Effect of spinning on the shape and stability of a pendent drop

Assuming that we wish to measure the surface tension between two liquids by running a pendent drop experiment, we present calculations supporting the case for spinning the drop. For bridges, jets, etc., spinning a heavy fluid surrounded by a lighter fluid is strictly destabilizing. But we find that spinning a drop may be stabilizing and, if this is so, it leads to larger critical volumes, volumes where stability is lost, and thus more accurate measurements of surface tension. There are two observable patterns, one symmetric and the other unsymmetric, at the point of instability. The symmetric pattern leads to larger critical volumes. Our aim is to show how spinning can be used to achieve the symmetric pattern.

Figure 1

A sketch of the problem. The entire apparatus is spinning. The volume of the drop is controlled by adding heavy fluid and removing an equal volume of light fluid. A series of experiments can be run by increasing the angular speed at constant volume. The drop shape is denoted by $Z(r,\theta )$, where $Z<0$.

Figure 2

The base shape plus a small symmetric perturbation at constant volume.

Figure 3

Drop shapes, $Z\left(r\right)$, at $B=12.5$ and $V=0$ (a) and $V=\frac{2}{5}$ (b), upon increasing ${\mathrm{\Omega }}^2$ from 0 to 1 to 2.

Figure 4

$V$ vs $P$ curves at $B=8$ and ${\mathrm{\Omega }}^2=0$, 1, 2.5, and 4. $\circ$, bifurcation point; $\times$, turning point.

Figure 5

The values of $B$ at crossover, denoted as $B_{*}$, vs ${\mathrm{\Omega }}^2$. Turning points appear before bifurcation points below the curve. Bifurcation points appear before turning points above the curve.

Figure 6

Plots of $B_{\mathrm{crit}}$ vs ${\mathrm{\Omega }}^2$ at a series of $V$'s, $m=1$. The $·$ is at ${\mathrm{\Omega }}^2=5\frac{1}{2}$ and $B=13.9$.

Figure 7

Plots of $B_{\mathrm{crit}}$ vs ${\mathrm{\Omega }}^2$ at a series of $V$'s, $m=0$.

Figure 8

$V$ vs $P$ curves at $B=13.9$ and ${\mathrm{\Omega }}^2=0$ and $5\frac{1}{2}$. $\circ$, bifurcation point; $\times$, turning point. Volumes between the two bifurcation points at ${\mathrm{\Omega }}^2=5\frac{1}{2}$ are stable.

Figure 9

An unstable drop at $B=12.5$ can be made stable by spinning but not too fast. $\circ$, bifurcation point; $\times$, turning point.

Figure 10

The drop pinned to a circle whose shape is $Z(r,\theta )$.