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.