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.

中文翻译：

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纤维混合物水力渗透率的混合规则

我们通过方形横截面纤维的有序阵列对斯托克斯流进行数值模拟。单尺寸光纤阵列的渗透率结果与 Jackson 和 James 众所周知的相关性非常吻合，前提是使用相同横截面积的等效圆柱形光纤的有效半径。然后，我们考虑了不同尺寸纤维的混合物阵列，并测试了几种混合规则，用于估计复合材料的渗透性。最著名的两个，即未加权和体积平均电阻率规则对不太大的尺寸比给出了很好的预测，但它们分别系统地高估和低估了数值结果。然后，似乎几乎不言而喻，以上两种混合规则的组合将提供更好的预测。事实上，几何平均值在较大的尺寸比和相对体积分数范围内与数值结果非常吻合。