The slow viscous flow around doubly-periodic arrays of infinite slender cylinders

The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media, this solution is important for many real-world systems. We asymptotically determine the flow around a doubly-periodic array of infinite slender cylinders, by placing doubly-periodic two-dimensional singularity solutions within the cylinder and expanding the no-slip condition on the cylinder's surface in powers of the cylinder radius. The asymptotic solution provides a closed-form estimate for the flow and forces as a function of the radius and the dimensions of the cell. The force is compared to results from lattice-Boltzmann simulations of low-Reynolds-number flows in the same geometry, and the accuracy of the no-slip condition on the surface of the cylinder, predicted by the asymptotic theory, is checked. Finally, the behaviour of the flow, flux, force and effective permeability of the cell is investigated as a function of the geometric parameters. The structure of the asymptotic permeability is consistent with other models for the flow parallel to an array of rods. These models could be used to help understand the flows within porous systems composed of fibres and systems involving periodic arrays such as deterministic lateral displacement.