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.
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March 2018
Research Article|
February 14 2018
On the macroscopic modeling of dilute emulsions under flow in the presence of particle inertia
Special Collection:
Papers from the 8th Meeting of the Hellenic Society of Rheology
Paul M. Mwasame;
Paul M. Mwasame
Center for Molecular and Engineering Thermodynamics, Department of Chemical and Biomolecular Engineering, University of Delaware
, Newark, Delaware 19716, USA
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Norman J. Wagner
;
Norman J. Wagner
Center for Molecular and Engineering Thermodynamics, Department of Chemical and Biomolecular Engineering, University of Delaware
, Newark, Delaware 19716, USA
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Antony N. Beris
Antony N. Beris
a)
Center for Molecular and Engineering Thermodynamics, Department of Chemical and Biomolecular Engineering, University of Delaware
, Newark, Delaware 19716, USA
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a)
Author to whom correspondence should be addressed: beris@udel.edu. Tel.: 1-302-831-8018.
Physics of Fluids 30, 030704 (2018)
Article history
Received:
August 24 2017
Accepted:
October 26 2017
Citation
Paul M. Mwasame, Norman J. Wagner, Antony N. Beris; On the macroscopic modeling of dilute emulsions under flow in the presence of particle inertia. Physics of Fluids 1 March 2018; 30 (3): 030704. https://doi.org/10.1063/1.5001477
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