Fragment Formation in the Break-up of a Drop Falling in a Miscible Liquid

When falling in a lighter miscible solvent, a drop of liquid deforms to a torus which then breaks up into several fragments or just disappears by diffusion. By using liquids of different compositions we show the universal behaviour of the phenomenon, and its dependence on two nondimensional numbers, the fragmentation number F , and the Schmidt number S. While F marks the transition from diffusion to splitting, here we show the role of S in controlling the number of horizontal fragments after the first break-up. The process is explained in terms of competitions of different time scales. The simplest example of a hydrodynamical instability in an open system is represented by the fragmentation of a drop falling in a lighter miscible liquid. Empirical reports on this phenomenon were given by several authors in the last century[l]. When the drop is falling, it is deformed into a torus. Kojima et al. [2] made some experiments and calculations on the deformation of the miscible drop and the expansion of the torus. They attributed the disagreement between calculations and experiments to a transient interfacial tension [3]. However a general understanding of the fragmentation process still lacks, because of complex intermingling of Kelvin-Helmholtz (KH) and Rayleigh-Taylor (RT) instabilities. The torus can eventually become destabilized in some points by a RT instability, breaking up into several horizontal fragments. In the case of miscible liquids considered here, each of these fragments may undergo a similar break-up process, splitting again into secondary fragments. However, the number of successive vertical break-ups remains limited to three. A detailed description of these phenomena including a preliminary explanation was recently presented [4]. Reference [4] is focused on the transition between no splitting and the break-up of the torus into two fragments. This is controlled by a nondimensional (*) Also with Dipartimento di Fisica, Universita di Firenze, Firenze, Italy. (**) Permanent address: Department of Physics, University of Cape Coast, Cape Coast, Ghana. (***) Permanent address: Departamento de Fisica, Universidad de Navarra, 31080 Pamplona and Universidad Autonoma de Barcelona, 08193 Bellaterra (Barcelona), Catalonia, Spain.