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.

Figure 1

(Color online) Model of a porous media with obstacles of different sizes based on an AP, together with typical flow pattern through the packing for $s=0.5$ and $g=3$. The colors ranging from blue (dark) to red (light) correspond to low and high velocity magnitudes, respectively. Preferential channels depend more strongly on $g$ than on $s$.

Figure 2

(Color online) Dependence of porosity $\varphi$ with respect to the reduction factor $s$ for several values of $g$. The differences in the values of $\varphi$ for fixed values of $s$ decreases as $g$ increases, as the radii of latter generations circles become smaller.

Figure 3

(Color online) The hydraulic resistance $G$ as a function of Re for different values of $s$ and $g=5$. Symbols indicate the values at the left-hand side of Eq. (6), while lines indicate the values obtained by quadratic expansion. To plot all curves in the same axis, $G$ has been divided by $\alpha$. Better accordance is achieved when $s$ increases and channels become narrower.

Figure 4

(Color online) The main panel shows the dependence of $k/k_0$ with respect to $\varphi$ for successive packing generations $g=1$ (circles), 2 (triangles up), 3 (triangles down), 4 (diamonds), and 5 (triangles left). In the inset, dependence of $k{S}_0^2/k_0$ with respect to ${\left(\varphi -{\varphi }_0\right)}^3/{\left(1-\varphi +{\varphi }_0\right)}^2$ for $g=3$,4, and 5, where ${\varphi }_0=\varphi \left(s=1,g\right)$. The straight line with slope 1.0 clearly shows that Eq. (7) holds with very high degree of accuracy. Symbols for $g=3$,4, and 5 are, respectively, square, circle, and diamond. In both plots, $\varphi =\varphi \left(s,g=\text{const}\right)$.