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 b ...
Highlights
There exist several mathematical approaches, collectively referred to as homogenization theory, for deriving Darcy’s law, as well as methods based on phase averaging (Whitaker 1986)
Each cube can be divided into a fluid part and a solid part, where the solid part has the shape of a vertical cylinder
We have considered flow in a thin porous medium with two small parameters ε and δ, related to the microstructure and the thickness of the domain
Summary
There exist several mathematical approaches, collectively referred to as homogenization theory, for deriving Darcy’s law (see e.g. Allaire 1989; Hornung 1997; Lions 1981; SanchezPalencia 1980; Tartar 1980 and the references therein), as well as methods based on phase averaging (Whitaker 1986). The VTPM regime is analogous to flow in a Hele-Shaw cell This approximation is only valid for λ 1, i.e. when the distance between the plates is much smaller than the interspatial distance between the obstacles. Their analysis is based on an approximate series solution of the Stokes equation They claim that this solution describes the transition from the Hele-Shaw potential flow limit (corresponding to VTPM) to the viscous two-dimensional limiting case (corresponding to HTPM). Their analysis does not give a distinct characterization of the PTPM regime, which is rigorously defined here. Their method is restricted to the particular geometry of circular cylinders, whereas our method can be applied to other geometries as well (see Remark 1)
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