Static stability of pendent drops pinned to arbitrary closed curves

We compare the stability of a static pendent drop under two types of control, volume control and pressure control. The drops are taken to be pinned to curves of arbitrary shape. The two types of control introduce integrals into the eigenvalue problems that determine the points of instability. We show that these integrals are solely responsible for the possible occurrence of bifurcation points, depending only on the Bond number. We then show that the points of instability for either type of control can be related to one another and predicted precisely from the eigenvalues of, yet, a third problem, one that is devoid of any integrals. If the curves of attachment are symmetric, we can derive a result that predicts the instability point and associated pattern, all without solving any eigenvalue problem.

Figure 1

A sketch of a drop pinned along a curve, $\mathcal{C}$, bounding a cross section, $\mathcal{A}$, under volume control, experiment I.

Figure 2

A sketch of a drop under pressure control, experiment II.

Figure 3

Graph of Eq. (28).

Figure 4

Graph of Eq. (29).

Figure 5

Graph of Eq. (28) depicting two cases: (a) ${\mu }_1^2<{\lambda }_1^2<{\mu }_2^2,{\lambda }_2^2={\mu }_2^2$ and (b) ${\lambda }_1^2={\mu }_2^2,{\mu }_2^2<{\lambda }_2^2$.

Figure 6

Graph of $V$ vs $P$ depicting ${\lambda }^2$: (a) $B=9.5$, (b) $B=8$; o = bifurcation point, $*$ = turning point.

Figure 7

Graph of $V$ vs $P$ at $B=B★=5$; o = bifurcation point, $*$ = turning point.

Figure 8

Graph of $V$ vs $P$ at $B<B★,B=4$; $*$ = turning point.

Figure 9

Graphs showing the intersection of $2P+BV=0$ and $V$ vs $P$: (a) large $B$, (b) small $B$; o = bifurcation point, $*$ = turning point.

Figure 10

Graphs of $V$ vs $P$ showing that experiment II is less stable than experiment I; $B=7,B*=\frac{11}{10}{\pi }^2$.

Figure 11

The $V$ vs $P$ curve for a circular cross section showing the straight line construction; $B=13$ (a), B = 9 (b), $B★=10$; o = bifurcation point, $*$ = turning point.

Figure 12

A sketch of experiment I.

Figure 13

Sketch of a pressure-controlled drop experiment.