Ellipsoidal model for droplet deformation in emulsions

An ellipsoidal model for droplet deformation in mixtures of Newtonian fluids is proposed. The model makes a bridge between the phenomenological description of the ellipsoidal deformation of the droplet [Eshelby (1957); Maffetonne and Minale (1998); Jackson and Tucker (JT model) (2003); Wetzel and Tucker (WT model) (2001)] and the interfacial velocity calculation between two Newtonian liquids. The bridging was obtained by the use of the general boundary integral formalism. The velocity at the interface was decomposed into a flow dominated term and an interfacial term. The flow term is the same as in JT and WT models and arises from Eshelby’s theory, while the interfacial term was calculated by assuming a linear, uniform velocity gradient tensor over the entire surface of the droplet. The model for droplet deformation is applicable to mixtures of two Newtonian liquids with arbitrary viscosity ratio and nonzero interfacial tension. The predictions of the present model in terms of shape evolution of the droplet agree well with many experiments and numerical simulations including transient deformation for small and large capillary numbers, transient shear widening for small viscosity ratio and large capillary numbers, and steady shear deformation. The rheology of dilute emulsions based on the morphological predictions by the present model was calculated according to Batchelor’s formalism (1970). The predicted rheological material functions agree reasonably well with the experimental data. The limitations of the present model are also discussed.