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.
Electrospray deposition device used to precisely control the matrix crystal to improve the performance of MALDI MSI
