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

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 $$\Omega _{\varepsilon }$$ Ω ε of thickness $$\varepsilon$$ ε which is perforated by periodically distributed solid cylinders of size $$a_\varepsilon$$ a ε . 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 $$H^{-1}$$ 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 $$a_{\varepsilon }$$ a ε with respect to $$\varepsilon$$ ε . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

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

, Volume 15 (2) – Feb 24, 2018
21 pages

/lp/springer_journal/the-transition-between-the-navier-stokes-equations-to-the-darcy-uTJvBeGybg
Publisher
Springer International Publishing
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-018-1086-z
Publisher site
See Article on Publisher Site

### Abstract

We consider a Newtonian flow in a thin porous medium $$\Omega _{\varepsilon }$$ Ω ε of thickness $$\varepsilon$$ ε which is perforated by periodically distributed solid cylinders of size $$a_\varepsilon$$ a ε . 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 $$H^{-1}$$ 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 $$a_{\varepsilon }$$ a ε with respect to $$\varepsilon$$ ε .

### Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Feb 24, 2018

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