Homogenization of Two Phase Emulsions with Surface Tension Eeects

We consider an emulsion of two Stokes uids, one of which is periodically distributed in the form of small spherical bubbles. The eeects 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 eeective viscosity tensor is consistent with the bulk stress formula obtained by Batchelor 2]. We nd a new dual variational principle describing the eeective viscosity and obtain bounds on the energy dis-sipation rate for periodic anisotropic emulsions.