  We study the rheology of a two-fluid emulsion in semi-concentrated
conditions; the solute is Newtonian while the solvent an inelastic power law
fluid. The problem at hand is tackled by means of direct numerical simulations
using the volume of fluid method. The analysis is performed for different
volume fractions and viscosity ratios under the assumption of negligible
inertia and zero buoyancy force. Several carrier fluids are considered
encompassing both the shear-thinning and thickening behaviors. We show that the
effective viscosity of the system increases for shear-thickening fluids and
decreases for the shear-thinning ones for all the viscosity ratios considered.
The changes in the emulsion viscosity are mainly due to modifications of the
coalescence in the system obtained by changing the carrier fluid property:
indeed, local large and low shear rates are found in the regions between two
interacting droplets for shear-thickening and thinning fluids, respectively,
resulting in increased and reduced local viscosity which ultimately affects the
drainage time of the system. This process is independent of the nominal
viscosity ratio of the two fluids and we show that it can not be understood by
considering only the mean shear rate and viscosity of the two fluids across the
domain, but the full spectrum of shear rate must be taken into account.
