The flow of ordered and random suspensions of two-dimensional drops in a channel

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Abstract
The flow of a periodic suspension of two-dimensional viscous drops in a closed channel that is bounded by two parallel plane walls executing relative motion is studied numerically using the method of interfacial dynamics. Ordered suspensions where at the initial instant the drops are arranged in several layers on a hexagonal lattice are considered for a variety of physical conditions and geometrical configurations. It is found that there exists a critical capillary number below which the suspensions exhibit stable periodic motion, and above which the drops elongate and tend to coalesce, altering the topology of the initial configuration. At sufficiently large volume fractions, a minimum drop capillary number exists below which periodic motion is suppressed owing to the inability of the drops to deform and bypass other neighbouring drops in adjacent layers. This feature distinguishes the motion of dense emulsions from that of foam. The effects of capillary number, viscosity ratio, volume fraction of the dispersed phase, lattice geometry, and instantaneous drop shape, on the effective stress tensor of the suspension are illustrated and the results are discussed with reference to theories of foam. Two simulations of a random suspension with 12 drops per periodic cell are performed, and the salient features of the motion are identified and discussed. These include pairing, tripling, and higher-order interactions among intercepting drops, cluster formation and destruction, and drop migrations away from the walls. The macroscopic features of the flow of random suspensions are shown to be significantly different from those of ordered suspensions and quite independent of the initial condition. The general behaviour of suspensions of liquid drops is compared to that of suspensions of rigid spherical particles, and some differences are discussed.

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