(387e) Extensional Flows and General Rheology of Concentrated Emulsions of Deformable Drops

An outstanding problem for any non-Newtonian liquid is in formulation of a constitutive equation for the stress tensor valid in a broad range of kinematic conditions, not necessarily those encountered in rheological experiments (typically limited to shear flows). Knowing the constitutive equation would allow the solution of many problems of non-Newtonian hydrodynamics of technological interest. As a step in this direction for a particular microstructure, we develop a constitutive model for a highly-concentrated emulsion of deformable drops based on rigorous large-scale multidrop simulations. The idea is to combine the results from two base flows –steady simple shear and planar extension (PE)  at arbitrary flow intensities, in order to construct a generalized Oldroyd equation with variable coefficients, which exactly fits both viscometric  (for simple shear) and extensiometric (for PE) functions at arbitrary deformation rates. The Oldroyd coefficients are chosen to be functions of the instantaneous energy dissipation rate, which gives, at an arbitrary flow intensity, five nonlinear equations for five unknown coefficients; this system may be solved by Newton – Raphson iterations with three viscometric and two extensiometric functions taken from numerical simulations.

 For the first base flow -steady shear, a multipole-accelerated boundary-integral algorithm for many deformable drops with periodic boundary conditions (BC) has already been developed [1], but the present work goes much farther by considering larger systems (up to N=400 drops in a periodic cell), much larger strains (to  ~1000, when necessary) for time-averaging of the shear viscosity and normal stress differences, high surface resolutions (2000-4000 triangular boundary elements on each  drop surface), broad drop-to-medium viscosity ratios λ (from 0.25 to 10) at drop volume fractions c=0.45-0.55. Large scale simulations with extensive averaging are particularly important at c =0.55 and small-to-moderate viscosity ratios λ=0.25 and 1, owing to the dynamical “phase transition” to a partially-ordered state; this transition  could not be studied in Ref. [1] in any detail. The main difficulty of phase-transitional simulations, especially at N>>1, is that the system first remains in a metastable state for long simulation times, with the viscosity significantly larger than the ultimate equilibrium value. When the capillary number Ca is reduced from large values, a familiar increase in the steady-state emulsion shear viscosity is first observed due to shear-thinning. Further decrease in Ca reduces the emulsion viscosity due to phase transition, until the viscosity rises again. Complex behavior of the average normal stress differences is also observed in the transition range. Increasing the viscosity ratio λ from 1 to 3 increases the average drop deformation, makes the system less jammed, and strongly shifts the phase transition range to much smaller Ca.

To obtain substantially new extensiometric functions for flowing concentrated emulsions (the extensional viscosity and stress cross-difference) in steady Planar Extension  v= (Gx₁, -Gx₂, 0), the multidrop algorithm of Zinchenko and Davis [1] has been adapted to new periodic BC, using the dynamics of replicable periodic lattices for PE from Kraynik & Reinelt [2]. Unlike for shear flow, we found no phase transition whatsoever for PE, at least, for drop volume fractions up to 0.55, regardless of Ca and λ. As the drop deformation is reduced, the extensional emulsion viscosity becomes a strong function of Ca, greatly exceeding the shear viscosity. In contrast, for large deformations, the extensional viscosity (unlike shear viscosity) of concentrated emulsions c=0.45-0.55 is almost independent of Ca. Near drop breakup in PE, a large increase in the stress cross-difference, relative to the extensional viscosity, is observed.

The only other type of macroscopic flow with a replicable periodic lattice is a mixed flow (MF) v= (Gx₂, fGx₁, 0), where f=0 for simple shear, and f=1 gives a flow equivalent to PE.  Reproducible dynamics of lattice vectors, necessary for periodic BC in our multidrop algorithm, was obtained using Kraynik-Reinelt [2] dynamics combined with matrix transformations [3]. Precise time-averaged rheological functions for MF were obtained for c=0.45-0.55 and λ=0.25, 1 and 3 up to drop breakup. There is no phase transition in MF, at least up to c=0.55, regardless of Ca and λ. The precise results for MF are compared with predictions from the generalized Oldroyd equation with our variable coefficients. For f=0.25 (a rheologically strong flow), the constitutive model is very accurate, especially, at λ=1 and 3. For a less typical f= -0.25 (a “weak”, elliptical type of flow), the Oldroyd model is found to be less successful. In contrast, for dilute emulsions of deformable drops, the generalized Oldroyd model with variable coefficients is accurate [4] for both strong and weak MF. The principal difference between dilute and concentrated emulsions is that the latter do not obey smoothness hypotheses postulated in Rivlin-Ericksen’s phenomenological theory in the limit of slow flows, due to close drop interactions.

[1] Zinchenko A.Z. and Davis R.H. 2002 “Shear flow of highly concentrated emulsions of deformable drops by numerical simulations”. J. Fluid Mech., vol. 455, pp.21-62.

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

 [3] Hunt T.A., Bernardi S. and Todd B.D. 2010 “A new algorithm for extended nonequilibrium molecular dynamics simulations of mixed flow”. J. Chem. Phys., vol. 133, p.154116.

[4] Martin R., Zinchenko A. and Davis R. 2014 “A generalized Oldroyd's model for non-Newtonian liquids with applications to a dilute emulsion of deformable drops”. J. Rheol., vol. 58, p. 759.