2.1 Family of solution curves

As shown in figure 1, there are two directions of gravity for the configuration: the downward and upward directions corresponding to positive loads $\unicode[STIX]{x1D700}=+1$ and negative loads $\unicode[STIX]{x1D700}=-1$, respectively. It is well known that the profile of the equilibrium meniscus is governed by the Young–Laplace equation, which can be expressed in parametric form (see e.g. Huh & Scriven Reference Huh and Scriven1969; Boucher & Evans Reference Boucher and Evans1975; Finn Reference Finn1986):

(2.1a-c)$$\begin{eqnarray}\frac{\text{d}r}{\text{d}s}=\cos \unicode[STIX]{x1D713},\quad \frac{\text{d}z}{\text{d}s}=\sin \unicode[STIX]{x1D713},\quad \frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}s}=\unicode[STIX]{x1D700}z-\frac{\sin \unicode[STIX]{x1D713}}{r},\end{eqnarray}$$

where $\unicode[STIX]{x1D713}$ is the inclination angle of the meniscus. The initial conditions for the simply connected meniscus are

(2.2a,b)$$\begin{eqnarray}r=0,\quad z=z_{0}\quad \text{at }s=0.\end{eqnarray}$$

In practice, the solutions of the linearized Young–Laplace equation $r^{2}z^{\prime \prime }+rz^{\prime }-\unicode[STIX]{x1D700}r^{2}z=0$ (valid at small inclination angles) satisfying (2.2) (Siegel Reference Siegel1980; Finn Reference Finn1986),

(2.3a)$$\begin{eqnarray}\displaystyle & z=\tan \unicode[STIX]{x1D713}^{\ast }\text{I}_{0}(r)/\text{I}_{1}(r^{\ast }),\quad \text{for }\unicode[STIX]{x1D700}=+1, & \displaystyle\end{eqnarray}$$

(2.3b)$$\begin{eqnarray}\displaystyle & z=-\text{tan}\,\unicode[STIX]{x1D713}^{\ast }\text{J}_{0}(r)/\text{J}_{1}(r^{\ast }),\quad \text{for }\unicode[STIX]{x1D700}=-1, & \displaystyle\end{eqnarray}$$

$r=0$

$\text{J}$

$\text{I}$

$r^{\ast }$

$\unicode[STIX]{x1D713}^{\ast }$

$r^{\ast }$

$\unicode[STIX]{x1D713}^{\ast }$

$0\leqslant r\leqslant r^{\ast }$

$r=r^{\ast }$

$(r,z,\unicode[STIX]{x1D713})=(r^{\ast },z^{\ast },\unicode[STIX]{x1D713}^{\ast })$

$\unicode[STIX]{x1D713}^{\ast }$

$+0.01^{\circ }$

$z^{\ast }\approx z_{0}$

$\unicode[STIX]{x1D713}^{\ast }=-0.01^{\circ }$

$z=0$

Integrating (2.1) with the initial conditions $(r^{\ast },z^{\ast },\unicode[STIX]{x1D713}^{\ast })$ for a set of values of $r^{\ast }$ and $\unicode[STIX]{x1D713}^{\ast }=0.01^{\circ }$, two families of solution curves under positive and negative loads are determined, as shown in figure 2. The solution curves for $\unicode[STIX]{x1D700}=+1$ can also be determined by integrating a different parametric form of the Young–Laplace equation (see e.g. Zhang & Zhou Reference Zhang and Zhou2020), where the parameter is not the arc length $s$ but the inclination angle $\unicode[STIX]{x1D713}$. However, this parametric form is not suitable for the case of $\unicode[STIX]{x1D700}=-1$, because of the singularity of $(\text{d}/\text{d}\unicode[STIX]{x1D713})$ (i.e. the curvature $\text{d}\unicode[STIX]{x1D713}/\text{d}s=0$) at the inflection points (e.g. the red point in figure 2b).

In figure 2(a,b), a pair $(\unicode[STIX]{x1D713},z_{0})$ can be uniquely determined by its coordinates $(r,z)$ in the regions $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ (Wente Reference Wente2011), because the coordinates $(r,z)$ can uniquely determine a solution curve and an inclination angle in $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$. These two regions (grey shaded areas) in the quadrant of the $r$–$z$ plane are bounded by the coordinate axes and the envelopes ($\unicode[STIX]{x1D701}_{1}$ and $\unicode[STIX]{x1D701}_{2}$ in figure 2). In these regions, the Jacobian determinants of $(r(\unicode[STIX]{x1D713},t),z(\unicode[STIX]{x1D713},t))$,

(2.4)$$\begin{eqnarray}F=\frac{\unicode[STIX]{x2202}r}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}\frac{\unicode[STIX]{x2202}z}{\unicode[STIX]{x2202}t}-\frac{\unicode[STIX]{x2202}r}{\unicode[STIX]{x2202}t}\frac{\unicode[STIX]{x2202}z}{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}},\end{eqnarray}$$

are non-zero and keep the same sign (Finn Reference Finn1986; Wente Reference Wente2011). The vanishing of $F$ locates the envelopes $\unicode[STIX]{x1D701}_{1}$ and $\unicode[STIX]{x1D701}_{2}$. As shown in figure 2(a), the envelope $\unicode[STIX]{x1D701}_{1}$ is represented by a function $z=e_{1}(r)$, decreasing with $r$ with $\lim _{r\rightarrow 0^{+}}e_{1}(r)=+\infty$ and $\lim _{r\rightarrow +\infty }e_{1}(r)=2$ (Wente Reference Wente2011). In figure 2(b), the envelope $\unicode[STIX]{x1D701}_{2}$ is represented by a function $z=e_{2}(r)$, increasing with $r$ with $\lim _{r\rightarrow 0^{+}}e_{2}(r)=-\infty$ and $e_{2}(r_{0})=0$ (see the blue point). The endpoint $(r_{0},0)$ of $\unicode[STIX]{x1D701}_{2}$ can be determined by the asymptotic solution of the Young–Laplace equation $G(z,r,t)=z-t\text{J}_{0}(r)=0$, i.e. (2.3b). By substituting the asymptotic solution into $\unicode[STIX]{x2202}G/\unicode[STIX]{x2202}t=0$ for determining the envelope (Lawrence Reference Lawrence2013), we have $\text{J}_{0}(r_{0})=0$, where $r_{0}\approx 2.405$ is the first zero of $\text{J}_{0}(r)$.

Zhang & Zhou (Reference Zhang and Zhou2020) have shown that the envelopes $\unicode[STIX]{x1D701}_{1}$ and $\unicode[STIX]{x1D701}_{2}$ are closely related to meniscus stability. It is known that an envelope of a family of curves is defined as a curve that touches every member of the family tangentially (Lawrence Reference Lawrence2013). The envelopes also bound the maximal stability regions ($\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$) which correspond to the maximal possible profiles of stable menisci, called the maximal stable profiles by Slobozhanin & Alexander (Reference Slobozhanin and Alexander2003). This means that only the meniscus in $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ is likely to be stable, for reasons discussed in § 3.2. Therefore, the points on the envelope are conjugate points of the Jacobi equation, i.e. the instability points. This can be explained in that the envelopes can be regarded as ECTs with zero contact angle, and the menisci in the ECTs correspond to the instability points to pressure disturbances (Zhang & Zhou Reference Zhang and Zhou2020). This implies that the shape of the ECT is closely related to meniscus stability. In the calculus of variations, a similar fact is that if a family of extremals to a functional through a fixed point has an envelope, then a point where an extremal intersects the envelope is a conjugate point to the fixed point (see e.g. Gulliver & Hildebrandt Reference Gulliver and Hildebrandt1986).

Figure 2.

Two families of solution curves for (2.1): (a) $\unicode[STIX]{x1D700}=+1$ for $\unicode[STIX]{x1D713}\in [0,\unicode[STIX]{x03C0}]$ and (b) $\unicode[STIX]{x1D700}=-1$ for $s\in [0,3]$. The shaded areas $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ represent the maximal stability regions. The black lines are the solution curves, which are divided by the envelopes into two parts: the lower part is denoted by the solid line in the maximal stability regions and the upper part is denoted by the dashed line. The blue point ($r_{0},0$) is the endpoint of the envelope $\unicode[STIX]{x1D701}_{2}$.

2.2 Determination of the shape of ECTs

The shape of the ECT with contact angle $\unicode[STIX]{x1D703}$ is the integral curve of the slope field $\{\cos (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}),\sin (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703})\}$, where the scalar fields $\unicode[STIX]{x1D713}(r,z)$ for $\unicode[STIX]{x1D700}=\pm 1$ are determined in the regions $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$, respectively (see figure 2) (Wente Reference Wente2011). Meanwhile, we know that the integral curve of the slope field $\{\cos \unicode[STIX]{x1D713},\sin \unicode[STIX]{x1D713}\}$ is the solution curve of the Young–Laplace equation. Thus, following the parametric form (2.1a–c), we can construct

(2.5a-c)$$\begin{eqnarray}\left.\frac{\text{d}x}{\text{d}s}\right|_{S}=\cos (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}),\quad \left.\frac{\text{d}y}{\text{d}s}\right|_{S}=\sin (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}),\quad \left.\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}s}\right|_{S}=\tilde{K}(x,y;\unicode[STIX]{x1D703})\end{eqnarray}$$

to determine the ECT, where $(x,y)$ denotes a point on the generatrix of the ECT in the $r$–$z$ plane, $\tilde{K}(x,y;\unicode[STIX]{x1D703})$ is the curvature of the integral curve of $\{\cos (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}),\sin (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703})\}$ ($\tilde{K}<0$ if the solid is convex to the liquid) and $(\text{d}/\text{d}s)|_{S}$ denotes the directional derivative along the solid surface, i.e. in the direction $(\cos (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}),\sin (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}))$.

Then we have

(2.6a)$$\begin{eqnarray}\displaystyle & \displaystyle K=\left.\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}s}\right|_{L}=\unicode[STIX]{x1D713}_{r}\cos \unicode[STIX]{x1D713}+\unicode[STIX]{x1D713}_{z}\sin \unicode[STIX]{x1D713}, & \displaystyle\end{eqnarray}$$

(2.6b)$$\begin{eqnarray}\displaystyle & \displaystyle \tilde{K}=\left.\frac{\text{d}\unicode[STIX]{x1D713}}{\text{d}s}\right|_{S}=\unicode[STIX]{x1D713}_{r}\cos (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703})+\unicode[STIX]{x1D713}_{z}\sin (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}), & \displaystyle\end{eqnarray}$$

$K$

$\{\cos \unicode[STIX]{x1D713},\sin \unicode[STIX]{x1D713}\}$

$r$

$z$

$(\text{d}/\text{d}s)|_{L}$

$(\cos \unicode[STIX]{x1D713},\sin \unicode[STIX]{x1D713})$

$\unicode[STIX]{x1D713}_{r}$

$\unicode[STIX]{x1D713}_{z}$

(2.7)$$\begin{eqnarray}\unicode[STIX]{x1D713}_{r}\cos \unicode[STIX]{x1D713}+\unicode[STIX]{x1D713}_{z}\sin \unicode[STIX]{x1D713}=\unicode[STIX]{x1D700}z-\frac{\sin \unicode[STIX]{x1D713}}{r}.\end{eqnarray}$$

Once $\unicode[STIX]{x1D713}_{r}$ or $\unicode[STIX]{x1D713}_{z}$ is determined, $\tilde{K}$ in (2.5c) is determined by (2.6b), and then the shape of the ECT is obtained by integrating (2.5).

The partial derivative $\unicode[STIX]{x1D713}_{r}$ can be calculated by integrating (2.1) together with

(2.8)$$\begin{eqnarray}\left.\frac{\text{d}\unicode[STIX]{x1D713}_{r}}{\text{d}s}\right|_{L}=\frac{\sin \unicode[STIX]{x1D713}}{r^{2}}+\frac{\unicode[STIX]{x1D713}_{r}(\unicode[STIX]{x1D713}_{r}-\unicode[STIX]{x1D700}z\cos \unicode[STIX]{x1D713})}{\sin \unicode[STIX]{x1D713}}.\end{eqnarray}$$

The derivation of (2.8) is shown in appendix A. To determine $\tilde{K}$, the boundary conditions for (2.1) and (2.8) are

(2.9a,b)$$\begin{eqnarray}r(0)=0,\quad \unicode[STIX]{x1D713}(0)=0\quad \text{and}\quad r(s_{1})=x,\quad z(s_{1})=y.\end{eqnarray}$$

In summary, there are two systems to be solved for the shapes of the ECTs. The first system consists of the initial-value problem: equations (2.5) with appropriate initial conditions, where $\tilde{K}(x,y;\unicode[STIX]{x1D703})$ is determined by solving the second system. The second system consists of the boundary-value problem: equations (2.1) and (2.8) with the boundary conditions (2.9).

The second system is solved here by the shooting method. Equation (2.1) is integrated first, using (2.9a) and a guessed value of $z(0)$ as initial conditions, until $r=x$. At this point, the boundary condition $z=y$ is generally not satisfied. Then, the value of $z(0)$ is adjusted by the secant method and this process is repeated until $z=y$ is satisfied to the desired accuracy. Thus, the appropriate value of $z(0)$ is found to accommodate the boundary conditions (2.9). Then, we integrate (2.1) and (2.8), using $(\unicode[STIX]{x1D713},r,z)=(0,0,z(0))$ and the value of $\unicode[STIX]{x1D713}_{r}$ at the point $(0,z(0))$ as initial conditions, to obtain the value of $\unicode[STIX]{x1D713}_{r}$ at $(x(s_{1}),y(s_{1}))$. Finally, $\tilde{K}(x,y;\unicode[STIX]{x1D703})$ is determined by (2.6b) and (2.7).

In the above computations, to avoid the singularity of (2.1) and (2.8) at $r=0$, the asymptotic solutions (2.3) are used for determining the initial conditions $(\unicode[STIX]{x1D713}^{\ast },r^{\ast },z^{\ast })$ at $s=r^{\ast }$ (due to small inclination angles), where the value of $\unicode[STIX]{x1D713}^{\ast }$ is chosen as $+0.01^{\circ }$. The initial condition for $\unicode[STIX]{x1D713}_{r}$ is determined by differentiating (2.3) with $r^{\ast }$ and $\unicode[STIX]{x1D713}^{\ast }$ replaced, respectively, by $r$ and $\unicode[STIX]{x1D713}$ with respect to $r$.

2.3 Relation to meniscus stability

As discussed in § 1, the unconstrained axisymmetric meniscus is less stable to axisymmetric perturbations than to non-axisymmetric perturbations (i.e. $\unicode[STIX]{x1D706}_{01}<\unicode[STIX]{x1D706}_{11}$). Thus, only axisymmetric perturbations are considered in this section. Because the meniscus is unconstrained, the meniscus stability is with regard to pressure disturbances. Following the method described by Slobozhanin & Alexander (Reference Slobozhanin and Alexander2003), the stability of the configuration in figure 1 is investigated by solving the Jacobi equation in the axisymmetric case:

(2.10)$$\begin{eqnarray}-\unicode[STIX]{x1D711}_{0}^{\prime \prime }-\frac{r^{\prime }}{r}\unicode[STIX]{x1D711}_{0}^{\prime }+a(s)\unicode[STIX]{x1D711}_{0}=0,\end{eqnarray}$$

with

(2.11)$$\begin{eqnarray}a(s)\equiv \cos \unicode[STIX]{x1D713}-K^{2}-\frac{\sin ^{2}\unicode[STIX]{x1D713}}{r^{2}},\end{eqnarray}$$

and the initial conditions

(2.12a,b)$$\begin{eqnarray}\unicode[STIX]{x1D711}_{0}(0)=1,\quad {\unicode[STIX]{x1D711}_{0}}^{\prime }(0)=0.\end{eqnarray}$$

The points letting $\unicode[STIX]{x1D711}_{0}(s_{c})=0$ are conjugate points on the boundaries of the regions $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$, i.e. the envelopes of the solution curves (see figure 2). In $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$, the boundary parameter of the solid (figure 1) is defined as

(2.13)$$\begin{eqnarray}\unicode[STIX]{x1D712}_{1}\equiv \frac{K(s_{1})\cos \unicode[STIX]{x1D703}-\tilde{K}}{\sin \unicode[STIX]{x1D703}},\end{eqnarray}$$

where $K(s_{1})$ is the curvature of the meniscus profile at which the meniscus meets the solid. The critical value of $\unicode[STIX]{x1D712}_{1}$ is calculated by

(2.14)$$\begin{eqnarray}\unicode[STIX]{x1D712}_{1}^{\ast }=-\frac{{\unicode[STIX]{x1D711}_{0}}^{\prime }}{\unicode[STIX]{x1D711}_{0}}.\end{eqnarray}$$

By comparing the boundary parameter $\unicode[STIX]{x1D712}_{1}$ and its critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$, the stability of menisci to pressure disturbances is determined: the equilibrium within the regions $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ is stable if $\unicode[STIX]{x1D712}_{1}>\unicode[STIX]{x1D712}_{1}^{\ast }$, and is unstable if $\unicode[STIX]{x1D712}_{1}<\unicode[STIX]{x1D712}_{1}^{\ast }$ (Slobozhanin & Alexander Reference Slobozhanin and Alexander2003). For a fixed meniscus shape, the meniscus stability is sensitive to the boundary conditions for the disturbance. For free disturbances (with a free contact line), we conclude from (2.13) that a convex solid ($\tilde{K}<0$) is relatively stable to a planar solid ($\tilde{K}=0$), which is relatively stable to a concave solid ($\tilde{K}>0$). For pinned disturbances (with a fixed contact line), the meniscus is in contact with a sharp solid edge (i.e. $\tilde{K}\rightarrow -\infty$) and therefore the meniscus has the boundary parameter $\unicode[STIX]{x1D712}_{1}\rightarrow \infty$. If the meniscus is in the maximal stability regions $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$, the meniscus with $\unicode[STIX]{x1D712}_{1}\rightarrow \infty$ is stable.

The ECT has the boundary parameter $\unicode[STIX]{x1D712}_{1}=\unicode[STIX]{x1D712}_{1}^{\ast }$ corresponding to each meniscus because of its ‘exotic’ property, as discussed in § 1. We have verified that $\unicode[STIX]{x1D712}_{1}={\unicode[STIX]{x1D712}_{1}}^{\ast }$ for exotic cylinders with $\unicode[STIX]{x1D700}=+1$ by numerically solving (2.10) in a previous study (Zhang & Zhou Reference Zhang and Zhou2020). The Jacobi equation (2.10) for $\unicode[STIX]{x1D700}=+1$ is numerically solved using the spectral parameter power series method (Kravchenko & Porter Reference Kravchenko and Porter2010), which expresses the general solution of the Sturm–Liouville equation as a spectral parameter power series.

Substituting (2.6) into (2.13), we obtain the boundary parameter of the ECT:

(2.15)$$\begin{eqnarray}\unicode[STIX]{x1D712}_{1}=\unicode[STIX]{x1D712}_{1}^{\ast }=\unicode[STIX]{x1D713}_{r}\sin \unicode[STIX]{x1D713}-\unicode[STIX]{x1D713}_{z}\cos \unicode[STIX]{x1D713},\end{eqnarray}$$

where $\unicode[STIX]{x1D713}_{r}$ and $\unicode[STIX]{x1D713}_{z}$ are determined as in § 2.2. This relation provides a convenient method for calculating the critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$ of the boundary parameter for a simply connected axisymmetric meniscus without solving the Jacobi equation (2.10). This method was proposed earlier for $\unicode[STIX]{x1D700}=+1$ in Zhang & Zhou (Reference Zhang and Zhou2020), where the parameter is the inclination angle $\unicode[STIX]{x1D713}$ instead of the arc length $s$ in this work.

There is also a rich geometry surrounding the critical value ${\unicode[STIX]{x1D712}_{1}}^{\ast }$, which is closely related to the curvature $\tilde{K}_{N}$ of the generatrix of the ECT with $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$:

(2.16)$$\begin{eqnarray}\unicode[STIX]{x1D712}_{1}^{\ast }=-\tilde{K}_{N},\end{eqnarray}$$

which can be easily derived by substituting $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ into (2.6b) or (2.13). From (2.13), we can see that the boundary parameter $\unicode[STIX]{x1D712}_{1}$ is related to the contact angle $\unicode[STIX]{x1D703}$, the curvature $K$ of the meniscus and the curvature $\tilde{K}$ of the solid. However, from (2.15) we have that the critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$ is independent of $\unicode[STIX]{x1D703}$ and can be determined uniquely by its location $(r,z)$ and $\unicode[STIX]{x1D700}$. Thus, if two different ECTs with different contact angles under the same loads have an intersection point $(r,z)$, their boundary parameters will be equal at the point $(r,z)$.

3.1 General ECTs

Wente (Reference Wente2011) investigated ECTs that permit a flat disc-shaped meniscus. These ECTs are proposed under the consideration that the ECTs are positioned in an infinite liquid bath under positive loads (i.e. the case of $\unicode[STIX]{x1D700}=+1$ in figure 1), and the ECT shapes are parts of the entire integral curves of $\{\cos (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}),\sin (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703})\}$ for $\unicode[STIX]{x1D700}=+1$. Following Wente (Reference Wente2011), we consider a non-uniform circular capillary tube (called a general ECT) with a specific shape that has constant pressure at the inlet and also permits an entire continuum of equilibrium menisci in itself. Thus, each integral curve for $\unicode[STIX]{x1D700}=+1$ and $\unicode[STIX]{x1D700}=-1$ corresponds to a general ECT.

Figure 3.

Integral curves of $\{\cos (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}),\sin (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703})\}$ (thick lines) and $\{\cos \unicode[STIX]{x1D713},\sin \unicode[STIX]{x1D713}\}$ (thin lines): (a) $\unicode[STIX]{x1D700}=+1$, $\unicode[STIX]{x1D703}=45^{\circ }$; (b) $\unicode[STIX]{x1D700}=+1$, $\unicode[STIX]{x1D703}=90^{\circ }$; (c) $\unicode[STIX]{x1D700}=-1$, $\unicode[STIX]{x1D703}=45^{\circ }$; (d) $\unicode[STIX]{x1D700}=-1$, $\unicode[STIX]{x1D703}=90^{\circ }$. There are five types of integral curves for $\unicode[STIX]{x1D700}=+1$ (a,b) and three types of integral curves for $\unicode[STIX]{x1D700}=-1$ (c,d). The regions of integral curves of different types are separated by their boundaries (blue dashed lines). Typical general ECT shapes (black solid lines) for (e) $\unicode[STIX]{x1D700}=+1$ and (f) $\unicode[STIX]{x1D700}=-1$ are depicted, which intersect the menisci (red dashed lines) at a constant angle $\unicode[STIX]{x1D703}$.

Figure 3(a–d) shows the integral curves (black thick lines) of the slope fields $\{\cos (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703}),\sin (\unicode[STIX]{x1D713}+\unicode[STIX]{x1D703})\}$ for $\unicode[STIX]{x1D700}=+1$, $-1$ and $\unicode[STIX]{x1D703}=45^{\circ }$, $90^{\circ }$. These integral curves intersect the menisci (thin lines) at a constant angle $\unicode[STIX]{x1D703}$ (i.e. the ‘exotic’ property). In figure 3(a,b), the integral curves for $\unicode[STIX]{x1D700}=+1$ and $\unicode[STIX]{x1D703}=45^{\circ }$, $90^{\circ }$ can be divided into five types depending on the number and locations of the endpoints: (i) one endpoint is on the positive half-axis of $z$ (region I); (ii) one endpoint is on the negative half-axis of $z$ (region II); (iii) two endpoints are on the negative half-axis of $z$ and the lower envelope (red thick line), respectively (region III); (iv) one endpoint is on an envelope (region IV); and (v) there is no endpoint (region V). The integral curves in II and V which have been studied by Wente (Reference Wente2011) intersect the $r$ axis and correspond to the ECTs that permit a flat disc-shaped meniscus. The integral curves in V which have no endpoint appear only when $\unicode[STIX]{x1D703}=90^{\circ }$. In I, III and IV, the integral curves do not intersect the $r$ axis and the corresponding ECTs are connected to a liquid which has a constant pressure at the inlet, rather than are vertically positioned in an infinite bath, to process the ‘exotic’ property (see figure 3e). Therefore, these general ECTs, extended from the ordinary ECTs studied by Wente (Reference Wente2011), can be seen as non-uniform circular nozzles connected to a liquid bath which has a constant pressure at the inlet. The general ECTs can also be constructed under negative loads ($\unicode[STIX]{x1D700}=-1$), as shown in figure 3(c,d,f). The integral curves for $\unicode[STIX]{x1D700}=-1$ and $\unicode[STIX]{x1D703}=45^{\circ }$, $90^{\circ }$ can be divided into three types depending on the number and locations of their endpoints: (i) two endpoints are on the positive half-axis of $z$ and the upper envelope, respectively (region $\text{I}^{-}$); (ii) two endpoints are on the negative half-axis of $z$ and the upper envelope, respectively (region $\text{II}^{-}$); and (iii) two endpoints are on two envelopes, respectively (region $\text{III}^{-}$). Only the curves in region $\text{I}^{-}$ do not intersect the $r$ axis. The above classifications for $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4$ can be extended to the case $\unicode[STIX]{x1D703}\in (0,\unicode[STIX]{x03C0}/2)$. The supplementary cases of $\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}$ can be obtained by reflecting the cases of $\unicode[STIX]{x1D703}$ about $z=0$.

In the study of Wente (Reference Wente2011), the ordinary ECTs (in II and V) have a surprising behaviour whereby lowering the ECT slightly from its natural position (as shown in figure 3e) causes the ECT to completely fill up, while raising the ECT slightly forces the ECT to drain out. Wente (Reference Wente2011) proved the above results by comparing the energies, $E(\unicode[STIX]{x1D6FA}_{1})$ and $E(\unicode[STIX]{x1D6FA})$, of the configurations with a meniscus $\unicode[STIX]{x1D6FA}_{1}$ (the profile of which is the integral curve of $\{\cos \unicode[STIX]{x1D713},\sin \unicode[STIX]{x1D713}\}$ and meets the ECT at a contact angle $\unicode[STIX]{x1D703}_{t}$) and an admissible meniscus $\unicode[STIX]{x1D6FA}$, where the ECT has a natural contact angle $\unicode[STIX]{x1D703}_{n}$. Here the contact angle $\unicode[STIX]{x1D703}_{t}$ denotes the intersection angle between the meniscus and the ECT and may not equal $\unicode[STIX]{x1D703}_{n}$. It was found that $E(\unicode[STIX]{x1D6FA}_{1})<E(\unicode[STIX]{x1D6FA})$ when $\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x1D703}_{n}$ and $\unicode[STIX]{x1D6FA}$ lies above $\unicode[STIX]{x1D6FA}_{1}$ (or when $\unicode[STIX]{x1D703}_{t}>\unicode[STIX]{x1D703}_{n}$ and $\unicode[STIX]{x1D6FA}$ lies below $\unicode[STIX]{x1D6FA}_{1}$). Wente (Reference Wente2011) also showed that if the ECT with contact angle $\unicode[STIX]{x1D703}_{n}$ is shifted upward then the new contact angle $\unicode[STIX]{x1D703}_{t}$ will satisfy $\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x1D703}_{n}$, and if the tube is shifted downward then we will have $\unicode[STIX]{x1D703}_{t}>\unicode[STIX]{x1D703}_{n}$. Therefore, when raising the ECT slightly, there is always a meniscus with $\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x1D703}_{n}$ and a lower energy below any admissible meniscus, which implies that the fluid is unable to remain inside the ECT and must flow out. An analogous situation that the fluid fills up the ECT occurs when lowering the ECT slightly.

Motivated by the above observations, the general ECTs also have a similar behaviour whereby lowering the pressure at the ECT inlet slightly from the value $p=-\unicode[STIX]{x1D700}z$ causes the ECT to completely drain out, while raising the pressure slightly forces the ECT to fill up. This is because raising and lowering the pressure are equivalent to lowering and raising the ECT for the case of $\unicode[STIX]{x1D700}=+1$, respectively. We note that the shifted ECT (e.g. in region III in figure 3a) may not locate in the extremal field (bounded by two envelopes and the $z$ axis) and the meniscus on the part not in the extremal field is always unstable and has a higher energy. Thus, the above results can be extended to the case of $\unicode[STIX]{x1D700}=+1$. For the case of $\unicode[STIX]{x1D700}=-1$, raising the pressure is equivalent to raising the ECT, while lowering the pressure corresponds to lowering the ECT. Contrary to the case of $\unicode[STIX]{x1D700}=+1$, raising the ECT with $\unicode[STIX]{x1D700}=-1$ will lead to $\unicode[STIX]{x1D703}_{t}>\unicode[STIX]{x1D703}_{n}$, and lowering the ECT corresponds to $\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x1D703}_{n}$. The conclusion that $E(\unicode[STIX]{x1D6FA}_{1})<E(\unicode[STIX]{x1D6FA})$ when $\unicode[STIX]{x1D703}_{t}<\unicode[STIX]{x1D703}_{n}$ and $\unicode[STIX]{x1D6FA}$ lies above $\unicode[STIX]{x1D6FA}_{1}$ (or when $\unicode[STIX]{x1D703}_{t}>\unicode[STIX]{x1D703}_{n}$ and $\unicode[STIX]{x1D6FA}$ lies below $\unicode[STIX]{x1D6FA}_{1}$) still holds true for the case of $\unicode[STIX]{x1D700}=-1$. Therefore, raising and lowering the pressure will lead to the filling and drainage in the ECT for both the cases $\unicode[STIX]{x1D700}=+1$ and $\unicode[STIX]{x1D700}=-1$, respectively. This surprising behaviour implies that the ECT is sensitive to pressure.

3.2 Critical values $\unicode[STIX]{x1D712}_{1}^{\ast }$ for $\unicode[STIX]{x1D700}=\pm 1$

Figure 4.

Contour lines (red lines) of $\unicode[STIX]{x1D712}_{1}^{\ast }$ for (a) $\unicode[STIX]{x1D700}=+1$ (see also figure 4 in Zhang & Zhou (Reference Zhang and Zhou2020)) and (b) $\unicode[STIX]{x1D700}=-1$. The black lines are the solution curves of the Young–Laplace equation. The values of $\unicode[STIX]{x1D712}_{1}^{\ast }$ for the contour lines along the arrow direction are given in each box from top to bottom.

Figure 4 shows the contour lines of $\unicode[STIX]{x1D712}_{1}^{\ast }$ (red lines) and the menisci (black lines). As discussed in § 2.3, the critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$ is related to the shape of the ECT with $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ by $\unicode[STIX]{x1D712}_{1}^{\ast }=-\tilde{K}_{N}$. Thus, the points on the line $\unicode[STIX]{x1D712}_{1}^{\ast }=0$ in figure 4(a) are the inflection points of the ECTs with $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ for $\unicode[STIX]{x1D700}=+1$, and there is no inflection point on the ECT with $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ for $\unicode[STIX]{x1D700}=-1$. The contour lines $\unicode[STIX]{x1D712}_{1}^{\ast }=+\infty$ are the envelopes of the solution curves because the envelopes can be seen as ECTs with $\unicode[STIX]{x1D703}=0$ which have the boundary parameters $\unicode[STIX]{x1D712}_{1}=+\infty$.

The critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$ can be used to determine the meniscus stability to pressure disturbances. For a simply connected axisymmetric meniscus with arbitrary contact angle $\unicode[STIX]{x1D703}$ wetting a solid, the boundary parameter $\unicode[STIX]{x1D712}_{1}$ is calculated by (2.13) and then is compared to its critical value $\unicode[STIX]{x1D712}_{1}^{\ast }$ (i.e. the value at the contact point in figure 4). It is shown that the equilibrium within regions $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ is stable if $\unicode[STIX]{x1D712}_{1}>\unicode[STIX]{x1D712}_{1}^{\ast }$, and is unstable if $\unicode[STIX]{x1D712}_{1}<\unicode[STIX]{x1D712}_{1}^{\ast }$ (Slobozhanin & Alexander Reference Slobozhanin and Alexander2003). It is noted that only the meniscus in the maximal stability regions $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ can be stable to pressure disturbances because the envelopes (i.e. the boundaries of $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$) correspond to $\unicode[STIX]{x1D712}_{1}^{\ast }=+\infty$.

To illustrate how to determine the meniscus stability by comparing $\unicode[STIX]{x1D712}_{1}$ and $\unicode[STIX]{x1D712}_{1}^{\ast }$, we examine the stability of the planar meniscus in a vertical straight circular tube with contact angle $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ and radius $R$ for both positive and negative loads. For this configuration, substituting $\tilde{K}=0$ and $K=0$ into (2.13) gives the boundary parameter $\unicode[STIX]{x1D712}_{1}=0$. By comparing $\unicode[STIX]{x1D712}_{1}$ and $\unicode[STIX]{x1D712}_{1}^{\ast }$ (see figure 4), it is found that the planar meniscus in the tube with an arbitrary radius $R$ is stable for positive loads and is unstable for negative loads.