Extremes of the pinch-off location and time in a liquid column by an accelerating solid sphere

When a solid body rises from a bath, a liquid column is formed and stretched until pinch-off occurs at different locations and times. In the present paper, we study the temporal and spatial evolution of a less viscous liquid column extracted by an accelerating sphere from a bath. At high acceleration, the observed liquid column has the shape of an up-pointing cone, which implies that the column near the sphere is stretched so quickly that the liquid cannot keep up with it. This causes the liquid column to pinch off at the upper location. At low accelerations, a liquid column pinches off at a lower location near the free surface. The shift in the pinch-off location is explained by the axial velocity-gradient profile in the liquid column. The numerical results show that the gradient of the axial velocity near the sphere increases when the solid sphere has a higher acceleration. As a result, at high accelerations, the axial velocity gradient is responsible for the necking and pinching off at the upper location of the liquid column. At low sphere accelerations, the column pinches off at the lower location by the strong Laplace pressure due to the higher gradient of the interfacial curvature between the column and the bath. Next the pinch-off time is observed to decrease as the acceleration increases regardless of the pinch-off location. We use a linear stability analysis to predict the pinch-off time as the inverse of the growth rate in the dispersion relation.

Figure 1

Shapes of a water column: (a) an upward-pointing cone column with upper pinch-off and (b) a uniform column diameter in the middle with lower pinch-off. Here a glass sphere of $D=6.35$ mm was withdrawn from a water bath with constant acceleration of (a) $a=2g$ or (b) $0.5g$. The pinch-off time was measured to be (a) 31.4 ms and (b) 43.2 ms. The inset in (a) illustrates the experimental schematic of ascending a solid sphere with a constant acceleration that is controlled by the linear stage. Here $z_p$ is the pinch-off location and $Z_p$ is the center position of the sphere from the free surface at the pinch-off moment.

Figure 2

(a) Temporal evolution of the water-air interface when a glass sphere of $D=3.2$ mm rises at $a=2g$. Experimental images and numerical results from gerris are compared side-to-side at different times: $t=0.3$, 4.3, 8.8, 13.3, and 18.3 ms. (b) Comparison of pinch-off time $t_p$ and normalized pinch-off location $z_p/(Z_p-0.5D)$ between the experiment and simulation results with a changing minimal mesh size. In the present study, the simulation results are obtained only with smallest mesh size $\le 6.3\mu \mathrm{m}$ after the data converge.

Figure 3

Shift in the pinch-off location by the acceleration-based Bond number ${\text{Bo}}_a={\rho }_{l}a{D}^2/\sigma$. The pinch-off location is presented by (a) the relative location in the liquid column, which is estimated by the normalized pinch-off location $z_p/(Z_p-0.5D)$, and (b) the absolute location from the free surface, which is estimated by $z_p/D$. The data are colored according to the acceleration of the solid sphere. Circles and triangles indicate experimental and simulation data, respectively.

Figure 4

Temporal and spatial evolution of the axial velocity, which is radially averaged in a water column formed by a rising solid sphere with different accelerations of $0.5g$ (blue) and $2g$ (red). The axial velocity at the highest $\tilde{z}$ equals the instantaneous velocity of an accelerating sphere $U_s\left(t\right)$, as shown in the subplot at $\tilde{t}=0.33$. The axial velocity is numerically calculated using the open-source code named gerris [21]. For simulations, the initial and boundary conditions are maintained with $D=3.2$ mm, ${\rho }_l=998\mathrm{kg}/{\mathrm{m}}^3$, and $\sigma =0.073$ N/m, except for the acceleration of the sphere. The shapes of the water-air interface are presented in the insets with blue and red borders for $a=0.5g$ and $2g$, respectively. The dark green regions indicate the decaying length defined in Sec. 3c.

Figure 5

Numerical data on the temporal evolution of the (a) dimensionless maximum velocity gradient and (b) decaying length for high accelerations of $2g$, $7g$, and $10g$ on the left and low accelerations of $0.5g$, $0.75g$, and $1g$ on the right. The black lines indicate (a) ${(\partial {\tilde{u}}_z/\partial \tilde{z})}_{\text{max}}=4.5{\tilde{t}}^{-1/2}$ and (b) $\tilde{\lambda }=(1/4.5){\tilde{t}}^{1/2}$.

Figure 6

Dimensionless pinch-off time ${\tilde{t}}_p$ versus acceleration-based Bond number ($={\rho }_{l}a{D}^2/\sigma$). The red line indicates the present theory of ${\tilde{t}}_p=\sqrt{8\pi /9}{\text{Bo}}_a^{-1/2}$ for the high acceleration of a sphere.