On the macroscopic modeling of dilute emulsions under flow in the presence of particle inertia

Abstract

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.

The relationship between viscoelasticity and elasticity

“…Only in the cases where a 2 = 1 do the viscoelastic constitutive relations exhibit a well-defined elastic limit, in the sense that their behaviours converge to that of an elastic solid in the limit of λ → ∞. The very same conclusions were reached in the context of emulsions, whose drops deform into ellipsoids—in that case the eigenvalues of A represent the square of the semi-axes of the deformed droplets [59,60]. Any other time derivative, which implies non-affine motion, does not correspond to any limit of the theory of elasticity.…”

“…In § 5 c, we supply a list of various models. Apart from the question of frame invariance, models have to be consistent with the requirements of thermodynamics [ 6 , 7 , 59 , 60 , 63 ].…”

The relationship between viscoelasticity and elasticity

“…The non-Newtonian contribution σ p originates from the stretching of the microstructure inside the fluid. Though we will refer to σ p as the "polymeric stress", having in mind dilute polymer suspensions, the concept equally applies to emulsions whose microstructure is described by droplet deformations [20,21]. The non-Newtonian contribution σ p is governed by a separate evolution equation, describing the state of the component governed by a long time scale λ, for example a polymer.…”

“…In Appendix A, we supply a list of various models. Apart from the question of frame invariance, models have to be consistent with requirements of thermodynamics [8,9,20,21,23].…”

“…The definition (21) has a natural interpretation. Since convection plays no role on the domain of material coordinates, one first projects the Eulerian tensor A back to the Lagrangian domain, using the inverse transformation…”