On the non‐stationary non‐Newtonian flow through a thin porous medium

Abstract

We consider a non‐stationary incompressible non‐Newtonian flow in a thin porous medium of thickness ε which is perforated by periodically distributed solid cylinders of size aε. The viscosity is supposed to obey the power law with flow index 32<p<2 (pseudoplastic fluids). The limit when the thickness tends to zero is considered. Time‐dependent Darcy's laws are rigorously derived from this model depending on the magnitude aε with respect to ε.

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]).…”

“…In [3], in particular, the flow of an incompressible stationary Stokes system with a nonlinear viscosity, being a power law, was studied. For non-stationary incompressible viscous flow in a thin porous medium see [1], where a non-stationary Stokes system is considered, and [2], where a non-stationary non-newtonian Stokes system, where the viscosity obeyed the power law, is studied. For the periodic unfolding method applied to the study of problems stated in other type of thin periodic domains we refer for instance to [18] for crane type structures and to [19], [20] for thin layers with thin beams structures, where elasticity problems are studied.…”

“…What distinguishes different fluids is the expression of the stress tensor σ. and the air. For a newtonian fluid, the entries of the stress tensor σ(p, u) are given by (σ(p, u)) ij = −pδ ij + 2µ(D(u)) ij , 1 ≤ i, j ≤ 3 (2) where δ ij is the Kronecker symbol, the real positive µ is the viscosity of the fluid and the entries of the strain tensor are (D(u)) ij = (∂ xj u i + ∂ xi u j )/2. If f belongs to (L 2 (Π)) 3 and the space V is defined by V = {v ∈ (H 1 0 (Π)) 3 | div v = 0}, then u and p satisfying (1) with (2) are such that (see for instance [17]):…”

“…where |D(u)| 2 = D(u) : D(u) and the positive number g represents the yield stress of the fluid. If g = 0, then (4) becomes (2). Viscoplastic Bingham fluids are quite often encountered in real life.…”

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

“…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]. In addition, for the case of non-Newtonian power-law fluid flows in thin porous media, we refer to Anguiano [4,5] and Anguiano and Suárez-Grau [11] and for the case of non-Newtonian Bingham fluid flows in thin porous media, we refer to Anguiano and Bunoiu [6,7].…”