Effects of inertia on the steady shear rheology of concentrated emulsions: sign reversal of normal stress differences*

The shear rheology of moderately concentrated emulsions (5-27{\%} volume fraction) in the presence of inertia is numerically investigated. Typically, an emulsion of viscous drops experiences positive first normal stress difference (N$_{1})$ and negative second normal stress difference (N$_{2})$, as has also been predicted by perturbative analysis (Choi-Schowalter model) and numerical simulation. However, recently using single drop results we have shown [Li and Sarkar, 2005, J. Rheo, 49, 1377] that introduction of inertia reverses the signs of the normal stress difference in the dilute limit. Here, we numerically investigate the effects of interactions between drops in a concentrated system. The simulation is validated against the dilute results as well as analytical relations. It also shows the reversal of signs for N$_{1}$ and N$_{2}$ for small Capillary numbers above a critical Reynolds number. The physics is explained by the inertia-induced orientation of the individual drops in shear. Increasing volume fraction increases the critical Reynolds number at which N$_{1}$ and N$_{2 }$change sign. The breakdown of linearity with volume fraction with increasing concentration is also analyzed.