Macroscopic modeling of microinertia effects in particulate flows*

A new approach to introduce micro-inertia into macroscopic models of particulate flows, based on an internal conformation tensor structural variable, is developed using the Non-Equilibrium Thermodynamics bracket formalism (Beris and Edwards, Thermodynamics of Flowing Systems, Oxford U. Press, 1994). The proposed approach is first applied to a dilute emulsion with the structural tensor parameter physically identified with the deformed ellipsoidal geometry of the dispersed phase. In addition to a new kinetic term in the Hamiltonian, micro-inertia effects also introduce a new non-affine term that couples the conformation and the vorticity tensors in the evolution equation for the conformation tensor. This derivation and resultant constitutive equation provide a new pathway to rigorously incorporate micro-inertia into general, conformation tensor-based, macroscopic models for multiphase systems. The model is consistent with previously developed in the literature asymptotic theories in the limit of small capillary, Ca, as well as small particle Reynolds, Re, numbers. These asymptotic solutions are also used to uniquely determine all the model parameters. Additional, more recent, applications to concentrated suspensions are also going to be outlined.