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

Homogenization of Bingham flow in thin porous media

“…We remark that in all three cases, the vertical componentŨ 3 of the velocity of filtration equals zero and this result is in accordance with the previous mathematical studies of the flow in this thin porous medium, for newtonian fluids (Stokes and Navier-Stokes equations) and for power law fluids (see [15], [1], [2], [3], [4]). Moreover, despite the fact that the limit pressure is not unique, the velocity of filtration is uniquely determined (see Section 4.3 in [24]).…”

“…Recently, the model of thin porous medium under consideration in this paper was introduced in [15], where the flow of an incompressible viscous fluid described by the stationary Navier-Stokes equations was studied by the multiscale expansion method, which is a formal but powerful tool to analyse homogenization problems. These results were rigorously proved in [4] using an adaptation introduced in [3] of the periodic unfolding method from [12]. This adaptation consists of a combination of the unfolding method with a rescaling in the height variable, in order to work with a domain of fixed height, and to use monotonicity arguments to pass to the limit.…”

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

“…In the critical case, the local problems are 3D, while they are 2D in the other cases, which is a considerable simplification. This result is proved in Fabricius et al [19] by using the multiscale expansion method, which is a formal but powerful tool to analyze homogenization problems, and later rigorously developed in Anguiano and Suárez-Grau [10] by using an adaptation of the periodic unfolding method, see Arbogast et al [13] and Cioranescu et al [15][16][17], which is introduced in Anguiano and Suárez-Grau [8]. For other related studies concerning Newtonian fluids in thin porous media such as the derivation of coupled Darcy-Reynolds for fluid flows in thin porous media including a fissure and the modelling of fluid flows in thin porous media with non-homogeneous slip boundary conditions on the cylinders, we refer to Anguiano [2,3] and Anguiano and Suárez-Grau [9,12].…”

Lower-Dimensional Nonlinear Brinkman’s Law for Non-Newtonian Flows in a Thin Porous Medium

“…Considering the change of variables given in (5), we obtain the following result in the domain Ω εδ . Lemma 3.6 (Estimates of dilated velocity and pressure in Ω εδ ).…”

“…On the other hand, the derivation of macroscopic laws for fluid flows in a thin porous medium has recently become of great interest, see Anguiano and Suárez-Grau [5] and Fabricius et al [13]. More precisely, a thin porous medium can be defined as a bounded perforated 3D domain confined between two parallel plates, where the distance between the plates is very small and the perforation consists of periodically distributed solid cylinders which connect the plates in perpendicular direction.…”