Fluid flow through Apollonian packings

This work considers the flow of a Newtonian fluid in a two-dimensional channel filled with an array of obstacles of distinct sizes that models an inhomogeneous medium. Obstacle sizes and positions are defined by the geometry of an Apollonian packing (AP). The radii of the circles are uniformly reduced by a factor s < 1 for assemblies corresponding to the five first AP generations. The region of validity of Darcy's law as a function of the channel Reynolds number is investigated for different values of s and the dependency of the flow pattern and permeability with respect to porosity is established. Our results show that the semiempirical Kozeny-Carman scaling relation is satisfied provided the effects of the apparent porosity and s-dependent formation factor are properly considered.