(419c) Applications of Conformation Tensor-Based Macroscopic Models to Particulate and Multiphase Systems

Particulate and multiphase flows are common in numerous industrial processes justifying the large amount of research dedicated to their study. Still most of their theoretical treatment relies on microscopic models that, albeit sophisticated enough to represent the often-encountered complex flow behavior, they have the inconvenience of requiring considerable computational resources and/or rather limited applicability to simple flow geometries. At the other extreme lie phenomenological continuum models that albeit easy to work with they do involve several adjustable parameters requiring many experiments to fit them and implying limited applicability to the range covered by those experiments. We propose here an intermediate approach, still at the continuum macroscopic level, but relying on structural-parameter based fluid model descriptions applied through a nonequilibrium thermodynamics-based approach. At that macroscopic level of description, a conformation tensor, C, is commonly used to represent the microstructure that characterizes the morphology of multiphase systems. For example, in dilute emulsions with droplet morphology, the dynamics of droplets can be simply represented through a contravariant conformation tensor of constant determinant, det(C)=1 [Maffetone Minale (1998)]. Such models have been previously postulated by various authors e.g. Maffetone Minale (1998) and Wetzel and Tucker (2001).

Recently, the Maffetone-Minale model has been recast and extended for arbitrary viscosity ratios (between the dispersed and the continuum phases) using the non-equilibrium thermodynamics bracket framework of Beris & Edwards (1994) [Mwasame et al., 2017]. In that work all the model parameters have been obtained through comparison against available asymptotic analysis results from the literature. Moreover, a further advantage of the bracket formalism, is that it most recently naturally led to a generalization of the dilute emulsion model under conditions under which particle inertia effects are important [Mwasame et al., 2018]. Again, all the model parameters have been obtained based on comparisons against previous asymptotic microscopic theory results [Raja et al. (2010)]. In this way, for the first time, one is able to macroscopically predict unique signature rheological features of emulsions seen only in the presence of particle (micro) inertia, such as negative first normal stress differences and positive second normal stress differences, as for example revealed by the microscopic simulations of Li and Sarkar (2005).

In the present work we offer two additional extensions of the conformation tensor-based multiphase modeling. In the first one, we describe how a two conformation tensors-based model can describe the rheology of emulsions in the presence of Ostwald ripening. The resultant model allows for effective mass transfer effects to be systematically incorporated into the emulsion multiphase model. We are thus able the describe a population of droplets and their evolution in time along with the evolution of the rheology. In the second application, a model for the rheology of concentrated non-Brownian suspensions is presented following previous work of Phan-Thien (1995). Similar to Phan-Thien (1995), the microstructure in a concentrated suspension is represented through a conformation tensor that represents now the second moment of the unit vector along the center to center line connecting two generic spheres. However, unlike that work, the thermodynamically-based model formulated here following an extension of the bracket approach of [Beris and Edwards, 1994] is consistent with all viscometric functions in non-Brownian suspensions. In shear flows, the model predicts negative first and second normal stress differences that have been observed both experimentally and in simulation studies. These are accompanied by microstructure orientation and localization along the compressional axis of the shear flow field. Those two new applications of conformation-based theories to particulate/multiphase flows show the potential benefit drawn from the systematic, nonequilibrium thermodynamics-based approach followed here that may therefore prove useful to other future applications as well.

Acknowledgment

This material is based upon work supported by the National Science Foundation under Grant No. CBET 312146. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

References

Beris, A.N. and Edwards, B.J., 1994. Thermodynamics of flowing systems: with internal microstructure.

Li, X. and Sarkar, K., 2005. Effects of inertia on the rheology of a dilute emulsion of drops in shear. Journal of Rheology, 49(6), pp.1377-1394.

Maffettone, P.L. and Minale, M., 1998. Equation of change for ellipsoidal drops in viscous flow. Journal of Non-Newtonian Fluid Mechanics, 78(2), pp.227-241.

Mwasame, P.M., Wagner, N.J., Beris, A.N. 2017. On the macroscopic modeling of dilute emulsions under flow. Journal of Fluid Mechanics, 831, 433-473.

Mwasame, P.M., Wagner N.J., Beris, A.N., 2018. On the macroscopic modeling of dilute emulsions under flow in the presence of particle inertia. Physics of Fluids, 30: 030704.

Phan‐Thien, N., 1995. Constitutive equation for concentrated suspensions in Newtonian liquids. Journal of Rheology, 39(4), pp.679-695.

Raja, R.V., Subramanian, G. and Koch, D.L., 2010. Inertial effects on the rheology of a dilute emulsion. Journal of Fluid Mechanics, 646, pp.255-296.

Wetzel , E. D. & Tucker R III, C. L. 2001 Droplet deformation in dispersions with unequal viscosities and zero interfacial tension. J. Fluid Mech. 426, 199-228.