Mixtures of immiscible liquids display a wide spectrum of behaviors, and thereby offer a means of achieving tunable material properties. Often they consist of small fraction of a specialized additive in a less expensive bulk liquid. The liquids phase separate into an emulsion containing discrete droplets of various sizes dispersed in a continuous phase. In industrial processing, the flow continually deforms the suspended drops leading to their coalescence and breakup. The evolving microstructure also results in stresses modifying the flow and the finished product. We investigate the dynamics of microstructure and its effects on the overall response (rheology) of the emulsion through direct numerical simulation and analytical techniques at finite Reynolds number. Tol date, research on drop deformation and rheology has mostly been restricted to inertia-less flows and small deformation. We use Front-tracking method to compute deformation of arbitrary magnitude at finite inertia. Stresses are computed from the computed microstructure. The relation between excess stress and imposed strain rate are investigated varying interfacial tension, inertia and frequency. For steady shear, shear thickening and change of sign of normal stress differences are observed with increased inertia. For oscillating extensional flows, the stress-strain relation is a function of the phase between the drop deformation and the imposed flow. At low Reynolds number, the simulation recovers the linear oscillatory rheology (loss and storage moduli) of Oldroyd and Bousmina. At low surface tension, stress is predominantly elastic, while at high surface tension it is viscous. Increased drop inertia leads to resonance and complex phase in deformation. The resulting excess interfacial stress displays a non-monotonic variation with frequency and obtains a negative elastic modulus at low frequency.

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