Darcy’s Law for Flow in a Periodic Thin Porous Medium Confined Between Two Parallel Plates

Abstract

We study stationary incompressible fluid flow in a thin periodic porous medium. The medium under consideration is a bounded perforated 3D-domain confined between two parallel plates. The distance between the plates is δ, and the perforation consists of ε-periodically distributed solid cylinders which connect the plates in perpendicular direction. Both parameters ε, δ are assumed to be small in comparison with the planar dimensions of the plates. By constructing asymptotic expansions, three cases are analysed: (1) ε δ, (2) δ/ε ∼ constant and (3) ε δ. For each case, a permeability tensor is obtained by solving local problems. In the intermediate case, the cell problems are 3D, whereas they are 2D in the other cases, which is a considerable simplification. The dimensional reduction can be used for a wide range of ε and δ with maintained accuracy. This is illustrated by some numerical examples.

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]).…”

“…The model of thin porous medium of thickness much smaller than the distance between the pores was introduced in [27], where a stationary incompressible Navier-Stokes flow was studied. 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].…”

Transitional and Turbulent Flow in a Bed of Spheres as Measured with Stereoscopic Particle Image Velocimetry

“…As in Andersson et al (2009), no index matching needs to be performed and results are compared to 2D simulations with computational fluid dynamics. Although the experimental flow will be 3D due to the bounding walls (Fabricius et al 2016) experimental results and results from simulations compare well to each other. Two-dimensional etched models were also studied with µPIV in Blois et al (2013).…”

“…Many researchers have modelled pore level flow, as such, theoretically or numerically, e.g. Geller and Hunt (1993), Hellström et al (2010b), Ljung et al (2012), Jourak et al (2014) and Fabricius et al (2016) and Pérez-Ràfols et al 2016). Also, experiments have been performed on simplified geometries like parallel tubes (Khayamyan et al 2014;Khayamyan and Lundström 2015).…”

Measurements of Transitional and Turbulent Flow in a Randomly Packed Bed of Spheres with Particle Image Velocimetry

“…A comprehensive investigation of the flow at the pore level may result in more sophisticated models with heterogeneous distributions of variables instead of the common assumption of homogeneous profiles (Andrigo et al 1999;Eigenberger 1992). To reveal the detailed flow within porous media, the pore level flow has been modelled theoretically and numerically (Geller and Hunt 1993;Hill et al 2001;Powers et al 1994;Payatakes 1982;Hellström et al 2010;Jourak et al 2013Jourak et al , 2014Fabricius et al 2016;Burström et al 2016). There is also some experimental work on the detailed scale.…”