Journal
JOURNAL OF FLUID MECHANICS
Volume 455, Issue -, Pages 21-62Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/S0022112001007042
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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 N-Delta similar to 10(3) 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 lambda. Extensive long-time simulations for N = 100-200 and N-Delta = 1000-2000 are performed up to c = 0.55 and drop-to-medium viscosity ratios up to lambda = 5, to calculate the nondimensional emulsion viscosity mu* = Sigma(12)/(mu(e)(gamma)over dot), and the first N-1 = (Sigma(11) - Sigma(22))/(mu(e)\(gamma)over dot\) second N-2 = (Sigma(22) - Sigma(33))/(mu(e)\(gamma)over dot\) normal stress differences, where (gamma)over dot is the shear rate, mu(e) is the matrix viscosity, and Sigma(ij) is the average stress tensor. For c = 0.45 and 0.5, mu* is a strong function of the capillary number Ca = mu(e)\(gamma)over dot\a/sigma (where a is the non-deformed drop radius, and a is the interfacial tension) for Ca much less than 1, so that most of the shear thinning occurs for nearly non-deformed drops. For c = 0.55 and lambda = 1, however, the results suggest phase transition to a partially ordered state at Ca less than or equal to 0.05, and mu* becomes a weaker function of c and Ca; using lambda = 3 delays phase transition to smaller Ca. A positive first normal stress difference, N-1, is a strong function of Ca; the second normal stress difference, N-2, is always negative and is a relatively weak function of Ca. It is found at c = 0.5 that small systems (N similar to 10) fail to predict the correct behaviour of the viscosity and can give particularly large errors for N-1, while larger systems N greater than or equal to O(10(2)) show very good convergence. For N similar to 10(2) and N-Delta similar to 10(3), the present algorithm is two orders of magnitude faster than a standard boundary-integral code, which has made the calculations feasible.
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