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471988 A New Conformation Tensor Based Macroscopic Model for Emulsions at Finite Reynolds Numbers

**C**, in the Maffetone and Minale (1998) model. This model describes the dynamics of droplets in dilute emulsions through a conformation tensor of constant determinant (det(

**C**)=1) and employs the Gordon-Schowalter time derivative. Recently, there has been renewed interest in describing the rheological behavior of emulsions at finite Reynolds number. Li and Sarkar (2005) have carried out simulations of single droplets in shear flows at finite Reynolds number. Unlike zero Reynolds flows where the droplets are observed to orient in the flow direction, in the presence of inertia, the droplets orient increasingly in the velocity gradient direction. Furthermore, negative first normal stress differences and positive second normal stress differences are seen. The observation of negative first normal stress differences is also characterized by orientation of the major axis of the droplets at angles, θ, greater than 45

^{0}above the flow direction. The findings of Li and Sarkar (2005) have also been demonstrated using analytic calculations by Raja et al. (2010) who associate the emergence of negative first normal stresses to a critical Ohnesorge number. This dimensionless group describes the competing effect of viscous forces on one hand and inertia and interfacial forces on the other. The effect of this dimensionless group has not been described by a macroscopic model to the best our knowledge.

In this work, we describe the systematic development of an extended Maffetone-Minale model using the General Equation for Reversible Irreversible Coupling (GENERIC) revealing a new dissipative term in the time derivative leading to a new codeformational time derivative incorporating two parameters. In addition, the extended Maffetone Minale model also includes a stress tensor, defined self consistently from the GENERIC framework. The new time derivative includes an additional parameter beyond the non-affine parameter,ξ , typically associated with the Johnson-Segalman model for polymers. For specific values of the new parameter, ζ, the macroscopic model allows us to predict negative first normal stresses. Futhermore, the droplet morphology associated with the negative first normal stresses is oriented increasingly in the velocity gradient direction with θ>45^{o}. Therefore, we are now able to parameterize the macroscopic model in terms of the Ohnesorge number allowing us to describe the effects of inertia on the droplet dynamics in the extended Maffetone Minale model and describe the results of Li and Sarkar (2005). Finally, by examining the asymptotic solutions of the extended Maffetone-Minale model, and comparing against analytic solutions by Raja et al. (2010) we are able to elucidate the role of the Ohnesorge number in controlling the onset of negative normal stresses associated with inertia.

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**References**

- 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. - Ait-Kadi, A., Ramazani, A., Grmela, M. and Zhou, C., 1999. “Volume preserving” rheological models for polymer melts and solutions using the GENERIC formalism.
*Journal of Rheology (1978-present)*,*43*(1), pp.51-72. - 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. - 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. - Beris, A.N. and Edwards, B.J., 1994. Thermodynamics of flowing systems: with internal microstructure.
- Grmela, M. and Öttinger, H.C., 1997. Dynamics and thermodynamics of complex fluids. I. Development of a general formalism.
*Physical Review E*,*56*(6), p.6620.

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