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Derivation of a coupled Darcy–Reynolds equation for a fluid flow in a thin porous medium including a fissure

Abstract

We study the asymptotic behavior of a fluid flow in a thin porous medium of thickness \(\varepsilon \), which is characteristic size of the pores \(\varepsilon \) and contains a fissure of width \(\eta _\varepsilon \). We consider the limit when the size of the pores tends to zero, and we find a critical size \(\eta _\varepsilon \approx \varepsilon ^{2\over 3}\) in which the flow is described by a 2D Darcy law coupled with a 1D Reynolds problem. We also discuss the other cases.

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Correspondence to María Anguiano.

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María Anguiano has been supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466, and in part by European Commission, Excellent Science-European Research Council (ERC) H2020-EU.1.1.-639227. Francisco Javier Suárez-Grau has been supported by Ministerio de Economía y Competitividad (Spain), Proyecto Excelencia MTM2014-53309-P.

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Anguiano, M., Suárez-Grau, F.J. Derivation of a coupled Darcy–Reynolds equation for a fluid flow in a thin porous medium including a fissure. Z. Angew. Math. Phys. 68, 52 (2017). https://doi.org/10.1007/s00033-017-0797-5

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  • DOI: https://doi.org/10.1007/s00033-017-0797-5

Mathematics Subject Classification

  • 75A05
  • 76A20
  • 76M50
  • 35B27

Keywords

  • Stokes equation
  • Darcy’s law
  • Reynolds equation
  • Thin porous medium
  • Fissure