2.1. Equilibrium shapes of pendant drops

Consider a pendant drop suspended from a plane as shown in figure 3, whose equilibrium interface is governed by the (dimensionless) Young–Laplace equation (see e.g. Padday Reference Padday1971; Del Rıo & Neumann Reference Del Rıo and Neumann1997):

(2.1a–d)\begin{equation} \left.\begin{gathered} \displaystyle{\frac{{{\rm{d}}r}}{{{\rm{d}}s}} = \cos \beta ,}\\ \displaystyle{\frac{{{\rm{d}}z}}{{{\rm{d}}s}} = \sin \beta ,}\\ \displaystyle{\frac{{{\rm{d}}\beta }}{{{\rm{d}}s}} ={-}Bo \times z - \frac{{\sin \beta }}{r} + \mu ,}\\ \displaystyle{\frac{{{\rm{d}}V}}{{{\rm{d}}s}} = {\rm \pi}{r^2}\sin \beta ,} \end{gathered}\right\} \end{equation}

where $(r(s),z(s))$ is a parametrization of the interface generatrix by its arc length $s$, $\beta$ is the inclination angle measured counterclockwise, $Bo\equiv \rho g {l_*}^{2}/{\sigma }$ is the Bond number with gravity $g$, surface tension $\sigma$ and the characteristic length $l_*$, $\mu$ is a Lagrange multiplier, and $V$ is the dimensionless volume. Note that (2.1d) is not necessary for defining the equilibrium, which is included to determine the drop volume (Del Rıo & Neumann Reference Del Rıo and Neumann1997). In the following, unless otherwise stated, the dimensionless drop volume is always set to $2{\rm \pi} /3$ (equal to the volume of a hemispherical drop of radius 1) to exclude the effect of drop volume. This helps us to focus on the effects of gravity. Thus the characteristic length is defined as $l_*=l_v\equiv (3 v / 2 {\rm \pi})^{1/3}$ based on the actual drop volume $v$. Accordingly, the Bond number is given as

(2.2)\begin{equation} Bo =\frac{\rho g }{\sigma}\left(\frac{3v}{2{\rm \pi}}\right)^{2 / 3}. \end{equation}

Figure 3. (a) Schematic diagram of a pendant drop suspended from a plane under gravity $g$, and (b) the rendered perturbed surface $\bar {\varGamma }$. For comparison, a gravity-free drop (dashed line) of the same volume is drawn.

The system of (2.1) generally has no closed-form solutions, and must be integrated numerically together with the initial conditions

(2.3)\begin{equation} \displaystyle{r(0) = z(0) = \beta (0) = V(0) = 0.} \end{equation}

To begin integration, the relation $\sin \beta /r = \mu /2$ is used to remove the singularity of (2.1c) at $r=0$ (Del Rıo & Neumann Reference Del Rıo and Neumann1997). For given $Bo$ and $\mu$, we can determine uniquely a solution curve $(r(s),z(s))$ and then obtain a drop shape by using the volume condition

(2.4)\begin{equation} V(s_{c})=2{\rm \pi}/3, \end{equation}

where $s_c$ denotes the arc length at the CL. However, we usually use the geometric condition (instead of unknown a priori $\mu$)

(2.5a,b)\begin{equation} \beta(s_c)=\alpha \quad \text{or} \quad r(s_c)=R \end{equation}

to determine a drop with given $Bo$. The boundary value problem (2.1) with (2.3)–(2.5) for the equilibrium shape of pendant drops can be solved by the shooting method (see e.g. Del Rıo & Neumann Reference Del Rıo and Neumann1997).

There may exist multiple equilibrium shapes for given parameters $Bo$ and $\alpha$ (or $R$), some of which are stable, while others are unstable and cannot exist stably (Pozrikidis Reference Pozrikidis2012). For given $Bo$ and $\alpha$, there may be two distinct stable pendant drops with pinned CLs (Pozrikidis Reference Pozrikidis2012), which will be discussed in § 2.2.1. This suggests that $(\alpha,Bo)$ cannot be used as control parameters for uniquely determining the base state with a pinned CL. This issue will be addressed in the next section.

2.2. Static stability

2.2.1. Axisymmetric perturbations

The static stability of pendant drops can be examined by solving a series of intractable eigenvalue problems (Myshkis et al. Reference Myshkis, Babskii, Kopachevskii, Slobozhanin and Tyuptsov1987, pp. 129–130). For axisymmetric perturbations, an alternative is the PM method (Maddocks Reference Maddocks1987) that determines stability from a family of equilibria. This method only requires solving the equilibrium problem (2.1) with (2.3)–(2.5) to obtain the diagram of volume $\tilde {V}$ against pressure $\tilde {P}$. The PM theory indicates the occurrence of instability along the counterclockwise direction through a TP (Lowry & Steen Reference Lowry and Steen1995). To draw the curve of $\tilde {V}$ versus $\tilde {P}$, here all drops are scaled by the capillary length $l_c=\sqrt {\sigma /{\rho g}}$ so that the volume is no longer fixed at $2{\rm \pi} /3$. Due to the different scalings used here and in § 2.1, the relations between dimensionless parameters are given as follows:

(2.6a–d)\begin{equation} \tilde{V}=\frac{2{\rm \pi}}{3}\,{Bo}^{3/2},\quad \tilde{P}=\frac{P}{\sqrt{Bo}} ,\quad \tilde{\alpha}=\alpha,\quad \tilde{R}=\sqrt{Bo}\,R, \end{equation}

where quantities with and without tildes correspond to the characteristic lengths of $l_c$ and $l_v$, respectively. Tildes of $\alpha$ are omitted hereafter for simplicity.

Using the capillary length $l_c$ as the characteristic length, the Young–Laplace equation (2.1) will change to a different dimensionless form (e.g. Sumesh & Govindarajan Reference Sumesh and Govindarajan2010; Zhang & Zhou Reference Zhang and Zhou2020):

(2.7a–d)\begin{equation} \frac{{{\rm{d}}\tilde{r}}}{{{\rm{d}}\tilde{s}}} = \cos \tilde{\beta} ,\quad \frac{{{\rm{d}}\tilde{z}}}{{{\rm{d}}\tilde{s}}} = \sin \tilde{\beta},\quad \frac{{{\rm{d}}\tilde{\beta} }}{{{\rm{d}}\tilde{s}}} ={-} \tilde{z} - \frac{{\sin \tilde{\beta} }}{\tilde{r}} + \tilde{\mu},\quad \frac{{{\rm{d}}\tilde{V}}}{{{\rm{d}}\tilde{s}}} = {\rm \pi}{\tilde{r}^2}\sin \tilde{\beta}, \end{equation}

where variables with tildes are dimensionless counterparts of variables in (2.1). With a variable $\tilde {\mu }$, (2.7) has a family of solutions (see the grey lines in figures 4a,d) for the initial conditions $\tilde {r}(0)=\tilde {z}(0)=\tilde {\beta }(0)=\tilde {V}(0)=0$. In this case, one can easily impose a fixed contact angle condition $\tilde \beta (\tilde {s}_c)=\alpha$ (free CL) or a fixed CL boundary condition $\tilde r(\tilde {s}_c)=\tilde R$ (pinned CL).

Figure 4. (a,d) The family of solution curves (grey lines) for the Young–Laplace equation (2.7), where five typical solution curves (black solid lines) are plotted for $\tilde {\mu }=1, 2, 3, 4, 5$. The red and blue lines denote the boundary conditions (a) $\alpha =70^\circ$ and (d) $\tilde {R}=1$, respectively, where the arrows indicate the direction of instability. (b,c,e–h) The curves of volume $\tilde {V}$ versus $\tilde {\mu }$, pressure $\tilde {P}$ versus $\tilde {\mu }$, and $\tilde {V}$ versus $\tilde {P}$ for (b,c,g) $\alpha =70^\circ$ and (e,f,h) $\tilde {R}=1$. For $\alpha =70^\circ$, the solid and dashed lines correspond to the first and second curves of CL points (yellow dots), respectively. The inset in (g) compares our results (red line) with those of Sumesh & Govindarajan (Reference Sumesh and Govindarajan2010) ($\bigcirc$) for $\tilde {V}/{\rm \pi}$ versus $H$ with $\alpha =70^\circ$. The inset in (h) shows a pendant drop scaled by $l_c$, where $H$ is the drop height.

Figures 4(a) and 4(d) show the curves of CL points for the boundary conditions $\alpha =70^\circ$ and $\tilde {R}=1$, respectively. Each point on the CL curves determines a pendant drop with a given $\alpha$ or $\tilde {R}$, whose shape is the portion of the corresponding solution curve with a specific $\tilde {\mu }$ (see the green lines). For $\alpha =70^\circ$, there can be multiple CL points (see the yellow dots) for one solution curve (e.g. with $\tilde {\mu }=5$). The first and second CL points belong to the first CL curve, while the third and fourth points belong to the second CL curve. Accordingly, we can plot the curves of volume $\tilde {V}$ versus $\tilde {\mu }$, and pressure $\tilde {P}$ versus $\tilde {\mu }$, as shown in figures 4(b,c,e,f). Additionally, we compare our results with those of Sumesh & Govindarajan (Reference Sumesh and Govindarajan2010) for $\tilde {V}/{\rm \pi}$ versus drop height $H$, and find excellent agreement, as shown in the inset of figure 4(g). Here, the pressure $\tilde {P}$ is the pressure of the liquid on the wall, and $\tilde {\mu }$ is the pressure at the drop apex (see the inset of figure 4h), so the pressure $\tilde {P}$ required by the PM theory is given by $\tilde {P}=\tilde {\mu }-H$. From figures 4(b,e), we can see that there are multiple equilibria for certain volumes $\tilde {V}$, consistent with the results of Sumesh & Govindarajan (Reference Sumesh and Govindarajan2010). Although we are only concerned with stable equilibrium, however, there may be two stable equilibria for pinned drops (Pozrikidis Reference Pozrikidis2012). Multiple stable equilibria render it difficult to choose the control parameters to determine the pendant drop uniquely. Fortunately, by choosing the appropriate parameter space, only one of these equilibria is stable, which is related to the PM theory (Maddocks Reference Maddocks1987).

Finally, the curves of $\tilde {V}$–$\tilde {P}$ can be drawn (see figures 4(g,h) for $\alpha =70^\circ$ and $\tilde {R}=1$). For fixed contact angle, the drops corresponding to the second $\tilde {V}$–$\tilde {P}$ curve (see figure 4g) are always unstable to both volume and pressure disturbances due to their multiple inflection points ($\textrm {{d}}\beta / \textrm {{d}} s=0$) (Wente Reference Wente1980; Sumesh & Govindarajan Reference Sumesh and Govindarajan2010). For simplicity in discussing stability, parts of the $\tilde {V}$–$\tilde {P}$ curves in figures 4(g,h) are reproduced in figure 5. Figure 5(a) shows the $\tilde {V}$–$\tilde {P}$ curves for drops with free CLs and $\alpha =70^{\circ }$, where the horizontal and vertical TPs correspond to the stability changes to volume and pressure disturbances, respectively. Therefore, for volume disturbances, the equilibria are $v$-stable on the right-hand side of the horizontal TP, and $v$-unstable on the left-hand side, as indicated by the PM theory. For pressure disturbances, all pendant drops with free CLs are $p$-unstable in the same way as gravity-free drops, while pendant drops can be $p$-stable due to the stabilisation of CL pinning. Figure 5(b) shows the results for pinned CLs and $\tilde {R}=1$. Pressure instability occurs at the first vertical TP (the right-hand one), and then there is a volume stability change through the horizontal TP. We recall that the gravity-free drops are always $v$-stable due to the absence of a horizontal TP (see figure 2). In contrast, pendant drops can be $v$-unstable under the destabilising effect of gravity. The stabilisation of CL pinning delays the onset of volume instability. For example, the drop $\tilde {A}$ with a free CL is $v$-unstable, while the drop $\tilde {D}$ with a pinned CL (having the same shape as the drop $\tilde {A}$) is $v$-stable.

Figure 5. Volume–pressure ($\tilde {V}$–$\tilde {P}$) diagrams of pendant drops with (a) free CLs and $\alpha =70^{\circ }$, and (b) pinned CLs and $\tilde {R}=1$. Instability occurs at the TPs (hollow circles) along the counterclockwise direction. In (a), drops $\tilde {A}$ and $\tilde {B}$ are $v$-unstable and $v$-stable, respectively, but both are $p$-unstable. In (b), the drop $\tilde {D}$ with the same shape as $\tilde {A}$ ($v$-unstable), however, is $v$-stable due to the stabilising effect of the pinned CL. All gravity-free drops are $v$-stable. Here, the characteristic length $l_*$ is the capillary length $l_c=\sqrt {\sigma /{\rho g}}$.

Pozrikidis (Reference Pozrikidis2012) has demonstrated that there may exist two distinct $v$-stable drops with pinned CLs for given $\alpha$ and $\tilde {V}$ (e.g. pinned drops with the same shape as $\tilde {A}$ and $\tilde {B}$, denoted by $\tilde {A}_p$ and $\tilde {B}_p$, respectively). Note that though the drop $\tilde {A}$ is $v$-unstable, the drop $\tilde {A}_p$ with pinned CL (i.e. $\tilde {D}$) is $v$-stable due to the stabilisation of CL pinning (see figure 5b). Therefore, the distinct drops $\tilde {A}_p$ and $\tilde {B}_p$ are both $v$-stable and have the same contact angle $\alpha$ and volume $\tilde {V}$ (or $Bo$). This implies that the parameter pair $(\alpha,Bo)$ cannot define uniquely the base state with a pinned CL. Knowing that only one branch of solutions in the fold bifurcation is stable (Maddocks Reference Maddocks1987), this indicates that the parameter pair $(R,Bo)$ can determine uniquely a stable drop with pinned CL. Thus in the following, the base states (stable) with free and pinned CLs for dynamic stability are governed by the control parameters $(\alpha,Bo)$ and $(R,Bo)$, respectively.

To summarize briefly, the PM theory states that the pendant drop is $v$-stable ($p$-stable) to axisymmetric perturbations before its volume (pressure) reaches a maximum. In contrast to sessile drops, pendant drops can be $v$-unstable destabilised by gravity. Note that static stability is maintained for different scalings, thus these conclusions hold for drops with fixed volume in § 2.1.

2.2.2. Non-axisymmetric perturbations

The PM method described above deals with only static stability for axisymmetric perturbations (Bostwick & Steen Reference Bostwick and Steen2015). Fortunately, for non-axisymmetric perturbations, the static stability of a pendant drop can be determined by comparing the boundary parameter $\chi$ with the critical boundary value $\chi _{1,1}^{*}$ (Myshkis et al. Reference Myshkis, Babskii, Kopachevskii, Slobozhanin and Tyuptsov1987; Zhang & Zhou Reference Zhang and Zhou2020), referred to as the boundary parameter comparison approach. The critical boundary value $\chi _{1,1}^{*}$ corresponds to the $\{1,1\}$ mode, which is the most dangerous among non-axisymmetric perturbations, given by (see Myshkis et al. Reference Myshkis, Babskii, Kopachevskii, Slobozhanin and Tyuptsov1987, p. 141)

(2.8)\begin{equation} \chi_{1,1}^{*}={-}{ \frac{r'(s_c)}{z'(s_c)}\,K_1(s_c) } ={-}{{K_1(s_c)}\cot \alpha }, \end{equation}

where the prime denotes the derivative with respect to arc length $s$, $s_c$ is the arc length at the CL, and $K_1=\textrm {d}\beta /\textrm {d}s$ is the signed curvature of the drop profile. The boundary parameter $\chi$ measures the degree of CL pinning, expressed in terms of the included angle ${\rm \pi} -\alpha$ between $\boldsymbol { n}$ and $\boldsymbol {n}_s$ (figure 3a), and the liquid $K_1$ and solid $\bar {K}$ curvatures at the CL, as

(2.9)\begin{equation} \chi = {\frac{{{K_1}(s_c)\cos ({\rm \pi}-\alpha) - \bar{K}}}{{\sin ({\rm \pi}-\alpha) }}}, \end{equation}

where the curvature of the solid surface is $\bar {K}<0$ if the solid is convex to the liquid, $\bar {K}=0$ for a flat solid surface, and $\bar {K}>0$ otherwise (Myshkis et al. Reference Myshkis, Babskii, Kopachevskii, Slobozhanin and Tyuptsov1987).

We determine the stability as follows: the drop is stable if $\chi > \chi _{1,1}^{*}$, is marginally stable (i.e. with a zero eigenvalue) when $\chi = \chi _{1,1}^{*}$, and otherwise unstable. Note that all non-axisymmetric perturbations automatically satisfy volume conservation, so volume and pressure disturbances are not distinguishable in this case. We have the solid curvature $\bar {K}=0$ for free CLs and $\bar {K}=-\infty$ for pinned CLs, since the pinned CL can be regarded as a CL at a convex edge with an infinitesimal curvature radius (Myshkis et al. Reference Myshkis, Babskii, Kopachevskii, Slobozhanin and Tyuptsov1987). Comparing (2.9) and (2.8) yields

(2.10)\begin{equation} \chi =\chi_{1,1}^{*}={-}{{K_1(s_c)}\cot \alpha } \end{equation}

for free CLs, and

(2.11)\begin{equation} \chi ={+}\infty >\chi_{1,1}^{*} \end{equation}

for pinned CLs. Thus for non-axisymmetric perturbations, pendant drops with pinned CLs are stable, while drops with free CLs are marginally stable, leading to a zero frequency mode $\{1,1\}$ (see § 3.4). This is consistent with the physical intuition that drops with free CLs on a plane have horizontal translational invariance (Zhang et al. Reference Zhang, Zhou and Ding2023).

2.3. Dynamic stability

2.3.1. Eigenvalue problems for natural oscillations

We restrict ourselves to the dynamic stability of stable drops with a volume constraint. In this case, stable drops exhibit natural oscillations, which reflect their dynamic stability. To study natural oscillations of pendant drops (figure 3a), we need to solve the following functional eigenvalue problem (Myshkis et al. Reference Myshkis, Babskii, Kopachevskii, Slobozhanin and Tyuptsov1987; Zhang et al. Reference Zhang, Zhou and Ding2023):

(2.12a)\begin{gather} \frac{1}{r}\,\frac{\partial }{{\partial r}}\left( {r\,\frac{{\partial \phi }}{{\partial r}}} \right) + \frac{{{\partial ^2}\phi }}{{\partial {z^2}}} - \frac{{{l^2}}}{{{r^2}}}\,\phi = 0\quad [D], \end{gather}

(2.12b)\begin{gather}\frac{{\partial \phi }}{{\partial n}} = 0\quad [\partial {D^s}], \end{gather}

(2.12c)\begin{gather}{\left( {\frac{{\partial \phi }}{{\partial n}}} \right)^{\prime \prime }} + \frac{{r'}}{r}{\left( {\frac{{\partial \phi }}{{\partial n}}} \right)^\prime } + \left[ {Bo \times r' + \left( {{K_1}^2 + {K_2}^2} \right) - \frac{{{l^2}}}{{{r^2}}}} \right]\frac{{\partial \phi }}{{\partial n}} ={-} {\lambda ^2}\phi \quad [\partial {D^f}], \end{gather}

(2.12d)\begin{gather}\int_{\varGamma} {\frac{{\partial \phi }}{{\partial n}}\,{\rm{d}}\varGamma } = 0\quad [\partial {D^f}], \end{gather}

(2.12e)\begin{gather}{\left. {{{\left( {\frac{{\partial \phi }}{{\partial n}}} \right)}^\prime } + \chi\,\frac{{\partial \phi }}{{\partial n}} = 0} \right|_{s = {s_c}}}\quad [\gamma ], \end{gather}

where $\phi$ and ${\partial \phi }/{\partial n}$ are the potential function and its normal derivative, respectively, $\lambda$ is the dimensionless frequency (scaled by the characteristic time ${t_ * } = \sqrt {\rho {l_ *^3}/\sigma }$ with $l_ *=l_v$), and $K_1=\textrm {d}\beta /\textrm {d}s$ and $K_2=\sin \beta /r$ are the two principle curvatures of the drop surface.

Substituting (2.10) and (2.11) into (2.12e), we obtain the free CL condition

(2.13)\begin{equation} {\left. {{{\left( {\frac{{\partial \phi }}{{\partial n}}} \right)}^\prime } - {K_1}\cot \alpha\, \frac{{\partial \phi }}{{\partial n}} = 0} \right|_{s = {s_c}}}, \end{equation}

and the pinned CL condition

(2.14)\begin{equation} {\left. {\frac{{\partial \phi }}{{\partial n}} = 0} \right|_{s = {s_c}}}, \end{equation}

respectively.

The eigenvalue problem (2.12) is derived from potential flow theory and linearised governing equations on the boundary via the normal mode decomposition and separation of variables method (see e.g. Bostwick & Steen Reference Bostwick and Steen2014; Zhang et al. Reference Zhang, Zhou and Ding2023). Equation (2.12a) is Laplace's equation written in cylindrical coordinates $(r,z)$, (2.12b) is the no-penetration condition on the solid surface, (2.12d) is the condition of volume conservation, (2.12c) is the free-surface governing equation derived from the kinematic condition and the dynamic pressure balance, and (2.12e) is the CL condition. The solutions of the eigenvalue problem (2.12) give the dimensionless frequencies $\lambda$ and mode shapes ${\partial \phi }/{\partial n}$. Since the frequency squared $\lambda ^2$ is inversely proportional to the drop volume (Noblin et al. Reference Noblin, Buguin and Brochard-Wyart2004), we adopt a characteristic length $l_v=(3 v / 2 {\rm \pi})^{1/3}$, fixing the drop volume at $2{\rm \pi} /3$ to exclude the volume effect. Therefore, the real frequency $f \equiv {\lambda } /({2{\rm \pi} t_*})$ (in Hz) is given as

(2.15) \begin{equation} f^{2}=\Biggl(\frac{\rho}{\sigma}\Biggr)^{1 / 2}\Biggl(\frac{g}{B o}\Biggr)^{3 / 2}\Biggl(\frac{\lambda}{2 {\rm \pi}}\Biggr)^{2}=\frac{\sigma}{6 {\rm \pi}\rho {v}}\,\lambda^{2}. \end{equation}

2.3.2. Boundary element model and mode classification

Theoretical methods (e.g. Lyubimov et al. Reference Lyubimov, Lyubimova and Shklyaev2006; Bostwick & Steen Reference Bostwick and Steen2014; Sharma & Wilson Reference Sharma and Wilson2021) for solving the eigenvalue problem (2.12) usually require drops to be hemispheres or spherical caps. The BE model developed in Zhang et al. (Reference Zhang, Zhou and Ding2023) can deal with drops of arbitrary shape and has been applied successfully to gravity-flattened sessile drops, so it is also applicable to our problem. We adopt the BE model to solve numerically the eigenvalue problem (2.12). The fundamental procedures for constructing the BE model can be found in Appendix A. One can refer to Zhang et al. (Reference Zhang, Zhou and Ding2023) for more details on the formulation of the BE model.

The solutions of the BE model give the natural frequencies and mode shapes. Analogous to spherical harmonics, these modes can be categorised into zonal $[k,l=0]$, sectoral $[k,l=k>0]$ and tesseral $[k>l,l>0]$ modes according to polar $k$ and azimuthal $l$ wavenumbers (Bostwick & Steen Reference Bostwick and Steen2014). In experiments (Chang et al. Reference Chang, Bostwick, Steen and Daniel2013, Reference Chang, Bostwick, Daniel and Steen2015), another more intuitive layer–sector classification $\{n,l\}$ is usually used for identifying the number $n=(k-l)/2+1$ of vertical layers, so that zonal, sectoral and tesseral modes are labelled as $\{n\geqslant 2,l=0\}$, $\{n= 1,l>0\}$ and $\{n> 1,l>0\}$, respectively. In terms of mode shape, the zonal mode is axisymmetric, the sectoral mode is star-shaped, and the tesseral mode has a complex wave pattern, as shown in figure 6. One of the rocking $l=1$ modes with one sector rocking from side to side is also drawn, and the rocking modes are particularly easy to identify in experiments. Besides, the directional movement of drops on a vibrating plate is attributed to the rocking modes (Ding et al. Reference Ding, Zhu, Gao and Lu2018; Costalonga & Brunet Reference Costalonga and Brunet2020).

Figure 6. Typical mode shapes for a pendant drop with $Bo=1.4$, $\alpha =90^{\circ }$ and $R=0.7$ in (a) cross-sectional and three-dimensional (c,e) oblique and (b,d) bottom views. The modes with free and pinned CLs are indicated by the red and green lines, respectively.

3.1. Verification

The BE model has been well validated for gravity-free drops and gravity-flattened sessile drops in Zhang et al. (Reference Zhang, Zhou and Ding2023). The natural frequencies of pendant drops are verified here. Figure 7 compares the experimental results of DePaoli, Scott & Basaran (Reference DePaoli, Scott and Basaran1992) with the inviscid results of the BE model for zonal mode $\{2,0\}$ with pinned CL. It is shown that for CL diameter $D= 0.16$ cm, the BE results of small drops are overpredicted, while the results of large drops agree well. This is due to the great viscous effect of small drops, which significantly reduces the resonant frequency (Lyubimov et al. Reference Lyubimov, Lyubimova and Shklyaev2006; Chang et al. Reference Chang, Bostwick, Daniel and Steen2015), while the viscous effect is negligible for large drops with low Ohnesorge numbers. However, for the CL 0.068 cm, our predictions are underestimated for large drops. A reasonable explanation is that the drop volume might be measured inaccurately in the experiment, because volume values of large drops in experimental data exceed the volume of the maximum drop (yellow dot), indicating that these drops cannot exist stably.

Figure 7. Frequency of zonal mode $\{2,0\}$ with pinned CL against equivalent radius ${\tilde {r}}= ({3{v}}/{4{\rm \pi} })^{1/3}$. The numerical results of the BE model (solid lines) are compared with the experimental results of DePaoli et al. (Reference DePaoli, Scott and Basaran1992) (dots) for CL diameters $D=0.068$ cm (red) and $0.16$ cm (black). The BE results without gravity (dashed lines) are drawn for comparison. For $D=0.068$ cm, the maximum drop (yellow dot) corresponds to the horizontal TP (the right-hand inset; see also figure 5). The physical parameters used here are $\rho =997\ \textrm {{kg}}\ \textrm {{m}}^{-3}$, $\sigma =0.072\ \textrm {{N}}\ \textrm {m}^{-1}$ and $g=9.81\ \textrm {m}\ \textrm {s}^{-2}$. A sharp fall in frequency is highlighted by the cyan region.

Interestingly, our results show that there is a sharp fall in frequency to zero near the maximum drop point. This suggests that the frequency of pendant drops near the maximum drop point is highly sensitive to the drop volume. Furthermore, the numerical results show that the zonal mode $\{2,0\}$ of the maximum drop has zero frequency. This is consistent with the maximum drop being marginally stable indicated by the PM theory (see the inset in figure 7), further validating the accuracy of the BE model.

3.2. Stability limits: maximum $Bo$

As shown above, the maximum drop (with the maximum Bond number $Bo_{max}$) determines the stability limit. Figures 8(a,b) plot the maximum Bond number $Bo_{max}$ against contact angle $\alpha$ and CL radius $R$, respectively, for free and pinned CLs. Results show that for free CLs, $Bo_{max}$ decreases with increasing $\alpha$, consistent with the physical intuition that more hydrophilic surfaces can suspend larger drops. For pinned CLs, however, $Bo_{max}$ increases and then decreases with increasing $R$ (figure 8b). At $R=R^*=1.5445$, $Bo_{max}$ reaches its maximum, and the corresponding drop (labelled $G$) has a zero contact angle, which has the same shape as the maximum drop $A$ (with $\alpha =0^{\circ }$) for free CLs. This is due to the indistinguishability of the free and pinned CLs for $\alpha =0^{\circ }$, both having the boundary parameter $\chi =+ \infty$. For $R>R^*$, the marginally $v$-stable drops with $\alpha <0^{\circ }$ are non-physical (see the drop ${\bar {H}}$) so that $Bo_{max}$ corresponds to $v$-stable drops with $\alpha =0^{\circ }$ (e.g. ${H}$). Thus the static stability limits are determined by the maximum Bond numbers $Bo_{max}$ corresponding to marginally $v$-stable drops, except for $R>R^*$. The curves of $Bo_{max}$ versus $\alpha$, and $Bo_{max}$ versus $R$, divide the parameter spaces into regions where equilibrium exists and regions where it does not, labelled as equilibrium and non-equilibrium, respectively. In the equilibrium region, multiple equilibria exist and the only $v$-stable equilibrium can be determined through the PM theory, as shown in figures 8(c–f). We are concerned only with the dynamic stability of $v$-stable equilibria in the following.

Figure 8. (a,b) Maximum Bond number $Bo_{max}$ versus (a) contact angle $\alpha$ for free CLs, and (b) CL radius $R$ for pinned CLs. The black solid and black dashed lines correspond to marginally $v$-stable drops with $\alpha \geqslant 0^{\circ }$ and $\alpha < 0^{\circ }$, respectively. In (b), the red solid line corresponds to drops with $\alpha = 0^{\circ }$. (c–f) The curves of $Bo$ versus $\tilde {P}$ for (c) drop $A$ with $\alpha =0^\circ$, (d) drop $B$ with $\alpha =50^\circ$, (e) drop $F$ with $R=1$, and ( f) drop $\bar {H}$ with $R=2$. The maximum Bond numbers $Bo_{max}$ correspond to the horizontal TPs of the $\tilde {P}$–$\tilde {V}$ curves due to $Bo=(3\tilde {V}/2{\rm \pi} )^{2/3}$ (see (2.6a–d)).

Figure 9(a) plots the frequency $\lambda _{2,0}$ of zonal mode $\{2,0\}$ against $Bo$ for free CLs with four different contact angles. For contact angles $\alpha =50^{\circ }$, $100^{\circ }$ and $150^{\circ }$, the frequencies decrease with increasing $Bo$, consistent with the numerical results of DePaoli et al. (Reference DePaoli, Scott and Basaran1992). However, for a very small contact angle $\alpha =0^{\circ }$, the frequency initially increases and then decreases. The initial increase of frequency seems counterintuitive, because gravity reduces the restoring force and makes the pendant drop slender, both of which decrease frequency in most cases (Basaran Reference Basaran1992; Ibrahim Reference Ibrahim2005; Temperton et al. Reference Temperton, Smith and Sharp2015). This suggests that the deformation of pendant drops with small $\alpha$ increases the frequency at low gravity. For pinned CLs, the frequencies always decrease with increasing $Bo$ regardless of the CL radius $R$, as shown in figure 9(b). We also observe zero frequency modes $\{2,0\}$ at $Bo_{max}$, except for the non-marginally stable drop $H$. Note that although the numerical results of $\lambda _{2,0}$ cannot be exactly zero, one can verify our results converging to zero by grid refinement, as will be discussed in § 3.4. Therefore, the static stability of pendant drops can be recovered by letting $\lambda _{2,0}=0$ in the dynamic stability, and the critical conditions for static and dynamic stability are essentially the same.

Figure 9. Frequency of zonal mode $\{2, 0\}$ against the Bond number $Bo$ for (a) free and (b) pinned CLs. The eight points $A$–$H$ correspond to the maximum Bond number $Bo_{max}$ (i.e. drops $A$–$H$ in figure 8). The cyan region highlights a sharp fall in frequency.

Furthermore, we observe sharp frequency falls when approaching $Bo_{max}$, except for the drop $H$. The sharp fall in $\lambda _{2,0}$ may be explained by the fact that the base state near $Bo_{max}$ (see the horizontal TP in figure 5) is sensitive to volume or gravity (i.e. a small change in volume leads to a large change in pressure). When approaching $Bo_{max}$, other modes are also sensitive to $Bo$, leading to similar sharp changes in frequency (see figure 13). We note that the sharp change in frequency with respect to wavenumber $k$ is also observed for the rotating Rayleigh–Plateau instability near stability thresholds (Dubey, Roy & Subramanian Reference Dubey, Roy and Subramanian2022). The sharp frequency changes may lead to inaccurate theoretical predictions near $Bo_{max}$, affecting relevant applications such as surface tension measurements via oscillation (Mollot et al. Reference Mollot, Tsamopoulos, Chen and Ashgriz1993; Chang & Franses Reference Chang and Franses1994). Therefore, for better predictions, experiments on pendant drops that are close to marginally $v$-stable should be avoided.

3.3. Frequency shifts due to gravity

To measure how gravity affects the frequency spectrum, we define the shift factor $S_{n,l}$ of frequency for each mode $\{n,l\}$ as follows:

(3.1a)\begin{gather} S^f_{n,l}(\alpha,Bo)=\frac{\lambda^f_{n,l}(\alpha,Bo)}{\lambda^f _{n,l}(\alpha,0)}-1,\quad \text{free CLs}, \end{gather}

(3.1b)\begin{gather}S^p_{n,l}(R,Bo)=\frac{\lambda^p_{n,l}(R,Bo)}{\lambda^p _{n,l}(R,0)}-1,\quad {\text{pinned CLs}}. \end{gather}

The value of $S_{n,l}$ denotes the relative change of frequency and $S_{n,l}>0$ (${<}0$) indicates an upward (downward) shift of frequency due to gravity. In (3.1), the control parameters are chosen as $(\alpha,Bo)$ and $(R,Bo)$ for free and pinned CLs, respectively, to determine the base state uniquely (see § 2.2.1).

Figure 10 shows five typical phase diagrams of how gravity shifts the natural frequencies of pendant drops. For free CLs, there are three types (I, II and T) of contour diagrams of $S^f_{n,l}$, as shown in figures 10(a–c). For type I, there is a longitudinally inclined critical line $S^f_{n,l}=0$, a region $S^f_{n,l}>0$ (red) on the left-hand side of the critical line, and a region $S^f_{n,l}<0$ (blue) on the right-hand side. This means that the frequency of the mode of type I with small (large) contact angles increases (decreases) under gravity. For type II, we observe $S^f_{n,l}>0$ regardless of $\alpha$ and $Bo$, which implies that gravity always increases the frequency. The diagram of type T presents a complex pattern of gravity affecting frequency, which is a transition between types I and II. The three patterns are generally opposite to those of sessile drops (Zhang et al. Reference Zhang, Zhou and Ding2023), as the direction of gravity is reversed for pendant drops and sessile drops. For pinned CLs, there are two types (A and B) of diagrams of $S^p_{n,l}$ (figures 10d,e). The diagram of type A indicates that gravity decreases (increases) the frequency for modes with small (large) CL radii $R$ (figure 10d). For type B, gravity always decreases the frequency (figure 10e).

Figure 10. Five typical contour diagrams of shift factor $S_{n,l}$. (a–c) Types I, T and II correspond to modes $\{10,0\}$, $\{6,9\}$ and $\{1,2\}$ with free CLs, respectively. (d,e) Types A and B correspond to modes $\{1,10\}$ and $\{2,0\}$ with pinned CLs, respectively. The black dashed and red dashed lines indicate $Bo_{max}$ (see figure 8) and $S_{n,l}=0$, respectively.

Figure 11 presents the classifications of the $S_{n,l}$ diagrams according to the mode numbers $\{n,l\}$. For free CLs, modes with larger $n$ ($l$) are more likely to be type I (II), and the remaining modes in the middle are type T. As a consequence, all zonal modes are of type I, and most sectoral modes are of type II. For pinned CLs, the modes with $l>n$ ($l< n$) are of type A (B), and modes $\{n,l=n\}$ are possible for both. Therefore, all zonal and sectoral modes with pinned CLs are of type B and A, respectively. The above findings provide an overview of gravity affecting natural frequencies for pendant drops, and complement the study of the effects of gravity on partially wetting drops (Zhang et al. Reference Zhang, Zhou and Ding2023).

Figure 11. Types of contour diagrams of $S_{n,l}$ for modes with (a) free and (b) pinned CLs, where $n=1,2,\ldots,10$ and $l=0,1,\ldots,10$.

The above findings show that the frequency shifts due to gravity compared to the gravity-free case are complex, depending on the modal pair $\{n,l\}$, the Bond number $Bo$, and the contact angle $\alpha$ or CL radius $R$. Intuitively, gravity is supposed to decrease the frequency of pendant drops due to gravity acting as a destabilising force, in agreement with the experimental and numerical observations (DePaoli et al. Reference DePaoli, Scott and Basaran1992; Basaran & DePaoli Reference Basaran and DePaoli1994). However, gravity also distorts the base state (elongated) and thus affects the frequency indirectly. Since the vibrations of sessile/pendant drops might be analogous to liquid sloshing in an upright cylindrical container (Noblin, Buguin & Brochard-Wyart Reference Noblin, Buguin and Brochard-Wyart2009; Temperton et al. Reference Temperton, Smith and Sharp2015), the drop elongation can be characterised by an increase in mean height of the liquid in the cylinder, which increases the frequencies, while gravity as a destabilising force always decreases the frequencies (Temperton et al. Reference Temperton, Smith and Sharp2015). Therefore, the competing effects of the reduced restoring force and base state elongation due to gravity lead to the above complex dependence of the frequencies on gravity.

3.4. Zero frequency modes $\{2,0\}$ and $\{1,1\}$

Two modes of pendant drops can have zero frequency: the zonal $\{2, 0\}$ and Noether $\{1, 1\}$ modes. The results of $\lambda _{1,1}=0$ for free CLs are not surprising, because the Noether mode $\{1, 1\}$ reflects the horizontal displacement of the drop and has zero frequency according to Noether's theorem when its base state has the horizontal translational invariance. Bostwick & Steen (Reference Bostwick and Steen2014) noticed that the symmetry of translational invariance can be broken by CL pinning or changing the wetting parameter, resulting in a non-zero frequency of the Noether mode. However, varying contact angle or exerting gravity does not break this symmetry, so that gravity-flattened sessile drops with free CLs always have the zero frequency mode $\{1, 1\}$ (Zhang et al. Reference Zhang, Zhou and Ding2023).

Obviously, pendant drops with free CLs on a plane also have horizontal translational invariance, indicating that their Noether mode $\{1,1\}$ has zero frequency regardless of $\alpha$ and $Bo$. For verification purposes, figure 12 plots the BE results of $|\lambda _{1,1}^2|$ against the inverse $1/N$ of the grid number in log-log scale. Here, the grid number $N$ is the number of BEs of the liquid surface (see figure 15). The power laws $|\lambda _{1,1}^2|\propto (1/N)^{c}$ with exponents $c \sim 2$ are found for two cases, $(\alpha =50^\circ,Bo=2)$ and $(100^\circ,1)$. The exponents $c \sim 2$ reflect the second-order convergence of the BE model (Zhang et al. Reference Zhang, Zhou and Ding2023). Therefore, one can expect that the numerical results of $\lambda _{1,1}$ decay to zero as the grid number $N$ increases. The results of $\lambda _{1,1}=0$ are also consistent with the prediction by the boundary parameter comparison approach (§ 2.2.2), which suggests that pendant drops with free CLs are marginally stable to non-axisymmetric perturbations.

Figure 12. Numerical results of the absolute value $|\lambda ^2|$ of frequency squared versus the grid number inverse $1/N$ of the BE model for zonal $\{2,0\}$ ($\bigcirc$, $\square$) and Noether $\{1,1\}$ ($\bigtriangleup$, $\bigtriangledown$) modes. The frequencies decay to zero in power laws $|\lambda ^2| \propto N^{-c}$ with exponents $c \sim 2$ for these modes, as indicated by the lines fitted by least squares.

A new feature of pendant drops is their zero frequency mode $\{2,0\}$ at $Bo_{max}$. Figure 12 confirms $\lambda _{2,0}=0$ of the maximum drops $B$ and $F$ (figure 8) by the power laws $|\lambda _{2,0}^2| \propto N^{-c}$ with $c\sim 2$. Due to the destabilisation of gravity, pendant drops can detach from the wall or pinch off under gravity (Henderson et al. Reference Henderson, Pritchard and Smolka1997), reflecting the capillary instability related to the zonal mode $\{2,0\}$. This instability occurs when the drop volume exceeds the maximum volume (i.e. $Bo_{max}$). As stated in § 3.2, the pendant drop with $Bo_{max}$ corresponding to the horizontal TP is marginally $v$-stable (see figure 5) and has the zero frequency mode $\{2,0\}$. This suggests that the static stability of pendant drops can be recovered by letting $\lambda _{2,0}=0$ in the dynamic stability, and the critical conditions for static and dynamic stability are essentially the same.

Additionally, the zero frequency mode $\{2,0\}$ with $Bo_{max}$ can also be used to define the characteristic length scale of instability for pendant drops. For the Rayleigh–Plateau instability of a liquid cylinder of radius $r_0$, the critical wavelength $L_{RP}=2{\rm \pi} r_0$ (with dimensionless wavenumber $k=1$) is usually used to define the characteristic length scale of critical instability (Plateau Reference Plateau1873). Similarly, the pendant drops also have critical states of instability corresponding to the $\{2,0\}$ mode with $k=2$. Thus a characteristic length scale can be defined as $L=2{\rm \pi} l_*/k={\rm \pi} l_*$ based on the wavelength of the $\{2,0\}$ mode. The critical value of $l_*=\sqrt { \sigma Bo/\rho g}$ is reached at $Bo=Bo_{max}$. Therefore, for pendant drops, the characteristic length scale of critical instability is given as $L={\rm \pi} \sqrt { \sigma \,Bo_{max}/\rho g}={\rm \pi} \,Bo_{max}^{1/2}\,l_c$, where $Bo_{max}$ depends on $\alpha$ or $R$ (see figure 8). It can be seen that the characteristic length scale $L$ is related to the capillary length $l_c$ and the maximum Bond number $Bo_{max}$.

3.5. Breaking of spectral degeneracy and spectral ordering

For free semi-drops (with $\alpha =90^\circ$, $R=1$, $Bo=0$ and free CLs), CL pinning or varying contact angle breaks the spectral degeneracy inherited from the RL spectrum (Lyubimov et al. Reference Lyubimov, Lyubimova and Shklyaev2006; Bostwick & Steen Reference Bostwick and Steen2014). Figure 13 shows that gravity breaks the spectral degeneracy, where the spectral splitting exhibits the larger (smaller) frequency for pendant (sessile) drops with the higher $l$ (see the direction of the arrows). The spectral splitting due to gravity for $Bo>0$ (pendant) is similar to those of varying contact angle for $\alpha >90^{\circ }$ and CL pinning (Lyubimov et al. Reference Lyubimov, Lyubimova and Shklyaev2006; Bostwick & Steen Reference Bostwick and Steen2014). Specifically, there are two opposite kinds of spectral splitting: the higher the $l$, the larger the frequency for $Bo>0$, CL pinning and $\alpha >90^{\circ }$, and the reverse holds for $Bo<0$ and $\alpha <90^{\circ }$. The two kinds of spectral splitting arising from the breaking of spectral degeneracy are associated with the construction of periodic tables (PTs) of modes (Steen et al. Reference Steen, Chang and Bostwick2019).

Figure 13. Breaking of the spectral degeneracy due to gravity: frequency $\lambda _{n,l}$ versus the Bond number $Bo$ for (a) $k=2$, (b) $k=3$, (c) $k=4$, (d) $k=5$, (e) $k=6$, and ( f) $k=7$, where the layer number is $n=(k-l+2)/2$. The yellow dots indicate the spectral degeneracy of free semi-drops with $\lambda ^2=k(k-1)(k+2)$ (see also Bostwick & Steen Reference Bostwick and Steen2014, figure 2). The cyan region highlights sharp frequency changes near $Bo_{max}$.

Steen et al. (Reference Steen, Chang and Bostwick2019) used an analogy of the PT of chemical elements to categorise the modes of spherical cap drops into two standard PTs and other non-standard PTs with irregularities, where modes are ordered according to the kinetic energy (or frequency). In the two standard PTs, the modes are first sorted in ascending order of $k$, and then modes with the same $k$ are sorted in descending or ascending order of $l$ (Steen et al. Reference Steen, Chang and Bostwick2019). Figures 14(a,b) show two standard spectral orderings equivalent to the two standard PTs, which arise from the breaking of spectral degeneracy. For free semi-drops satisfying (1.1), modes are sorted in ascending order of $k$, but cannot be sorted by $l$ due to the spectral degeneracy. As described above, the CL pinning, varying $\alpha$ or exerting gravity leads to the spectral splitting, so that modes with the same $k$ can be sorted by $l$. For example, exerting low gravity (e.g. $Bo=0.3$) to free semi-drops allows modes to be ordered in ascending order of $l$ (see figure 13), and still maintains the ascending order of $k$, as shown in figure 14(a). Likewise, the CL pinning or slightly raising contact angle (e.g. $\alpha =91^\circ$) for free semi-drops can also result in the same standard spectral ordering (Steen et al. Reference Steen, Chang and Bostwick2019). Similarly, the other standard spectral ordering in figure 14(b) can be generated by slightly lowering $\alpha$ or exerting opposite low gravity for free semi-drops, where modes with the same $k$ are in descending order of $l$.

Figure 14. Spectral ordering of the first 19 modes. (a,b) In the two standard orderings (equivalent to the standard PTs proposed by Steen et al. Reference Steen, Chang and Bostwick2019), modes are first sorted in ascending order of $k$, and then modes with the same $k$ are sorted in (a) ascending or (b) descending order of $l$. (c–e) Three non-standard spectral orderings with irregularities arise from the breaking of the order of (c,e) $k$ or (d) $l$.

Figures 14(c–e) show several non-standard spectral orderings with irregularities when the order of $l$ or $k$ is broken. Exerting higher gravity, the order of $k$ of few modes can be broken, e.g. $\lambda _{[k = 6, l=0]}<\lambda _{[5, 5]}$ for $Bo=1$ shown in figure 14(c), while the order of $l$ still holds. For the extreme case of a drop with $Bo=Bo_{max}\simeq 2.9$, $R=1$ and a pinned CL, the order of $k$ can even be completely broken (figure 14e). Figure 14(d) shows the breaking of the order of $l$, where the contact angle and gravity of free semi-drops are both changed (e.g. $\alpha =85^\circ$ and $Bo=0.3$). This is because lowering the contact angle and exerting gravity have opposite effects on the order of $l$, the competition of which breaks the order of $l$. In summary, gravity can also produce the same spectral orderings as in gravity-free cases (Steen et al. Reference Steen, Chang and Bostwick2019), which provides an alternative to controlling the spectral ordering without varying contact angle.