Homogenization of two phase emulsions with surface tension e ects

We consider an emulsion of two Stokes uids, one of which is periodically distributed in the form of small spherical bubbles. The e ects of surface tension on the bubble boundaries are modelled mathematically, as in the work of G. I. Taylor, by a jump only in the normal component of the traction. For a given volume fraction of bubbles we consider the two-scale convergence, and in the ne phase limit we nd that the bulk ow is described by an anisotropic Stokes uid. The e ective viscosity tensor is consistent with the bulk stress formula obtained by Batchelor [2]. We nd a new dual variational principle describing the e ective viscosity and obtain bounds on the energy dissipation rate for periodic anisotropic emulsions.