  We present a modification of a recently developed volume of fluid method for
multiphase problems, so that it can be used in conjunction with a fractional
step-method and fast Poisson solver, and validate it with standard benchmark
problems. We then consider emulsions of two-fluid systems and study their
rheology in a plane Couette flow in the limit of vanishing inertia. We examine
the dependency of the effective viscosity on the volume-fraction (from 10% to
30%) and the Capillary number (from 0.1 to 0.4) for the case of density and
viscosity ratio 1. We show that the effective viscosity decreases with the
deformation and the applied shear (shear-thinning) while exhibits a
non-monotonic behavior with respect to the volume fraction. We report the
appearance of a maximum in the effective viscosity curve and compare the
results with those of suspensions of rigid and deformable particles and
capsules. We show that the flow in the solvent is mostly a shear flow, while it
is mostly rotational in the suspended phase; moreover this behavior tends to
reverse as the volume fraction increases. Finally, we evaluate the
contributions to the total shear stress of the viscous stresses in the two
fluids and of the interfacial force between them.
