On the macroscopic modelling of dilute emulsions under flow

Abstract

A new macroscopic model describing the rheology and microstructure of dilute emulsions with droplet morphology is developed based on an internal contravariant conformation tensor variable which is physically identified with the deformed ellipsoidal geometry of the dispersed phase. The model is consistent with existing first-order capillary number, $O(Ca)$, theory describing the microstructure as well as $O(Ca^{2})$ theory describing the emulsion-contributed extra stress. These asymptotic solutions are also used to determine all of the model parameters, making it the only macroscopic emulsion model that is consistent with all available asymptotic theories in the limit of small $Ca$. The governing equations are obtained from the Poisson and dissipation brackets, as developed for an incompressible fluid system endowed with an internal contravariant second-order tensor, subject to the imposition of the constraint of a unit determinant. First proposed by Maffettone & Minale (J. Non-Newtonian Fluid Mech., vol. 78, 1998, pp. 227–241), this constraint physically corresponds to conservation of the volume of the dispersed phase in the emulsion. The Hamiltonian of the emulsion is expressed through the surface energy of the dispersed phase, in addition to the kinetic energy, following previous work by Grmela et al. (J. Non-Newtonian Fluid Mech., vol. 212, 2014, pp. 1–12), but employing a more accurate evaluation of the surface area in terms of the internal contravariant conformation tensor. Structural predictions of the ellipsoid droplet morphology obtained with the new model are compared with classic experiments by Torza et al. (J. Colloid Interface Sci., vol. 38, 1972, pp. 395–411), showing good agreement.