Longitudinal Permeability of Spatially Periodic Rectangular Arrays of Circular Cylinders I. A Single Cylinder in the Unit Cell

Abstract

We study the longitudinal permeability of a spatially periodic rectangular array of circular cylinders, when a Newtonian fluid is flowing at low Reynolds number along the cylinders. The longitudinal component of the velocity obeys a Poisson equation which is transformed into a functional equation. This equation can be solved by the method of successive approximations. The major advantage of this technique is that the permeability of the array can be expressed analytically in terms of the radius of the cylinders and of the aspect ratio of the unit cell.

Basic sums and their random dynamic changes in description of microstructure of 2D composites

“…It is worth noting that (2.11) and (2.12) are different expansions of the effective conductivity in which only the first terms coincide [7,25,36] (see also formula e 2 ¼ p in (3.2)):…”

“…Every A j from (2.12) is a polynomial in q of the degree j with coefficients consisting of the finite linear combinations of e m [7,8,25]. The coefficients R j from (2.11) have more complicated structure; R 2 has the form (2.15), (2.16); the next R j can be computed by recurrent formulae [7,25,36].…”

“…In this paper, we develop the results [23,7,8,26,25,10,11] and propose the following choice of the geometric parameters:…”

A Spatially Non-Local Model for Flow in Porous Media

“…This method was pioneered by Rayleigh (1892) who calculated macroscopic properties of a continuous medium equivalent to a regular arrangement of disks. In recent years, many useful extensions, generalizations, and alternative calculations have been made (Mityushev and Adler 2002). This approach leads naturally to the concept of percolation which describes the flow of a fluid through a network of bonds, that can be thought of as pores, and links.…”

A Functional Analytic Approach for a Singularly Perturbed Dirichlet Problem for the Laplace Operator in a Periodically Perforated Domain

“…We also observe that boundary value problems in domains with periodic inclusions can be analysed with the method of functional equations (at least for the two dimensional case). Here we mention, e.g., the work of Mityushev and Adler [13], where a doubly periodic Dirichlet problem for the Poisson equation is studied in order to compute the longitudinal permeability of spatially periodic rectangular arrays of circular cylinders. We also mention Rogosin, Dubatovskaya, and Pesetskaya [14], where Eisenstein functions are used to construct the solution of a mixed boundary value problem for a doubly periodic multiply connected domain, in order to study effective properties of a doubly periodic 2D composite material.…”