Thinning and breakup of liquid filaments are central to dripping of leaky faucets, inkjet drop formation, and raindrop fragmentation. As the filament radius decreases, curvature and capillary pressure, both inversely proportional to radius, increase and fluid is expelled with increasing velocity from the neck. As the neck radius vanishes, the governing equations become singular and the filament breaks. In slightly viscous liquids, thinning initially occurs in an inertial regime where inertial and capillary forces balance. By contrast, in highly viscous liquids, initial thinning occurs in a viscous regime where viscous and capillary forces balance. As the filament thins, viscous forces in the former case and inertial forces in the latter become important, and theory shows that the filament approaches breakup in the final inertial-viscous regime where all three forces balance. However, previous simulations and experiments reveal that transition from an initial to the final regime either occurs at a value of filament radius well below that predicted by theory or is not observed. Here, we perform new simulations and experiments, and show that a thinning filament unexpectedly passes through a number of intermediate transient regimes, thereby delaying onset of the inertial-viscous regime. The new findings have practical implications regarding formation of undesirable satellite droplets and also raise the question as to whether similar dynamical transitions arise in other free-surface flows such as coalescence that also exhibit singularities.rop formation is ubiquitous in daily life, industry, and nature (1-3). The phenomenon is central to inkjet printing (4, 5), dripping from leaky faucets (6, 7), measurement of equilibrium and dynamic surface tension (8, 9), DNA arraying and printing of cells (10, 11), chemical separations and analysis (12, 13), production of particles and capsules (14, 15), printing of wires and transistors (16,17), and mist formation in waterfalls and fragmentation of raindrops (18,19). Fig. 1A shows an experimental setup for studying the dynamics of a drop of an incompressible Newtonian fluid of density ρ, viscosity μ, and surface tension σ forming from a tube of radius R (Fig. 1B and Drop Formation from a Tube and Filament Thinning). A salient feature of, and key to understanding, drop formation is the occurrence of a thin filament that connects an about-to-form primary drop to the rest of the fluid that is attached to the tube ( Fig. 1 B and C). Thus, it often proves convenient to study filament thinning in the idealized setup depicted in Fig. 1D (Drop Formation from a Tube and Filament Thinning). As time t advances and the filament radius decreases, curvature and capillary pressure, both of which are at leading order inversely proportional to radius, increase and fluid is expelled with increasing velocity from the neck. At the instant t = t b when the neck radius vanishes, a finite time singularity occurs and the filament breaks. When the filament breaks, one or more satellite droplets may also...
