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Asymptotic Behavior of a Bingham Flow in Thin Domains with Rough Boundary

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Abstract

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter \(\epsilon \) denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when \(\epsilon \) tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law.

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References

  1. Anguiano, M., Bunoiu, R.: Homogenization of Bingham flow in thin porous media. Netw. Heterogeneous Media 15(1), 87–110 (2020)

    Article  MathSciNet  Google Scholar 

  2. Anguiano, M., Suarez-Grau, F.J.: Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary. IMA J. Appl. Math. 84(1), 63–95 (2019)

    Article  MathSciNet  Google Scholar 

  3. Arrieta, J.M., Villanueva-Pesqueira, M.: Thin domains with non-smooth periodic oscillatory boundaries. J. Math. Anal. Appl. 446(1), 30–164 (2017)

    Article  MathSciNet  Google Scholar 

  4. Bayada, G., Chambat, M.: Homogenization of the Stokes system in a thin film flow with rapidly varying thickness. RAIRO Modél. Math. Anal. Numér. 23, 205–234 (1989)

    Article  MathSciNet  Google Scholar 

  5. Bingham, E.C.: Fluidity and Plasticity. McGraw-Hill, New York (1922)

    Google Scholar 

  6. Blanchard, D., Griso, G.: Microscopic effects in the homogenization of the junction of rods and a thin plate. Asympt. Anal. 56(1), 1–36 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Bourgeat, A., Mikelić, A.: A note on homogenization of Bingham flow through a porous medium. J. Math. Pures Appl. 72(9), 405–414 (1993)

    MathSciNet  MATH  Google Scholar 

  8. Bunoiu, R., Cardone, G.: Bingham flow in porous media with obstacles of different size. Math. Meth. Appl. Sci. 40(12), 4514–4528 (2017)

    Article  MathSciNet  Google Scholar 

  9. Bunoiu, R., Cardone, G., Perugia, C.: Unfolding method for the homogenization of Bingham flow. In: Ferreira, J., Barbeiro, S., Pena, G., Wheeler, M. (eds.) Modelling and Simulation in Fluid Dynamics in Porous Media. Springer Proceedings in Mathematics & Statistics, vol. 28, pp. 109–123. Springer, New York, NY (2013). https://doi.org/10.1007/978-1-4614-5055-9_7

  10. Bunoiu, R., Donato, P.: Unfolding homogenization in doubly periodic media and applications. Appl. Anal. 96(13), 2218–2235 (2017)

    Article  MathSciNet  Google Scholar 

  11. Bunoiu, R., Gaudiello, A., Leopardi, A.: Asymptotic analysis of a bingham fluid in a thin t-like shaped structure. J. Math. Pures Appl. 123, 148–166 (2019)

    Article  MathSciNet  Google Scholar 

  12. Bunoiu, R., Kesavan, S.: Fluide de Bingham dans une couche mince. Annals of the University of Craiova. Math. Comput. Sci. Ser. 30, 1–9 (2003)

    MATH  Google Scholar 

  13. Bunoiu, R., Kesavan, S.: Asymtotic behavior of a Bingham fluid in thin layers. J. Math. Anal. Appl. 293(2), 405–418 (2004)

    Article  MathSciNet  Google Scholar 

  14. Cardone G., Corbo Esposito A., Panasenko G.P.: Asymptotic partial decomposition for diffusion with sorption in thin structures. Nonlinear Anal. 65, 79–106 (2006)

  15. Cardone G., Fares R., Panasenko G.P.: Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure, J. Math. Phys. 53, 103702 (2012)

  16. Cardone G., Panasenko G.P., Sirakov Y.: Asymptotic analysis and numerical modeling of mass transport in tubular structures. Math. Models Meth. Appl. Sc. 20(4), 1–25 (2010)

  17. Cioranescu, D., Damlamian, A., Donato, P., Griso, G., Zaki, R.: The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44(2), 718–760 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620 (2008)

    Article  MathSciNet  Google Scholar 

  20. Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method. In: Theory and Applications to Partial Differential Problems, Vol. 3, Springer, Singapore (2018)

  21. Duvaut, G.: Lions, J.-L.: Les inéquations en mécanique et en physique. Travaux et Recherches Mathématiques 21, Dunod, Paris, (1972)

  22. Fabricius, J., Koroleva, Y.O., Tsandzana, A., Wall, P.: Asymptotic behaviour of Stokes flow in a thin domain with a moving rough boundary. Proc. Math. Phys. Eng. Sci. 470, 20130735 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Fabricius, J., Tsandzana, A., Pérez-Rafolds, F., Wall, P.: A comparison of the roughness regimes in hydrodynamic lubrication. J. Tribol. 139, 051702 (2017)

    Article  Google Scholar 

  24. Girault, V., Raviart, P.A.: Finite Element methods for Navier-Stokes equations. In: Theory and Algorithms. Springer Series in Computational Mathematics, Vol. 5. Springer-Verlag (1986)

  25. Hale, J.K., Raugel, G.: Reaction-diffusion equation on thin domains. J. Math. Pures Appl. 71(9), 33–95 (1992)

    MathSciNet  MATH  Google Scholar 

  26. Lions, J.L., Sánchez-Palencia, E.: Ecoulement d’un fluide viscoplastique de Bingham dans un milieu poreux. J. Math. Pures Appl. 60, 341–360 (1981)

    MathSciNet  MATH  Google Scholar 

  27. Mikelić, A., Bayada, G., Chambat, M.: Remark on the result on homogenization in hydrodynamical lubrication. RAIRO Modél. Math. Anal. Numér. 25, 363–370 (1991)

    Article  MathSciNet  Google Scholar 

  28. Pereira, M.C., Silva, R.P.: Error estimates for a Neumann problem in highly oscillating thin domains. Dis. Continuous Dyn. Syst. 33(2), 803–817 (2013)

    Article  MathSciNet  Google Scholar 

  29. Prizzi, M., Rinaldi, M., Rybakowski, K.P.: Curved thin domains and parabolic equations. Studia Math. 151, 109–140 (2002)

    Article  MathSciNet  Google Scholar 

  30. Raugel, G.: Dynamics of partial differential equations on thin domains. In: Johnson, R. (eds.) Dynamical Systems. Lecture Notes in Mathematics, vol. 1609, pp. 208–315. Springer, Berlin, Heidelberg (1995). https://doi.org/10.1007/BFb0095241

  31. Richardson, P., Karlstrom, L.: The multi-scale influence of topography on lava flow morphology. Bull. Volcanol. 81, 21 (2019)

    Article  Google Scholar 

  32. Tartar, L.: Incompressible fluid flow in a porous medium convergence of the homogenization process. In: Appendix to Lecture Notes in Physics, vol. 127. Springer-Velag, Berlin (1980)

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Acknowledgements

This paper was initiated during the visit of the last author at the University of Sannio, whose warm hospitality and support are gratefully acknowledged. G.C. and C.P. are members of GNAMPA (INDAM). The last author was partially supported by grant MTM2016-75465-P from the Ministerio de Economia y Competitividad, Spain and Grupo de Investigación CADEDIF, UCM.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università del Sannio, Corso Garibaldi, 107, 82100, Benevento, Italy

    Giuseppe Cardone

  2. Dipartimento di Scienze e Tecnologie, Università del Sannio, Via F. De Sanctis, Palazzo ex-Enel, 82100, Benevento, Italy

    Carmen Perugia

  3. Universidad Pontificia Comillas Departamento Matemática Aplicada, ICAI Alberto Aguilera 25, 28015, Madrid, Spain

    Manuel Villanueva Pesqueira

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  1. Giuseppe Cardone
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  2. Carmen Perugia
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  3. Manuel Villanueva Pesqueira
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Correspondence to Giuseppe Cardone.

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Cardone, G., Perugia, C. & Villanueva Pesqueira, M. Asymptotic Behavior of a Bingham Flow in Thin Domains with Rough Boundary. Integr. Equ. Oper. Theory 93, 24 (2021). https://doi.org/10.1007/s00020-021-02643-7

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  • DOI: https://doi.org/10.1007/s00020-021-02643-7

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