On mixing rules for the hydraulic permeability of fiber mixtures

We perform numerical simulations of Stokes flow through ordered arrays of fibers of square cross-section. The results for the permeability of single-size-fiber arrays are in very good agreement with the well-known correlation of Jackson and James, provided that an effective radius of equivalent cylindrical fibers is used with the same cross-sectional area. Then, we consider arrays of mixtures of different-sized fibers and test several mixing rules, proposed for estimating the composite permeability. The two most well-known, namely, the unweighted- and the volume-averaged resistivity rules give good predictions for not too large size-ratios, yet they systematically overestimate and underestimate the numerical results, respectively. Then, it appears almost self-evident that a combination of the above two mixing rules would provide a better prediction. Indeed, the geometric mean gives excellent agreement with the numerical results over an extended range of size ratios and relative volume fractions.