The Transition Between the Navier–Stokes Equations to the Darcy Equation in a Thin Porous Medium


We consider a Newtonian flow in a thin porous medium \(\Omega _{\varepsilon }\) of thickness \(\varepsilon \) which is perforated by periodically distributed solid cylinders of size \(a_\varepsilon \). Generalizing (Anguiano and Suárez-Grau, ZAMP J Appl Math Phys 68:45, 2017), the fluid is described by the 3D incompressible Navier–Stokes system where the external force takes values in the space \(H^{-1}\), and the porous medium considered has one of the most commonly used distribution of cylinders: hexagonal distribution. By an adaptation of the unfolding method, three different Darcy’s laws are rigorously derived from this model depending on the magnitude \(a_{\varepsilon }\) with respect to \(\varepsilon \).

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Corresponding author

Correspondence to Francisco Javier Suárez-Grau.

Additional information

María Anguiano has been supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466. 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. The Transition Between the Navier–Stokes Equations to the Darcy Equation in a Thin Porous Medium. Mediterr. J. Math. 15, 45 (2018).

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Mathematics Subject Classification

  • 76A20
  • 76M50
  • 35B27
  • 35Q30


  • Homogenization
  • Navier–Stokes equations
  • Darcy’s law
  • porous medium
  • thin-film fluids