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.
The present study develops an extension of the approach pioneered by Farris [Trans. Soc. Rheol. 12, 281-301 (1968)] to model the viscosity in polydisperse suspensions. Each smaller particle size class is assumed to contribute to the suspension viscosity through a weighting function in two ways: first, indirectly, by altering the background viscosity, and second, directly, by increasing the contribution of the larger particles to the suspension viscosity. The weighting functions are developed in a consistent fashion as a power law with the exponent, h, a function of the relative volume fraction ratio and the base, g, a function of the solid particle size ratio. The model is fit to available theoretical and experimental results for the viscosity of several binary suspensions and shows good to excellent agreement depending on the functions g and h chosen. Once parameterized using binary suspension viscosity data, the predictive capability to model the viscosity of arbitrary continuous size distributions is realized by representing such distributions with equivalent ternary approximations selected to match the first six moments of the actual size distribution. Model predictions of the viscosity of polydisperse suspensions are presented and compared against experimental data. V
Recently, Mwasame et al. [“On the macroscopic modeling of dilute emulsions under flow,” J. Fluid Mech. 831, 433 (2017)] developed a macroscopic model for the dynamics and rheology of a dilute emulsion with droplet morphology in the limit of negligible particle inertia using the bracket formulation of non-equilibrium thermodynamics of Beris and Edwards [Thermodynamics of Flowing Systems: With Internal Microstructure (Oxford University Press on Demand, 1994)]. Here, we improve upon that work to also account for particle inertia effects. This advance is facilitated by using the bracket formalism in its inertial form that allows for the natural incorporation of particle inertia effects into macroscopic level constitutive equations, while preserving consistency to the previous inertialess approximation in the limit of zero inertia. The parameters in the resultant Particle Inertia Thermodynamically Consistent Ellipsoidal Emulsion (PITCEE) model are selected by utilizing literature-available mesoscopic theory for the rheology at small capillary and particle Reynolds numbers. At steady state, the lowest level particle inertia effects can be described by including an additional non-affine inertial term into the evolution equation for the conformation tensor, thereby generalizing the Gordon-Schowalter time derivative. This additional term couples the conformation and vorticity tensors and is a function of the Ohnesorge number. The rheological and microstructural predictions arising from the PITCEE model are compared against steady-shear simulation results from the literature. They show a change in the signs of the normal stress differences that is accompanied by a change in the orientation of the major axis of the emulsion droplet toward the velocity gradient direction with increasing Reynolds number, capturing the two main signatures of particle inertia reported in simulations.
A population balance model appropriate for the shear flow of thixotropic colloidal suspensions with yield stress is derived and tested against experimental data on a model system available in the literature. Modifications are made to account for dynamic arrest at the onset of the yield stress, in addition to enforcing a minimum particle size below which breakage is not feasible. The resulting constitutive model also incorporates a structural based relaxation time, unlike existing phenomenological models that use the inverse of the material shear rate as the relaxation time. The model provides a reasonable representation of experimental data for a model thixotropic suspension in the literature, capturing the important thixotropic timescales. When compared to prevalent structure kinetics models, the coarse‐grained population balance equation is shown to be distinct, emphasizing the novelty of utilizing population balances as a basis for thixotropic suspension modeling. © 2016 American Institute of Chemical Engineers AIChE J, 63: 517–531, 2017
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