A Viscoelastic Model with Droplet Breakup Effects for Processing of Dense Emulsions*

In the processing of dense emulsions, complex flows strech and deform droplets to induce breakup, altering the overall distribution of droplet sizes and tuning the mechanical and rheological properties of the final product. For steady homogeneous flows, a droplet's shape and proclivity to breakup can be inferred on the basis of a Capillary number, but more generally a droplet's shape depends its whole history of deformation. This distinction can be particularly important for the complex flow geometries encountered in industrial processing of dense emulsions. To model history-dependence in droplet breakage dynamics, we have developed a new viscoelastic constitutive equation which we call the "shape tensor emulsion population balance" (STEPB) model, built upon an ellipsoidal approximation of droplet shapes. When breakup occurs, complexities of the intermediate breakup process are interpolated via a cascade of binary breakup events, with conservation of capillary stresses at each break. By incorporating population balance terms into the shape evolution equation, our model is the first to present a physical (as opposed to numerical) argument for the log-configuration tensor transform widely used in viscoelastic computational fluid dynamics calculations. In a first application of the STEPB model, we consider non-linear elasticity and droplet breakup in monodisperse dense emulsions under step shear, where we predict non-trivial strain-dependent final daughter droplet distributions. We also compare predictions from STEPB against experimental observations in steady shear flow, illuminating key weaknesses in the model due to neglected terms.