When a liquid drop falls from a fluid source with a slow rate, it remains attached to the source by an elongating liquid filament until the filament pinches off. For many fluids, this pinch-off occurs first near the end of the filament, where the filament joins to the liquid drop. For other fluids, the filament pinches off at one or more interior points. In this paper, we study the motion of this filament, and we make two points. First, the flow in this filament is not that of a uniform jet. Instead, we show experimentally that a different solution of the Navier-Stokes equations describes the motion of this filament before it pinches off. Second, we propose a criterion for the location of the first pinch-off. In particular, we analyze the linearized stability of the exact solution, both for an inviscid fluid and for a very viscous fluid. Our criterion for pinch-off is based on this stability analysis. It correctly predicts whether a given filament pinches off first near its ends or at points within its interior for all of our experimental data.

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