Coalescence, torus formation and breakup of sedimenting drops: experiments and computer simulations

Abstract

The motion and shape evolution of viscous drops made from a dilute suspension of tiny, spherical glass beads sedimenting in an otherwise quiescent liquid is investigated both experimentally and theoretically for conditions of low Reynolds number. In the (presumed) absence of any significant interfacial tension, the Bond number [Bscr ] = (Δρ)gR2/σ is effectively infinite. The key stages of deformation of single drops and pairs of interacting drops are identified. Of particular interest are (i) the coalescence of two trailing drops, (ii) the subsequent formation of a torus, and (iii) the breakup of the torus into two or more droplets in a repeating cascade. To overcome limitations of the boundary-integral method in tracking highly deformed interfaces and coalescing and dividing drops, we develop a formal analogy between drops of homogeneous liquid and a dilute, uniformly distributed swarm of sedimenting particles, for which only the 1/r far-field hydrodynamic interactions are important. Simple, robust numerical simulations using only swarms of Stokeslets reproduce the main phenomena observed in the classical experiments and in our flow-visualization studies. Detailed particle image velocimetry (PIV) for axisymmetric configurations enable a mechanistic analysis and confirm the theoretical results. We expose the crucial importance of the initial condition – why a single spherical drop does not deform substantially, but a pair of spherical drops, or a bell-shaped drop similar to what is actually formed in the laboratory, does undergo the torus/breakup transformation. The extreme sensitivity of the streamlines to the shape of the ring-like swarm explains why the ring that initially forms in the experiments does not behave like the slender open torus analysed asymptotically by Kojima, Hinch & Acrivos (1984). Essentially all of the phenomena described above can be explained within the realm of Stokes flow, without resort to interfacial tension or inertial effects.

An Eulerian-Eulerian Model for the Dispersion of a Suspension of Microscopic Particles Injected Into a Quiescent Liquid

“…The experimental results of Machu et al (2001) compared favourably with predictions of a Lagrangian numerical method that is valid for very small suspension drop Reynolds numbers. In contrast to the work of Adachi, Kiriyama and Yoshioka (1978), Machu et al (2001) examined the sub-structural effects that were taking place within the swarm. Effects such as the formation of a tail, torus formation and eventual breakup of the torus were seen both experimentally and numerically.…”

“…A number of authors (Adachi, Kiriyama and Yoshioka, 1978;Machu et al, 2001;Smith, 2000) have examined the dispersion of multiple particles injected into a liquid, dropped on to the surface of a liquid, or allowed to settle through a liquid under gravity. Experimental and theoretical work by Adachi, Kiriyama and Yoshioka (1978) considered the way in which a "swarm" or "plume" of microscopic particles behaved in a viscous fluid.…”

“…Although a theoretical sedimentation velocity for the particle swarm was developed by Adachi, Kiriyama and Yoshioka (1978), the transient shape evolution of the swarm was not modelled. Machu et al (2001) examined the motion and shape evolution of a dilute suspension of micronsized spherical glass beads injected into an initially quiescent liquid of the same composition as the carrier liquid in the injected suspension. The experimental results of Machu et al (2001) compared favourably with predictions of a Lagrangian numerical method that is valid for very small suspension drop Reynolds numbers.…”

“…Machu et al (2001) examined the motion and shape evolution of a dilute suspension of micronsized spherical glass beads injected into an initially quiescent liquid of the same composition as the carrier liquid in the injected suspension. The experimental results of Machu et al (2001) compared favourably with predictions of a Lagrangian numerical method that is valid for very small suspension drop Reynolds numbers. In contrast to the work of Adachi, Kiriyama and Yoshioka (1978), Machu et al (2001) examined the sub-structural effects that were taking place within the swarm.…”

“…This work was extended by Bosse et al (2005) who used an Eulerian-Lagrangian formulation to examine the evolution for finite "drop" Reynolds numbers, while retaining the assumption of very small particles for which inertia can be ignored. The models of both Machu et al (2001) and Bosse et al (2005) ignore the process of particle injection into the liquid, and specify the initial drop configuration immediately after injection is completed. Bosse et al (2005) found that the instability leading to the breakup of the torus at moderate drop Reynolds number was very sensitive to the initial conditions specified for the suspension drop.…”