The results show an interesting common pattern in all three examples. The Hashin-Shtrikman upper bound is always smaller, and therefore a better/tighter bound, than the upper FF bound. But the situation is more complicated for the lower bounds. Near the point where all the bounds cross, the lower Hashin-Shtrikman bounds are just slightly better for higher values of , but significantly better for the lower values. On the other hand, far from this convergence point the lower FF bound is clearly superior to Hashin-Shtrikman, both at quite high and quite low values of . In fact this is not surprising since it is in these asymptotic regimes that the FF bounds tend to become exact estimates. So a reasonable conclusion I reach from these observations is that the combination of the two Hashin-Shtrikman bounds and the lower FF bound provides quite accurate estimates of overall conductivity for the entire range of pore-fluid conductivities.
Future work along these lines will be directed towards improving the estimates obtained from the analytical method by making more direct use of various known constraints on the resonance density and its integral moments.