Bubble Pinch-Off in a Rotating Flow

We create air bubbles at the tip of a “bathtub vortex” which reaches to a finite depth. The bathtub vortex is formed by letting water drain through a small hole at the bottom of a rotating cylindrical container. The tip of the needlelike surface dip is unstable at high rotation rates and releases bubbles which are carried down by the flow. Using high-speed imaging we find that the minimal neck radius of the unstable tip decreases in time as a power law with an exponent close to $1/3$. This exponent was found by Gordillo et al. [Phys. Rev. Lett. 95, 194501 (2005)] to govern gas flow driven pinch-off, and indeed we find that the volume oscillations of the tip creates a considerable air flow through the neck. We argue that the Bernoulli pressure reduction caused by this air flow can become sufficient to overcome the centrifugal forces and cause the final pinch-off.

Figure 1

By increasing the rotation rate while keeping $R_{\mathrm{drain}}=1.2\mathrm{mm}$, the air-filled tip of the bathtub vortex goes from a stable state at 13.5 rpm (a) to an unstable bubbling state at 24 rpm (b) and 30.5 rpm (c). In the bubbling state the frequency of bubble formation increases and the bubble size decreases with increasing rotation rate as shown in (b) and (c) for typical cases.

Figure 2

Bubble pinch-off at the tip of the bathtub vortex for $R_{\mathrm{drain}}=1.2\mathrm{mm}$ and $\Omega =13.5\mathrm{rpm}$. The time to pinch-off in milliseconds is shown above each frame. Note that the axial downflow around the tip does not give rise to any apparent up-down asymmetry in the pinch-off region, and that we observe a symmetric pinch-off as expected for bubble formation.

Figure 3

Measured radius of the air-filled neck as a function of the time to pinch-off (circles) and power law with exponent $\alpha =1/3$ (solid line). $R_{\mathrm{drain}}$ is 1.0 mm and $\Omega$ is 42 rpm. Inset: Frequency histogram for the measured exponents with $\alpha =1/3$ indicated.

Figure 4

Measured azimuthal (a),(c) and axial (b) flow velocities. The measurements in (a) and (b) were made 1 mm above and below the minimal neck radius for $R_{\mathrm{drain}}=1.2\mathrm{mm}$ and $\Omega =13.5\mathrm{rpm}$ (diamonds) and $\Omega =19\mathrm{rpm}$ (circles, crosses), respectively. The shaded areas indicate the initial radius of the neck before pinch-off. The data within the shaded areas are therefore obtained below the tip, except the red crosses in (a) which are obtained by tracking bubbles on the surface. In (a) we observe solid body rotation around the tip (dash-dot line), while away from the center $v_{\theta }$ tends to the line vortex observed in the bulk (dashed line). Note that in (b) positive $v_z$ corresponds to downward flow. (c) shows $v_{\theta }$ of a bubble on the surface of the neck during the collapse ($\Omega =30.5\mathrm{rpm}$) [17] and agrees with Eq. (3), with ${\nu }_e\approx 1.8\times {10}^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ (red line).

Figure 5

Measured minimal neck radius (triangles) and tip volume (filled circles) as functions of time to pinch-off for $R_{\mathrm{drain}}=1.2\mathrm{mm}$ and $\Omega =13.5\mathrm{rpm}$. The compressibility of the air in the tip is negligible, since once the neck reaches the resolution of our recordings ($1\mu \mathrm{m}$) the air is found to be compressed less than 1%. The average flow velocity through the collapsing neck (inset) is therefore computed directly from the volume change.