A three-dimensional computer simulation of a concentrated emulsion in shear flow has been developed for low-Reynolds-number finite-capillary-number conditions. Numerical results have been obtained using an efficient boundary integral formulation with periodic boundary conditions and up to twelve drops in each periodically replicated unit cell. Calculations have been performed over a range of capillary numbers where drop deformation is significant up to the value where drop breakup is imminent. Results have been obtained for dispersed-phase volume fractions up to 30% and dispersed- to continuous-phase viscosity ratios in the range of 0 to 5. The results reveal a complex rheology with pronounced shear thinning and large normal stresses that is associated with an anisotropic microstructure that results from the alignment of deformed drops in the flow direction. The viscosity of an emulsion is only a moderately increasing function of the dispersed-phase volume fraction, in contrast to suspensions of rigid particles or undeformed drops. Unlike rigid particles, deformable drops do not form large clusters.

An algorithm is presented for the adaptive restructuring of meshes on evolving surfaces. The resolution of the relevant local length scale is maintained everywhere with prescribed accuracy through the minimization of an appropriate mesh energy function by a sequence of local restructuring operations. The resulting discretization depends on the instantaneous configuration of the surface but is insensitive to the deformation history. Application of the adaptive discretization algorithm is illustrated with three-dimensional boundary-integral simulations of deformable drops in Stokes flow. The results show that the algorithm can accurately resolve detailed features of deformed fluid interfaces, including slender filaments associated with drop breakup and dimpled regions associated with drop coalescence. Our algorithm should be useful in a variety of fields, including computational fluid dynamics, image processing, geographical information systems, and biomedical engineering problems.

A boundary integral formulation is used to investigate the interaction between a pair of deformable drops in a simple shear flow. The interactions do not promote appreciably the breakup of the drops. For certain ratios of the viscosities of the drops and the suspending fluid, the lubrication gap that separates the two drops can diminish rapidly in the extensional quadrant of the flow. Slight deformation endows the drops with an apparent short-range repulsive interaction: drop coalescence requires van der Waals attraction which was not included in this study. From the trajectories of different collisions, the self-diffusion coefficients that describe the cross-flow migration of the non-Brownian drops in a dilute sheared emulsion are obtained. The self-diffusivities are very anisotropic, depend strongly on the viscosity ratio, and depend modestly on the shear rate.

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