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

Abstract

In this paper we study the stationary incompressible power law fluid flow in a thin porous medium. The media under consideration is a bounded perforated 3D domain confined between two parallel plates, where the distance between the plates is very small. The perforation consists in an array solid cylinders, which connect the plates in perpendicular direction, distributed periodically with diameters of small size compared to the period. For a specific choice of the thickness of the domain, we found that the homogenization of the power law Stokes system results a lower-dimensional nonlinear Brinkman type law.

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

“…Observe that (K P ) i3 = (K P ) 3,j = 0, i, j = 1, 2, 3, so that Ṽ can be rewritten to have (15) with K P given by (16).…”

“…Finally, the divergence condition with respect to the variable x together with the expression of Ṽ gives the second line of (15), which has a unique solution P ∈ H 1 (Ω)∩L 2 0 (ω) since K P is positive definite (see [13, Theorem 2.1]).…”

“…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].…”

Carreau law for non-newtonian fluid flow through a thin porous media

“…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 pass to the limit. In particular, the generalized Newtonian fluids obeying the power law in the thin porous media Ω ε have been studied rigorously in Anguiano and Suárez-Grau [10] where we have obtained a 2D Darcy's law when the domain thickness tends to zero (see also [15] for the extension to the case of a thin porous media with an array of cylinders with small diameter). Also, the Bingham plastic behavior in the thin porous media Ω ε has been studied in [8,9].…”

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.…”