Abstract
A bubble is slowly grown from a vertical nozzle until it becomes unstable and pinches off. We use ultra-high-speed video imaging, at frame-rates up to
1
million
fps
, to study the dynamics and shape of the pinch-off neck region. For bubbles in water
(
Bo
≃
1.0
)
the radius of the neck reduces with a power law behavior
R
∼
t
α
, over more than 2 decades, with an exponent in the range
α
=
0.57
±
0.03
, in good agreement with other available studies, but which is slightly larger than
1
∕
2
predicted by Rayleigh-Plesset theory. The vertical curvature in the neck increases more slowly than the azimuthal curvature, making the neck profiles more slender as pinch-off is approached. Self-similar shapes are recovered by normalizing the axial coordinate by a separate length scale which follows a different power law,
L
z
∼
t
γ
, where
γ
=
0.49
±
0.03
. Results for air, He, and
S
F
6
gas are identical, suggesting that the gas density plays a minimal role in the dynamics. The pinch-off in water leaves behind a tiny satellite bubble, around
5
μ
m
in diameter and the flow-field inside the liquid is shown to be consistent with simple sink flow. The effects of liquid viscosity on the pinch-off speed and neck shapes, are also characterized. The speed starts to slow down at a viscosity of about 10 times that of water, which corresponds to
Re
μ
≃
2000
. This also changes the power law, increasing the exponent to
α
≃
1
for viscosities above
70
cP
(
Re
μ
≃
40
)
. For surrounding liquid of viscosity above
10
cP
, we observe just before pinch-off, that the neck is stretched into a thin filament of air, which then breaks into a stream of microbubbles. In some cases we observe a cascade of bubble sizes. While some of the details differ, our results are in overall agreement with those of Burton, Waldrep, and Taborek [Phys. Rev. Lett.
94, 184502 (2005)], except we do not observe the rupture of the air cylinder as it reduces to
50
μ
m
size. For water we observe a continuous necking down to the pixel-resolution of our optical system, which at the largest frame-rates is
∼
10
μ
m
.