(379j) A Generalized Oldroyd's Model for Dilute Emulsions of Deformable Drops

Formulating a constitutive equation for a non-Newtonian fluid valid in a broad class of kinematic conditions is an outstanding problem of great practical relevance. The focus in theoretical rheology so far has been on viscometric (shear) flows, and, to a lesser extent, on extensional flows (typically, uniaxial extension/compression). However, the information about the stress tensor in these types of flow alone cannot predict the material behavior for an arbitrary history of deformation. We propose a novel theoretical approach to constitutive modeling for non-Newtonian fluids based on a generalized Oldroyd’s equation. In its traditional form [1], this constitutive equation of differential type relates the instantaneous stress and its corotational time derivative to the rate-of–strain tensor and its derivate; all seven material parameters in this equation are assumed constant. We work instead with a generalized model, in which these parameters are allowed to be functions of the instantaneous second invariant of the rate-of-strain tensor; such a model still obeys the objectivity principle.  For a deviatoric stress, the number of material parameters reduces to five. The basic idea of the present approach is to consider two types of flow: (1) steady shear and (2) planar extension (PE) or hyperbolic flow, and then match all five theoretically-obtained rheological functions (three for shear flow , and two for PE ) to those predicted by Oldroyd’s equation. This match gives five non-linear equations for five material parameters solved by Newton iterations at every value of the second invariant. The choice of shear and PE is crucial, since these are the only two types of flow with a reproducible lattice [2] that allows for long-time simulations of concentrated systems (e.g., emulsions) with periodic boundaries.

A dilute emulsion of isolated deformable drops gives a rare opportunity to verify this approach to constitutive modeling, since the single-drop contribution to the average stress can be found precisely by boundary-integral (BI) solutions for arbitrary ambient-flow kinematics.  The generalized Oldroyd’s parameters were found by this approach for drop-to-medium viscosity ratios of 1 and 5; the solution was found to be always unique, regardless of an initial guess in the matching algorithm.  The second invariant  enters the equations through the effective capillary number Ca; the range of Ca is limited by breakup conditions for PE (Ca<Ca_cr). Unlike usual shear-thinning, the dilute emulsion is shown to be tension-thickening in PE.  

Predictions using the generalized Oldroyd model were compared with exact BI results for several types of flow kinematics. For a mixed linear flow between simple shear and PE and equal viscosities, the model predictions for all intrinsic stress components are practically indistinguishable from exact BI results for all Ca<Ca_cr . For viscosity ratio of 5, larger deviations are observed near Ca_cr, but our model is still much more accurate than small-deformation theories. The second example is the rheological response of a small drop submerged in a Stokes flow past a macroscopic sphere. Here, a fluid element trajectory resembles uniaxial compression near the stagnation point and extension in the wake, and is akin to unsteady shear flow at intermediate times. Again, the model has excellent accuracy, far exceeding the accuracy of small deformation theories. Another type of kinematics considered is finite-Reynolds-number flow in a rectangular cavity  with a moving wall. Here, the history of deformation along a fluid element trajectory is time- periodic, different from simple shear or extension. Most importantly, our approach to constitutive modeling can be extended to large-strain flows of concentrated emulsions, unlike small deformation theories limited to single drops.

Acknowledgment: This work was supported by NSF Grant No. CBET 1064132

1. Oldroyd J.B. 1958 Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Soc. Lond. A245 , 278-297.

2. A.M. Kraynik and D.A. Reinelt 1992 Extensional motions of spatially periodic lattices. Int. J. Multiphase Flow 18, 1045-1059.