We construct, through precise large-scale multi-drop dynamical simulations, a general constitutive model for highly-concentrated emulsions of deformable drops covered with an insoluble surfactant. An outstanding problem for any non-Newtonian liquid, including emulsions, is in formulation of a constitutive equation for the stress tensor valid in a broad class of kinematic conditions, not necessarily those encountered in rheological experiments; the later are typically limited to shear flows, steady or oscillatory. Knowing such an equation would allow the solution of boundary-value problems for non-Newtonian flow in complex geometries of practical interest.

Our emulsions are monodisperse, with the same equivalent (nondeformed) radius *a* and the same total amount *Q* of surfactant for each drop. A linear equation of state *σ=σ*_{0}*-RTΓ * is employed to relate the local values of the surface tension *σ* and surfactant concentration *Γ*, where *σ*_{0 }is the clean-surface value. Stokes equations apply on the microscale; still, the drops are large enough to neglect colloidal forces and Brownian fluctuations. Extending the approach [1-2] developed for clean drops, we combine the simulation results for two steady-state base flows to construct a generalized five-parameter Oldroyd equation which exactly fits the total of five rheological functions in the base flows 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 non-linear equations for five unknown coefficients solved numerically.

An extension of the multipole-accelerated boundary-integral algorithm [2] for many deformable drops in a periodic box has been made to account for Marangoni stresses and couple hydrodynamics to surfactant transport. Special transformations of the single-layer boundary integrals with variable surface tension are made to remove singularities and near-singularities in the kernels, based, in part, on the ideas from [3]. The surfactant diffusivity values are disputable, but are generally deems very small [4]; so, to avoid uncertainties, the Fokker-Planck equation for surfactant transport is solved on each drop surface in the zero-diffusion limit. The simulations include drop volume fractions up to 60% for matching (*λ*=1) and up to 55% for contrast viscosities (0.25<=*λ*<=3), where *λ* is the drop-to-medium viscosity ratio. The surfactant elasticity parameter *E=RTQ*/(4*πa*^{2}*σ*_{0}) is up to 0.2; small values are dictated by the linear equation of state.

The first base flow is Planar Extension (PE) (*Gx*_{1}, -*Gx*_{2}, 0) with _{ } two rheological functions to match (effective viscosity and stress cross-difference). Using simple shear as the second base flow (with shear viscosity and normal stress differences as three rheological functions) was found for clean drops [2] to meet severe limitations. Namely, a concentrated emulsion transitions to order in simple shear, with a kinked, non-smooth behavior of the rheological functions and severe ergodic difficulties necessitating very large times for averaging and very large system sizes. In contrast, this “phase transition” is not observed for other flows (e.g., PE), which puts the relevance of simple shear, as a building block for general rheology, into question. Instead, the second base flow herein is Planar Mixed (PM) flow (*Gx*_{2}, *Gχx*_{1}, 0) with a small parameter *χ*=0.16 making it different from simple shear; there are still three rheological functions to match in PM. Both PE and PM allow for periodic boundaries implementation using Kraynik-Reinelt’s [5] periodic lattices and their generalization [6]. Due to the absence of phase transition in PE and PM, strains to 100-200 and the system size of N=100-200 drops in a periodic box always suffice for both flows. Small-capillary number flows, though, require high surface resolutions of up to 4000 triangular boundary elements per drop. Even for small elasticities *E*, we see a strong effect of surface contamination on the rheology, especially for small viscosity ratios *λ*, high emulsion concentrations and smallest capillary numbers due to Marangoni stresses. Our converged results accurately describe the emulsion effective viscosities (in both PE and PM), even when they are an order of magnitude larger than the continuous-phase viscosity. The effective viscosity in PE is larger than in PM, but experiences less tension-thinning. The upper limit on the capillary numbers in the simulations is due to drop breakup/cusping.

From the database rheological functions, the variable Oldroyd parameters are computed, and the resulting generalized Oldroyd model is checked against boundary-integral simulations for two additional flows. For a PM with *χ*=0.5, very good agreement is seen in a broad range of capillary numbers. For a time-dependent PE, with strong harmonic oscillations in the value of the deformation rate, the database results from steady-state PE (used in a quasi steady-state manner) give a poor prediction. Remarkably, though, when these database results are combined with database steady-state PM results through the generalized Oldroyd equation, the prediction for the time-dependent PE is drastically improved. These tests validate the robustness of the proposed scheme of constitutive modeling for systems with strong hydrodynamical interactions.

[1] 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.

[2] Zinchenko A.Z. and Davis R.H. 2015 “Extensional, shear flows and general rheology of concentrated emulsions of deformable drops”. J. Fluid Mech. (in press).

[3] Klaseboer E., Sun Q. and Chan D.Y.C. 2012 “Non-singular boundary integral methods for fluid mechanics applications”. J. Fluid Mech, vol. 696, pp.468-478.

[4] Eggleton C.D., Pawar Y.P. and Stebe K.J. 1999 “Insoluble surfactants on a drop in an extensional flow: a generalization of the stagnated surface limit to deforming interfaces”. J. Fluid Mech., vol. 385, pp.79-99.

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

[6] 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.

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