Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by twodimensional slices of a threedimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient
Resistance scaling on 4 N carpets
Abstract The 4 N {4N} carpets are a class of infinitely ramified selfsimilar fractals with a large group of symmetries. For a 4 N {4N} carpet F , let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact prefractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ( u , v ) = ∫ F N ∇ u ⋅ ∇ v d x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ( u n , u n ) ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n . Such estimates have implications for the existence and scaling properties of Brownian motion on F .
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 Award ID(s):
 1659643
 NSFPAR ID:
 10413764
 Date Published:
 Journal Name:
 Forum Mathematicum
 Volume:
 34
 Issue:
 1
 ISSN:
 09337741
 Page Range / eLocation ID:
 61 to 75
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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