Abstract
We present the constitutive equation for the volume-averaged hydrodynamic stress for a dilute emulsion in a linear ambient flow, when there is slip at the liquid-liquid interface between the Newtonian drop and the suspending fluids. Slip is modeled using a simple Navier slip boundary condition. We provide analytical solutions in the limit of small capillary numbers for the shape deformation, viscosity, and normal stresses. Slip moderates these quantities, with changes from the no-slip case being more pronounced for large viscosity ratios of the drop relative to the suspending fluid. It has been suggested in the past that slip can explain the anomalously low viscosities of certain polymeric blends. Our analysis indicates that slip can only partially account for these deviations, and that other mechanisms should be explored to explain the residual discrepancy.
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