez-Grau. Estimates for the asymptotic expansion of a viscous fluid satisfying Navier’s law on a rugous boundary. Math. Meth. Appl. Sci., 34: 1553–1561, 2011.

    Article  MATH  Google Scholar 

  14. J. Casado-Díaz, M. Luna-Laynez, and F.J. Suárez-Grau. Asymptotic behavior of the Navier-Stokes system in a thin domain with the slip condition on a slightly rough boundary, submitted for publication.

  15. J. Casado-Díaz, M. Luna-Laynez, and F.J. Suárez-Grau. The homogenization of elliptic partial differential systems on rugous domains with variable boundary conditions, to appear in Proc. Royal Soc. Edinburgh A.

  16. D. Cioranescu, A. Damlamian, and G. Griso, Periodic unfolding and homogenization. C.R. Acad. Sci. Paris, Sér. I, 335: 99–104, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. Cioranescu, and F. Murat. Un terme étrange venu d’ailleurs, Nonlinear partial differential equations and their applications. Collège de France seminar, Vols II and III. Eds H. Brézis and J. L. Lions. Research Notes in Math. 60 and 70. Pitman, London, 98–138 and 154–78, 1982.

    Google Scholar 

  18. G. Dal Maso, A. Defranceschi, and E. Vitali. Integral representation for a class of C1-convex functionals, J. Math. Pures Appl. 9,73: 1–46, 1994.

    Google Scholar 

  19. G.P. Galdi. An introduction to the mathematical theory of the Navier-Stokes equations. Vol. 1, Springer-Verlag, New-York, 1994.

    Google Scholar 

  20. P. Grisvard. Elliptic problems in nonsmooth domains. Pitman, Boston, 1985.

    MATH  Google Scholar 

  21. Lenczner M., Homogénéisation d’un circuit électrique, C.R. Acad. Sci. Paris II, 324,b: 537–542, 1997.

    MATH  Google Scholar 

  22. C.L.M.H. Navier. Mémoire sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Paris 6: 389–416, 1823.

    Google Scholar 

  23. G. Nguetseng. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal., 20,3: 608–623, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  24. F.J. Suárez-Grau. Comportamiento asintótico de fluidos viscosos con condiciones de deslizamiento sobre fronteras rugosas. Ph.D. Dissertation, Universidad de Sevilla, 2011.

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Authors and Affiliations

  1. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain

    J. Casado-Díaz & M. Luna-Laynez

  2. Departamento de Matemáticas, Universidad de Huelva, Spain

    F. J. Suárez-Grau

Authors
  1. J. Casado-Díaz
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  2. M. Luna-Laynez
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  3. F. J. Suárez-Grau
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Correspondence to J. Casado-Díaz.

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Casado-Díaz, J., Luna-Laynez, M. & Suárez-Grau, F.J. On the Navier boundary condition for viscous fluids in rough domains. SeMA 58, 5–24 (2012). https://doi.org/10.1007/BF03322603

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  • DOI: https://doi.org/10.1007/BF03322603

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