The effective shear viscosity of a regular array of suspended spheres

A general expression is derived for the effective shear viscosity of a simple regular array of suspended spheres for arbitrary values of the filling fraction. The resulting direction dependent viscosities are evaluated explicitly for the s.c., f.c.c. and b.c.c. lattices. We find that these viscosities diverge at a value of the filling fraction below the closest packing density. While these results are not directly applicable to random distributions of suspended spheres there is qualitative agreement with experimental data for these systems. A detailed comparison is made. We find that, while the qualitative features are the same in all cases, the precise position of the divergence is rather dependent on the details of the distribution of the spheres. The same is found to be the case for the value of the Huggins coefficient.