(414e) A New Conformation Tensor Based Macroscopic Model for Emulsions with Particle Inertia

Dispersion are common in numerous industrial processes justifying the large amount of research dedicated to their study. At the macroscopic level of description, a conformation tensor, C, can be advantageously used to represent their microstructure. 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, developed when particle inertia is neglected, have been previously postulated by various authors e.g. Maffetone Minale (1998) and Wetzel and Tucker (2001).

This work makes a new advance in emulsion modeling through the development of a new, thermodynamically consistent, conformation tensor-based model that is able to describe unique signature rheological features of emulsion droplets seen only in the presence of particle (micro) inertia. This is motivated by recent microscopic studies of the rheological behavior of dilute emulsions at finite Reynolds number. Li and Sarkar (2005) have carried out simulations of single droplets in simple shear flows at finite particle Reynolds number. Unlike zero particle Reynolds flows where the droplets are observed to orient in the flow direction, in the presence of micro-inertia, characterized by particle Reynolds numbers of O(1), the droplets orient increasingly in the velocity gradient direction. Furthermore, negative first normal stress differences and positive second normal stress differences are seen. The emergence of negative first normal stress differences is also characterized by orientation of the major axis of the droplets at angles, θ, greater than 45⁰ above the flow direction. The findings of Li and Sarkar (2005) have been further validated by recent independent asymptotic microscopic theory by Raja et al. (2010). Raja et al. (2010) 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.

Our work builds upon these recent microscopic developments to develop a new macroscopic conformation tensor-based model that incorporates micro-inertia effects. It achieves that goal through the use of the non-equilibrium thermodynamics bracket framework [Beris & Edwards (1994)]. They key outcome of the developments here is the revelation of a new dissipative term arising from the presence of micro-inertia leading to a new co-deformational time derivative. The new time derivative that results includes an additional parameter, ζ, that is related to the Ohnesorge number. This parameter is beyond the standard non-affine parameter, ξ , that arises in the Gordon-Schowalter derivative used in zero Reynolds number emulsion flow models. For specific values of the new parameter, ζ, the macroscopic model now allows us to predict negative first normal stresses. Furthermore, the droplet morphology associated with the negative first normal stresses is oriented increasingly in the velocity gradient direction with θ>45^o, corresponding to behavior seen in simulation studies [Li and Sarkar (2005)]. The key development of this work is the use of microscopic theory [Raja et al. (2010)] to guide the development of a macroscopic emulsion model that allows for the effects of micro-inertia to be incorporated into macroscopic, continuum level, models.

 

References

1. 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.

2. 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.

3. 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.

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

5. 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.