Droplet Detachment and Satellite Bead Formation in Viscoelastic Fluids

The presence of a very small amount of high molecular weight polymer significantly delays the pinch-off singularity of a drop of water falling from a faucet and leads to the formation of a long-lived cylindrical filament. In this Letter, we present experiments, numerical simulations, and theory which examines the pinch-off process in the presence of polymers. The numerical simulations are found to be in good agreement with experiment. As a test case, we establish the conditions under which a small bead remains on the filament; we find that the presence of a bead is due to the asymmetry induced by the self-similar pinch off of the droplet.

Figure 1

(a) A drop of water falling from a faucet, the satellite drop is about to disintegrate into a number of even smaller droplets, $R=3\mathrm{mm}$. (b),(c) Close-up of the pinch region, with 100 ppm of PEO solution added. Left-hand part of the photographs: numerical simulations. Right-hand part (in gray with white spots due to lensing): experiment. (b) $R=3\mathrm{mm}$, $t_c-t=6,2,0,-3,-5\mathrm{ms}$; (c) $R=0.4\mathrm{mm}$ $t_c-t=1,0\mathrm{ms}$. Model parameters: ${\eta }_p=3.7\times {10}^{-4}\mathrm{Pa}\mathrm{s}$, $\lambda =1.2\times {10}^{-2}\mathrm{s}$, $b=2.5\times {10}^4$, ${\eta }_s=1\times {10}^{-3}\mathrm{Pa}\mathrm{s}$, $\gamma =6\times {10}^{-2}\mathrm{N}/\mathrm{m}$.

Figure 2

(a) The phase boundary for satellite bead formation, from experimental runs at polymer concentrations of 5, 10, 20, 50, 100, 200, 500, 1000, and 2000 ppm and nozzle radii $R=0.25$, 0.75, 1, 1.5, 2, 3, 4, and 5 mm (triangles). Between the dashed and the solid lines a bead is formed, but subsequently disappears. (b) The asymmetry parameter $\alpha$ for the 100 ppm runs measured from the last video frame before the appearance of a thread. The vertical lines correspond to the transition lines in (a).

Figure 3

The minimum neck radius versus time for nozzle radii $R=0.4\mathrm{mm}$ (circles), $R=1.5\mathrm{mm}$ (triangles), and $R=4\mathrm{mm}$ (squares), at a polymer concentration of 100 ppm, where $t_0$ is the extrapolated time at which pinch off would take place for a Newtonian liquid. The straight lines are fits to a $2/3$ power law, yielding a universal prefactor of 0.8 (note that the neck radius is reported in units of $R$). The dashed lines are exponential fits to the disturbance amplitude $1-h_{\min }/R$. The squares in the inset are the corresponding growth rates; the line is the theoretical prediction for the most unstable Rayleigh mode of an inviscid fluid.

Figure 4

The migration of the primary bead for the 50 ppm, $R=1.5\mathrm{mm}$ run (pictures are $1\times 4\mathrm{mm}$). The profiles are shown in time steps of 2 ms. The bead is pushed upward against the direction of gravity.