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

  • María Anguiano
  • Francisco Javier Suárez-GrauEmail author


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 \(\varepsilon \).


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

Mathematics Subject Classification

76A20 76M50 35B27 35Q30 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Departamento de Ecuaciones Diferenciales y Análisis Numérico Facultad de MatemáticasUniversidad de SevillaSevillaSpain

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