Experiments on the low-Reynolds-number settling of a sphere through a fluid interface

The low-Reynolds-number gravitational settling of a sphere through a fluid interface is investigated experimentally. By varying the viscosity ratio between the two fluids and the Bond number, two different modes of interfacial deformation are observed: a tailing mode and a film drainage mode. In the tailing mode, the interface deforms significantly as the sphere approaches, and the sphere becomes enveloped by a layer of the upper fluid. A tail forms, connecting the sphere to the bulk of the upper phase. In the film drainage mode, the interface deforms much less and the sphere impacts onto the interface, which either ruptures to form a contact line on the sphere or leaves a very thin wetting film. Additionally, two types of sinking profiles are observed: steady sinking, where the sphere velocity changes monotonically as it sinks, and stalled sinking, where the sphere's progress is inhibited by the interface, before it accelerates into the lower fluid. We present a regime diagram showing the different behaviors. Finally, the dependence of the sinking time on the Bond number and viscosity ratio is investigated. For the film drainage regime a simple scaling law is deduced; the tailing regime exhibits more complicated dynamics, possibly explained by a multistage sinking process.

Figure 1

Sketch of the physical parameters in the system. The sphere is denser than both fluids, i.e., ${\rho }_{\text{s}}>{\rho }_2>{\rho }_1$. If a three phase contact line exists on the surface of the sphere, the contact angle is defined as $\chi$.

Figure 2

Sketch showing modes of interfacial deformation. The upper is a tailing mode, the lower, film drainage. The tailing mode leads to significant entrainment of the upper phase into the lower layer.

Figure 3

(a) Aerial view of the experimental setup. The Image-J plug-in MtrackJ [52, 53] was used to track the sphere position in the images from the Sony Handycam. Higher resolution images from the Canon DSLR were used to determine both the mode of sinking and the sinking time. In some experiments, the sphere velocity was small enough that the whole tank could be viewed at the slower frame rate of the DSLR, allowing all measurements to be determined from a single set of images. (b) Sketch of the tank used in the experiments. By varying the water content of the syrup solution (0%–5%), the grade of PDMS and the temperature ($0–{32}^{\circ }$), the viscosity ratio was varied between ${10}^{-2}$ and ${10}^3$. By using spheres from 2 to 10 mm in diameter the Bond number varied from 0.1 to 5.

Figure 4

Regime diagram for floating-sinking. Points show our experimental data. There is a clear divide between floating and sinking, although we have not attempted to constrain the boundary. Curves are calculated from the numerical model of Vella et al. [16], see Eqs. (7)–(9) therein, which predicts the transition for a sphere instantaneously placed at the interface, for different values of the contact angle $\chi$. The experiments compare favourably with this model if $\chi \gtrsim 3\pi /5$.

Figure 5

Images of modes of deformation. (a) Film drainage ($\mathrm{Bo}=4.0, D=3.2, \lambda =260$; experiment F42 in the Supplemental Material [51]). (b) Tailing ($\mathrm{Bo}=2.4, D=3.5, \lambda =4.0$; experiment D24). (c) Tailing with a long tail ($\mathrm{Bo}=4.3, D=3.2, \lambda =1.7$; experiment I14). Videos of these experiments can be found in the Supplemental Material [51].

Figure 6

Two different shapes of position curve seen in experiments. The dashed line shows the initial interface position. Both cases show a transition from the terminal velocity in the upper fluid to a new terminal velocity in the lower fluid. (a) Steady sinking ($\mathrm{Bo}=3.6, D=3.38, \lambda =35$; experiment D15 in the Supplemental Material [51]). The sphere slows on approach to the interface and steadily decelerates to a new slower velocity. (b) Stalled sinking ($\mathrm{Bo}=2.5, D=3.31, \lambda =0.45$; experiment G15). The sphere slows while approaching the interface and appears to stall there before accelerating to a new terminal velocity as it moves away. In both cases the greatest deceleration occurs just below $z=0$.

Figure 7

Regime diagram for both mode of deformation (Fig. 5) and position curve shape (Fig. 6). Distinct regimes can be seen for film drainage (red) and tailing (blue). Points labeled transitional correspond to experiments where the optical resolution of the image meant it was difficult to identify the mode of deformation. The purple line is drawn to help visibly separate the regimes. The different shapes of position curve, steady (hollow symbols) and stalled (solid symbols) sinking, also separate into distinct regimes, with the black curve a fitted theoretical relationship for the transition [Eq. (13)].

Figure 8

For the film drainage experiments, the dependence of the dimensionless sinking time on (a) the Bond number and (b) the viscosity ratio. The lines are the result of a linear fit and the gradients are stated.

Figure 9

A log-log plot of the dimensionless sinking time against the viscosity ratio divided by the Bond number. The solid line has a gradient of 4/5, as calculated from the planar fit and does a good job of describing the data. The dashed line has a gradient of 1, as predicted by Lee and Kim [35], and has been positioned to provide the closest fit to our data.

Figure 10

Tailing mode was divided into three stages: (a) pushing, when the sphere traverses the plane of the initial interface, (b) thinning, when the tail thins until its thickness is equal to the capillary length, and (c) snapping, when the tail pinches off. The dependence of the dimensionless time of each stage on $\mathrm{Bo}$ and $\lambda$ is shown. (a) $t_{\text{p}}/t_{\text{cu}}$ decreases with $\mathrm{Bo}$ for $\mathrm{Bo}>1$ and is nonmonotinic with $\lambda$. (b) $t_{\text{t}}/t_{\text{cu}}$ decreases with $\mathrm{Bo}$ and increases with $\lambda$. Linear fits are provided for $\mathrm{Bo}>1$ and $\mathrm{Bo}\approx 1,\lambda <1$. Measurement uncertainty is comparable to the symbol size. One point (circled with a dashed line) has an anomalously high value. c) $t_{\text{s}}/t_{\text{cu}}$ depends only weakly on $\mathrm{Bo}$, as can be seen by the collapse of the curves on the plot against $\lambda$.