Shear flow of highly concentrated emulsions of deformable drops by numerical simulations

An efficient algorithm for hydrodynamical interaction of many deformable drops subject to shear flow at small Reynolds numbers with triply periodic boundaries is developed. The algorithm, at each time step, is a hybrid of boundary-integral and economical multipole techniques, and scales practically linearly with the number of drops N in the range N < 1000, for NΔ ∼ 103 boundary elements per drop. A new near-singularity subtraction in the double layer overcomes the divergence of velocity iterations at high drop volume fractions c and substantial viscosity ratio γ. Extensive long-time simulations for N = 100–200 and NΔ = 1000–2000 are performed up to c = 0.55 and drop-to-medium viscosity ratios up to λ = 5, to calculate the non-dimensional emulsion viscosity μ* = Σ12/(μeγ˙), and the first N1 = (Σ11−Σ22)/(μe[mid ]γ˙[mid ]) and second N2 = (Σ22−Σ33)/(μe[mid ]γ˙[mid ]) normal stress differences, where γ˙ is the shear rate, μe is the matrix viscosity, and Σij is the average stress tensor. For c = 0.45 and 0.5, μ* is a strong function of the capillary number Ca = μe[mid ]γ˙[mid ]a/σ (where a is the non-deformed drop radius, and σ is the interfacial tension) for Ca [Lt ] 1, so that most of the shear thinning occurs for nearly non-deformed drops. For c = 0.55 and λ = 1, however, the results suggest phase transition to a partially ordered state at Ca [les ] 0.05, and μ* becomes a weaker function of c and Ca; using λ = 3 delays phase transition to smaller Ca. A positive first normal stress difference, N1, is a strong function of Ca; the second normal stress difference, N2, is always negative and is a relatively weak function of Ca. It is found at c = 0.5 that small systems (N ∼ 10) fail to predict the correct behaviour of the viscosity and can give particularly large errors for N1, while larger systems N [ges ] O(102)show very good convergence. For N ∼ 102 and NΔ ∼ 103, the present algorithm is two orders of magnitude faster than a standard boundary-integral code, which has made the calculations feasible.