Effective viscosity of an ordered suspension of small drops

We employ a method of singularity distribution to determine the effective viscosity of a suspension of small neutrally buoyant drops of one fluid in another viscous fluid. We assume that the interfacial tension is relatively large so that the drops may be considered nearly spherical and consider an instantaneous configuration in which the centers of drops coincide with the lattice points of a periodic cubic array. Under these conditions, the effective viscosity tensor is characterized by only two scalars, α and β, which we determine for the complete range of c and K where c is the volume fraction of the drops and K is the ratio of viscosities. Our numerical results, given for the simple, body-centered and face-centered cubic arrays, are in excellent agreement with those obtained for the rigid particles (K=∞) by Nunan and Keller except for β for the face-centered cubic array where our results appear to be more accurate. The results are also in agreement with the asymptotic expressions for the dilute arrays (c≪1) for all K. The accuracy of the numerical results is also adequate in most cases to yield the formulas for concentrated arrays of very viscous drops (K≫1).