Penn State University

Abstract:

We show that an exact solution to the Navier-Stokes equation described the motion of the filament that connects a drop of viscous fluid as the drop falls under gravity from an orifice. This solution predicts the narrowing of the filament in time as the drop falls, in agreement with experiments. The filament's narrowing is a fundamental difference between the motion of the fluid in this problem and the corresponding motion in a classic Rayleigh-jet, for which the cross section is constant in time and space.

Experiments show that the filament either pinches off at the ends or pinches off in its interior at several locations. We compare this behavior to predictions based on a linearized stability analysis of the exact solution. A feature of this analysis is that the ":most unstable mode" changes with time, unlike the analogous situation for the jet problem. To use this analysis, we assume that the wavelength most likely to be observed at short times is the one with the largest growth rate initially. Two scenarios are possible: (1) If the most unstable mode is the longest mode that fits in the (changing) length of the filament, then the filament pinches off at its ends. (2) If the most unstable mode has a wavelength shorter than the length of the filament, then this shorter wavelength grows, and the filament pinches off in its interior at several locations. This prediction of whether the filament pinches off at its ends or in its interior agrees with the experimental results.