The asymptotic analysis of a Darcy-Stokes system coupled through a curved

• Morales, Fernando A. ^[1]
  1. [1] Universidad Nacional de Colombia

     Universidad Nacional de Colombia

     Colombia

     
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 37, Nº. 2, 2019 (Ejemplar dedicado a: Revista Integración, temas de matemáticas), págs. 261-297
• Idioma: inglés
• DOI: 10.18273/revint.v37n2-2019004
• Títulos paralelos:
  • Análisis asintótico de un sistema Darcy-Stokes acoplado a través de una interfaz curva
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En el trabajo se presenta el análisis asintótico de un sistemaDarcy-Stokes acoplado a través de una interfaz curva. El sistema modela el intercambio de fluido entre un canal angosto (flujo Stokes) y un medio poroso (flujo Darcy). El canal es un dominio cilíndrico definido entre la interfaz (Τ) y una traslación paralela de dicha superficie (Τ + ε eN, ε > 0). Utilizando un cambio de variables para fijar un dominio de referencia e introduciendo dos sistemas de coordenadas, el Cartesiano canónico y el local (consistente con la geometría de la superficie), es posible encontrar la forma límite cuando el ancho del canal tiende a cero (ε→ 0). El problema límite es un sistemaacoplado con flujo Darcy en el medio poroso y flujo Brinkman en la interfaz (Τ).

  • English

    We present the asymptotic analysis of a Darcy-Stokes coupledsystem, modeling the fluid exchange between a narrow channel (Stokes flow) and a porous medium (Darcy flow), coupled through a C2 curved interface. The channel is a cylindrical domain between the interface (Τ) and a parallel translation of itself (Τ + ε  eN, ε> 0). The introduction of a change variable (to fix the domain geometry) and the introduction of two systems of coordinates: the Cartesian and a local one (consistent with the geometry of the surface), permit to find the limiting form of the system when the width of the channel tends to zero (ε → 0). The limit problem is a coupled system with Darcy flow in the porous medium and Brinkman flow on the curved interface (Τ).

• Referencias bibliográficas
  • [1] Adams R.A., Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied...
  • [2] Allaire G., Briane M., Brizzi R. and Capdeboscq Y., “Two asymptotic models for arrays of underground waste containers”, Applied Analysis...
  • [3] Arbogast T. and Brunson D.S., “A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium”, Computational...
  • [4] Arbogast T. and Lehr H., “Homogenization of a Darcy-Stokes system modeling vuggy porous media”, Computational Geosciences 10 (2006), No....
  • [5] Babuska I. and Gatica G.N., “A residual-based a posteriori error estimator for the Stokes-Darcy coupled problem”, SIAM J. Numer. Anal....
  • [6] Cannon J.R. and Meyer G.H., “Diffusion in a fractured medium”, SIAM J. Appl. Math. 20 (1971), No. 3, 434–448.
  • [7] Dobberschütz S., “Stokes-Darcy coupling for periodically curved interfaces”, Comptes Rendus Mécanique 342 (2014), No. 2, 73–78.
  • [8] Dobberschütz S., “Effective behavior of a free fluid in contact with a flow in a curved porous medium”, SIAM J. Appl. Math. 75 (2015),...
  • [9] Gatica G.N., Meddahi S. and Oyarzúa R., “A conforming mixed finite-element method for the coupling of fluid flow with porous media flow”,...
  • [10] Girault V. and Raviart P.A., Finite element approximation of the Navier-Stokes equations, volume 749 of Lecture Notes in Mathematics....
  • [11] Lamichhane B.P., “A new finite element method for Darcy-Stokes-Brinkman equations”, ISRN Computational Mathematics 2013, 4 pages. http://dx.doi.org/10.1155/2013/798059
  • [12] Lesinigo M., D’Angelo C. and Quarteroni A., “A multiscale Darcy-Brinkman model for fluid flow in fractured porous media”, Numer. Math....
  • [13] Martin V., Jaffré J. and Roberts J.E., “Modeling fractures and barriers as interfaces for flow in porous media”, SIAM J. Sci. Comput....
  • [14] Morales F. and Showalter R.E., “A Darcy-Brinkman model of fractures in porous media”, J. Math. Anal. Appl. 452 (2017), No. 2, 1332–1358.
  • [15] Morales F.A., “The formal asymptotic expansion of a Darcy-Stokes coupled system”, Rev. Fac. Cienc. 2 (2013), No. 2, 9–24.
  • [16] Morales F.A., “Homogenization of geological fissured systems with curved non-periodic cracks”, Electronic Journal of Differential Equations...
  • [17] Neuss–Radu M., “A result on the decay of the boundary layer in the homogenization theory”, Asymptot. Anal. 23 (2000), No. 3-4, 313–328.
  • [18] Neuss–Radu M., “The boundary behavior of a composite material”, Math. Model. Numer. Anal. 35 (2001), No. 3, 407–435.
  • [19] Tartar L., An introduction to Sobolev spaces and interpolation spaces, volume 3 of Lecture Notes of the Unione Matematica Italiana, Springer-Verlag,...
  • [20] Temam R., Navier-Stokes equations, volume 2 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam,...
  • [21] Xie X., Xu J. and Xue G., “Uniformly-stable finite element methods for Darcy-Stokes- Brinkman models”, J. Comput. Math. 26 (2008), No....