Stability of a static liquid bridge knowing only its shape

The shape and the stability of a static liquid bridge under either pressure control or volume control can be derived by knowing only the shape of a master liquid bridge. No perturbation problems need be solved.

Figure 1

A sketch of a bridge spanning two disks.

Figure 2

A sketch of a bridge whose surface shape is being controlled by the pressure at the left-hand reservoir.

Figure 3

The $V$ vs $P_{*}$ curve on the right at $B_{*}>0$ is obtained from the $V$ vs $P$ curve on the left at $B_{*}=0$ without solving Eq. (7). $L=\frac{3}{4}\pi ,B=\frac{1}{100},B_{*}=\frac{1}{20}$.

Figure 4

Graphs of the solutions to Eq. (27), where the eigenvalues, ${\lambda }_1^2,{\lambda }_2^2,···$, at $B_{*}>0$ are obtained from the eigenvalues, ${\mu }_1^2,{\mu }_2^2,···$, at $B_{*}=0$. The values of $L,B,$ and $V$ are $L=\frac{3}{4}\pi ,B=\frac{1}{10},V=0.976$.

Figure 5

A sketch of the least eigenvalues along the $V$ vs $P$ curve at $B_{*}=0$, solid line, and along the $V$ vs $P_{*}$ curve at $B_{*}>0$, dashed line. The eigenvalues, ${\lambda }_1^2$, derive from the eigenvalues, ${\mu }_1^2$, as indicated in Fig. 4. Where ${\mu }_1^2$ or ${\lambda }_1^2$ is zero, turning points appear. The values of $L, B,$ and $B_{*}$ are $L=\frac{1}{2}\pi ,B=\frac{1}{100},B_{*}=\frac{1}{100}$.

Figure 6

Turning points and therefore points where the least eigenvalue is zero can be read directly off base curves, $B_{*}=0,L=\frac{1}{2}\pi ,B=\frac{1}{100}$.

Figure 7

Graphs of the solutions to Eq. (41), showing the first two eigenvalues, (${\lambda }_1^2, {\mu }_2^2$), at a small value at $B_{*}$ and (${\mu }_2^2, {\lambda }_1^2$) at a large value of $B_{*}$. $L=\frac{3}{4}\pi ,V=0.975$.

Figure 8

The solid curve is the base curve $V$ vs $P_{*}$ at $B_{*}>0$. The dot-dash curve is $\frac{1}{2L}{\int }_{-L}^L\frac{1}{R}\frac{1}{{(1+R_z^2)}^{\frac{1}{2}}}dz+B_{*}V\text{vs}{P}_{*}$ and the dot-dot curve is Eq. (49). Their intersection identifies the point $(P_{*},V)$ where ${\mu }_2^2=0$, i.e., a bifurcation point. The values of $L$ and $B_{*}$ are $L=\frac{3}{4}\pi$ and $B_{*}=\frac{1}{100}$.

Figure 9

At two values of $B_{*}$ viz., $B_{*}=0$ and $B_{*}=.01$, we obtain bifurcation points, ${\mu }_2^2=0$ at the same value of $V$. The value of $L$ is $\frac{3}{4}\pi$.

Figure 10

The least even eigenvalues for three values of $B_{*}$ and the least odd eigenvalue plotted vs $\frac{{\pi }^2}{L^2}$.

Figure 11

Volume vs Pressure Curves at $B_{*}=0,0.15,0.30$. A value of $V$ is chosen so that ${\mu }_2^2>0.$

Figure 12

Volume vs pressure curve at $B_{*}=0,0.1,0.25,0.30$. A value of $V$ is chosen so that ${\mu }_2^2=0.$

Figure 13

Volume vs pressure curve at $B_{*}=0,0.15,0.30$. A value of $V$ is chosen so that ${\mu }_2^2<0.$

Figure 14

A graph of $B_{*}^{†}$ as a function of $L$. Note that ${\lambda }_1^2=0={\mu }_2^2$ at each point on the curve and that $V$ changes along the curve.