Theoretical derivation of Darcy's law for fluid flow in thin porous media

Abstract

In this paper we study stationary incompressible Newtonian fluid flow in a thin porous media. The media under consideration is a bounded perforated 3𝐷 domain confined between two parallel plates. The description of the domain includes two small parameters: 𝜀 representing the distance between plates and 𝑎 𝜀 connected to the microstructure of the domain such that 𝑎 𝜀 ≪ 𝜀. We consider the classical setting of perforated media, i.e. 𝑎 𝜀 -periodically distributed solid (not connected) obstacles of size 𝑎 𝜀 . The goal of this paper is to introduce a version of the unfolding method, depending on both parameters 𝜀 and 𝑎 𝜀 , and then to derive the corresponding 2𝐷 Darcy law.

Sharp Pressure Estimates for the Navier–Stokes System in Thin Porous Media

“…To finish the introduction, we give a list of recent references concerning studies of partial differential equations in TPM. Studies related to Newtonian fluids through TPM can be found in Anguiano and Suárez-Grau [11,14], Armiti-Juber [16], Bayada and et al [17], Larsson et al [26], Suárez-Grau [31], Valizadeh and Rudman [32], Wagner et al [33] and Zhengan and Hongxing [36]. Concerning generalized Newtonian fluids see Anguiano and Suárez-Grau [7,12,15], for Bingham fluids see Anguiano and Bunoiu [8,9], for compressible and piezo-viscous flow see Pérez-Ràfols et al [27], and for micropolar fluids see Suárez-Grau [30] and for diffusion problems see Anguiano [5,6] and Bunoiu and Timofte [21].…”

On p-Laplacian Reaction–Diffusion Problems with Dynamical Boundary Conditions in Perforated Media

“…Another possible way is to study this parabolic model in a thin porous media (see, for instance, [2,8,10,12,42,43] for more details on the importance of this type of domains). Finally, another problem could be to consider a porous media containing a thin fissure.…”