  The shear induced collective diffusivity in an emulsion of viscous drops,
specifically as a function of viscosity ratio, was numerically computed. An
initially randomly packed layer of viscous drops spreading due to drop-drop
interactions in an imposed shear has been simulated. The shear induced
collective diffusivity coefficient was computed using a self-similar solution
of the drop concentration profile. We also obtained the collective diffusivity
computing the dynamic structure factor from the simulated drop positions--an
analysis typically applied only to homogeneous systems. The two quantities
computed using different methods are in agreement including their predictions
of nonmonotonic variations with increasing capillary number and viscosity
ratio. The computed values were also found to match with past measurements. The
gradient diffusivity coefficient computed here was expectedly one order of
magnitude larger than the self-diffusivity coefficient for a dilute emulsion
previously computed using pair-wise simulation of viscous drops. Although
self-diffusivity computed previously showed nonmonotonic variation with
capillary number, its variation with viscosity ratio is in contrast to
nonmonotonic variation of gradient diffusivity found here. The difference in
variation could arise from drops not reaching equilibrium deformation between
interactions--an effect absent in the pair-wise simulation used for computation
of self-diffusivity--or from an intrinsic difference in physics underlying the
two diffusivities. We offer a qualitative explanation of the nonmonotonic
variation by relating it to average nonmonotonic drop deformation. We also
provide empirical correlations of the collective diffusivity as a function of
viscosity ratio and capillary number.
