4.1. Convergence to the Rayleigh–Lamb spectrum

The previous section shows that modes of droplets on thin fibres resemble spherical harmonics. We conjecture that when the fibre radius trends to zero, the frequencies of these modes converge to the Rayleigh–Lamb spectrum (1.1).

We compute nine modes for a droplet on a thin fibre with free contact lines, $\alpha = 90^\circ$ and $R = 0.02$ , as shown in table 1. Due to the small fibre radius, the droplet’s base state is nearly spherical. The frequencies of these zonal, sectoral and tesseral modes are found to closely align with the Rayleigh–Lamb spectrum (1.1), with only minor discrepancies. The frequency of the zonal mode $\{2,0\}$ approaches zero ( ${\sim} 10^{-6}$ , attributed to numerical inaccuracies), as anticipated. This occurs because the zero frequency of this mode corresponds to translational invariance in the $z$ direction, a characteristic present for all droplets on fibres with free contact lines. A similar scenario arises for sessile droplets on flat surfaces with free contact lines, where the corresponding mode $\{1,1\}$ also exhibits zero frequency due to horizontal translational invariance, even in the presence of gravity (Zhang & Zhou Reference Zhang and Zhou2023; Zhang et al. Reference Zhang, Zhou and Ding2023).

Table 1.

Eigenvalues of the modes for a droplet on a thin fibre with free contact lines, $\alpha =90^\circ$ and $R=0.02$ . The squared eigenvalues of nine modes (zonal, sectoral, tesseral) are compared with the Rayleigh–Lamb frequencies $\lambda _{RL}$ of a unit-radius spherical droplet from (1.1) with $k=n+l-1$ . The differences $(\lambda _{n,l}/\lambda _{RL})^2 - 1$ are presented.

The $\{1,1\}$ mode with a squared eigenvalue slightly below zero also deserves attention. Note that this negative value ( ${\lambda }^2=-2.4044\times 10^{-3}$ ) does not result from numerical inaccuracies but rather reflects the inherent instability of the mode. The time evolution of modes, described by $\exp ({{\mathrm{i}}\lambda t})$ , indicates that for $\lambda ^2 \lt 0$ , the imaginary component of the eigenvalue denotes the growth rate of instability, whereas for $\lambda ^2 \gt 0$ , the real component corresponds to the vibration frequency. Consequently, $\lambda ^2 = 0$ represents the threshold for instability. This $\{1,1\}$ mode is linked to the barrel-to-clamshell transition (figure 1 b) (Carroll Reference Carroll1986; McHale et al. Reference McHale, Newton and Carroll2001; Eral et al. Reference Eral, de Ruiter, de Ruiter, Oh, Semprebon, Brinkmann and Mugele2011). A comprehensive analysis of the stability threshold under various contact line conditions will be presented in the next section.

Figure 5.

Spectral lines ( $\textrm {Re}(\lambda )$ ) of droplets on fibres with $ R \in [0.01, 1]$ for $ l = 0, 1, 2$ : ( $a$ – $c$ ) free contact lines with $ \alpha = 90^\circ$ ; ( $d$ – $f$ ) pinned contact lines with $\alpha = 90^\circ$ ; ( $g$ – $i$ ) mixed contact lines with $\alpha = 120^\circ$ . Red solid lines represent the first few modes for each $l$ -family ( $k\geq l$ ), with $k$ decreasing from 6 to $\max (l,1)$ , arranged from top to bottom in the plots. Black dashed lines represent the Rayleigh–Lamb spectrum (1.1). All spectral lines converge to the Rayleigh–Lamb spectrum as $R$ decreases, except for zonal modes in panels ( $d$ ) and ( $g$ ). Blue stars in panels ( $c$ ) and ( $i$ ) mark the stability threshold.

To investigate the influence of contact angle and contact line conditions on the conjecture, we analyse the frequency spectra for varying contact angles and three different contact line conditions (free, pinned and mixed) over the range $R \in [0.01, 1]$ . First, we present the spectrum (real part of the eigenvalues) for droplets on fibres with free contact lines and $\alpha = 90^\circ$ for $k$ up to 6 (see red solid lines in figure 5 a–c). As $R$ approaches zero, the frequencies of all modes converge to the Rayleigh–Lamb spectrum (dashed lines). As $R$ increases, the frequencies of zonal modes rise, while those of non-axisymmetric modes either decrease or show non-monotonic behaviour, initially decreasing and then increasing.

Figure 5( $d$ – $f$ ) shows the spectral lines for pinned contact lines with $\alpha =90^\circ$ , while figure 5(g–i) displays those for mixed contact lines with $\alpha =120^\circ$ . A slight deviation is observed between the spectral lines of zonal modes and the Rayleigh–Lamb spectrum at small $R$ (figure 5 d,g), indicating that the conjecture does not hold for zonal modes. However, for non-axisymmetric modes (e.g. $l = 1, 2$ ), varying the contact angle or the contact line condition, individually or simultaneously (figure 5 e,f,h,i), does not affect the convergence to the Rayleigh–Lamb spectrum. This suggests that only axisymmetric modes fail to converge, except in the specific case of free contact lines with $\alpha =90^\circ$ (figure 5 a). In contrast, non-axisymmetric modes generally converge to the Rayleigh–Lamb spectrum at small $R$ .

The distinction between axisymmetric and non-axisymmetric modes can be understood from the boundary conditions at the contact lines. Regardless of the contact angle $\alpha$ , the base-state shape on a very thin fibre closely resembles a sphere, except in the vicinity of the contact line. As shown in figure 3, in the vibrational modes of a spherical droplet, only the top and bottom vertices of axisymmetric modes exhibit displacement (highlighted by red and blue regions), whereas the corresponding vertices of non-axisymmetric modes remain immobile (white regions).

For axisymmetric modes, the displacement of the upper and lower vertices causes the local tangent at these points to form a $90^\circ$ angle with the fibre surface, which precisely matches the case of free contact lines with $\alpha =90^\circ$ . Any deviation in the contact angle or modification to the contact line condition disrupts this alignment, preventing axisymmetric modes from converging to the Rayleigh–Lamb spectrum.

In contrast, for non-axisymmetric modes, the displacements at the upper and lower contact lines of the fibre diminish to zero, regardless of the contact angle or contact line condition. This matches the zero displacement at the upper and lower vertices of the spherical harmonic functions, ensuring that non-axisymmetric modes generally converge to the Rayleigh–Lamb spectrum at small $R$ .

In figure 5(a,b,e,h), the lowest modes $\{2, 0\}$ and $\{1, 1\}$ have spectral lines where $\textrm {Re}(\lambda )=0$ . As previously discussed, the $\{2, 0\}$ mode has $\textrm {Re}(\lambda )=0$ due to translational invariance in the $z$ direction. The stabilising effect of contact line pinning leads to a non-zero frequency for the $\{2, 0\}$ mode when one or both contact lines are pinned (figure 5 d,g). The $\{1, 1\}$ mode, however, reaches $\textrm {Re}(\lambda )=0$ due to its instability, characterised by $\lambda ^2 \lt 0$ , as only the real part of the eigenvalue is shown. Interestingly, the $\{1, 1\}$ mode with pinned contact lines and $\alpha =90^\circ$ precisely reaches $\lambda = 0$ (figure 5 e), marking the threshold for the barrel-to-clamshell transition, which will be explored further in the next section. A similar instability is seen for the $\{1, 2\}$ mode in figure 5(c,i). As $R$ increases, this mode transitions from stable to unstable, with the instability threshold marked by blue stars.

4.2. Eigenvalue of the $\{1,1\}$ mode: threshold for barrel-to-clamshell transition

As discussed in § 1, the barrel-to-clamshell transition is a significant topic in the study of droplets on fibres. This section delves into this transition by examining the eigenvalues of mode $\{1,1\}$ . Since mode $\{1,1\}$ is the first to become unstable (when ${\lambda _{1,1}}^2\lt 0$ ), its stability governs the overall droplet morphology and serves as the threshold for the barrel-to-clamshell transition (see figure 6 a). For a fixed droplet volume, the eigenvalues of mode $\{1,1\}$ are determined by the parameter pair $(\alpha, R)$ and the contact line conditions. Our findings are presented separately for full-barrel-shaped droplets and half-barrel-shaped droplets. The latter analysis is particularly aimed at explaining the experimental anomaly reported by Sun et al. (Reference Sun, Bazilevsky and Kornev2023).

4.2.1. Full-barrel shapes

Figure 6.

( $a$ ) Dynamic behaviour of the $\{1,1\}$ mode based on its squared eigenvalue: ${\lambda _{1,1}}^2\gt 0$ indicates oscillation (left), while ${\lambda _{1,1}}^2\lt 0$ indicates instability (right). ( $b$ – $d$ ) Contour plots of the squared eigenvalues ${\lambda _{1,1}}^2$ for the $\{1,1\}$ mode in the parameter space $(\alpha, R) \in [10^\circ, 170^\circ ] \times [0.1, 1]$ under different contact line conditions: ( $b$ ) free, ( $c$ ) mixed and ( $d$ ) pinned contact lines. In panel ( $d$ ), the vertical line of $\lambda ^2 = 0$ coincides with the Steiner limit for liquid bridges (Russo & Steen Reference Russo and Steen1986).

Figure 6( $b$ – $d$ ) shows the squared eigenvalues of mode $\{1,1\}$ across the parameter range $(\alpha, R) \in [10^\circ, 170^\circ ] \times [0.1, 1]$ for free, mixed and pinned contact line conditions. The region where ${\lambda _{1,1}}^2 \gt 0$ (indicated in red) is largest for pinned contact lines, followed by mixed and then free contact lines. This suggests that, for a given $(\alpha, R)$ , the $\{1,1\}$ mode is most stable (with the highest ${\lambda _{1,1}}^2$ ) under pinned contact line conditions, less stable with mixed contact lines and least stable with free contact lines. This ranking of stability is attributed to the stabilising effect of contact line pinning, as noted by Bostwick & Steen (Reference Bostwick and Steen2015).

For both free and mixed contact line conditions, the critical stability line ( $\lambda _{1,1}=0$ ) lies within the region $\alpha \lt 90^\circ$ and slopes from the upper left to the lower right (figure 6 b,c). This suggests that barrel-shaped droplets are stable only on hydrophilic fibres. As $R$ increases, the contact angle $\alpha$ must decrease to maintain stability, consistent with previous findings (e.g. Eral et al. Reference Eral, de Ruiter, de Ruiter, Oh, Semprebon, Brinkmann and Mugele2011). In most cases, ${\lambda _{1,1}}^2$ decreases with increasing $R$ or $\alpha$ , indicating enhanced stability with smaller $R$ or $\alpha$ . For pinned contact lines (figure 6 d), the critical stability line is the vertical line at $\alpha =90^\circ$ , coinciding with the Steiner limit for liquid bridges (Russo & Steen Reference Russo and Steen1986), where the critical profile is horizontal at the contact line.

Figure 7.

Contour plots of the squared eigenvalues ${\lambda _{1,1}}^2$ in the parameter space $(\alpha, N)$ for free contact lines, corresponding to figure 6(b) in the $(\alpha, R)$ parameter space. Here, $N$ denotes the ratio of the droplet’s maximum radius to the fibre radius $R$ . The critical stability line ( ${ \lambda _{1,1}} = 0$ ) aligns with the inflection point condition (4.1) (McHale et al. Reference McHale, Newton and Carroll2001), while the stability threshold proposed by Carroll (Reference Carroll1986) lies within the region where ${\lambda _{1,1}}^2 \lt 0$ .

The above discussion of stability for the $\{1,1\}$ mode, based on the parameter space $(\alpha, R)$ for a fixed volume, is converted to the parameter space $(\alpha, N)$ for comparison with previous studies (Carroll Reference Carroll1986; McHale et al. Reference McHale, Newton and Carroll2001; Eral et al. Reference Eral, de Ruiter, de Ruiter, Oh, Semprebon, Brinkmann and Mugele2011). Here, $N$ represents the ratio of the droplet’s maximum radius to the fibre radius $R$ . Figure 7 show that the critical stability line matches the inflection point condition proposed by McHale et al. (Reference McHale, Käb, Newton and Rowan1997):

(4.1) \begin{equation} N=\frac {1+\sin \alpha }{\cos \alpha }. \end{equation}

This stability condition corresponds to the stability boundary of a barrel-shaped droplet on the fibre, where the inflection point ( $\beta '=0$ ) of the droplet profile comes into contact with the fibre surface. In contrast, Carroll (Reference Carroll1986)’s stability condition falls in the region where ${\lambda _{1,1}}^2\lt 0$ . The stability condition proposed by Carroll (Reference Carroll1986) is based on a perturbation approach that does not explicitly enforce a constant contact angle condition, which may introduce inaccuracies in the prediction of the stability boundary for free contact lines.

In summary, our modal analysis confirms that the threshold for the barrel-to-clamshell transition depends on the contact line conditions. For free contact lines, the stability threshold matches the inflection point condition (4.1), indicating that only unduloidal shapes possessing inflection points can be stable (McHale et al. Reference McHale, Newton and Carroll2001; Eral et al. Reference Eral, de Ruiter, de Ruiter, Oh, Semprebon, Brinkmann and Mugele2011). For pinned contact lines, the threshold occurs at $\alpha = 90^\circ$ (Steiner limit), independent of $R$ . In this case, droplets on fibres and liquid bridges share the same stability threshold when their liquid surface shapes are identical (Bostwick & Steen Reference Bostwick and Steen2015). This implies that fibre droplets with pinned contact lines are stable only when $\alpha \lt 90^\circ$ , without requiring an inflection point, due to the stabilising effect of contact line pinning (Bostwick & Steen Reference Bostwick and Steen2015). For mixed contact lines (one pinned and one free), the stability threshold lies between those for free and pinned contact lines.

4.3. Breaking of frequency degeneracy

As discussed in § 4.1, the modes of a droplet on a fibre converge towards the Rayleigh–Lamb spectrum of spherical droplets as $R\to 0$ , with minor deviations observed in zonal modes under pinned or mixed contact line conditions (figure 5). This convergence links their behaviour to the well-known Rayleigh–Lamb spectrum, providing a basis for exploring how fibre geometry and contact line dynamics influence the frequency spectrum.

An important feature of the Rayleigh–Lamb spectrum (1.1) is frequency degeneracy, where modes with the same $k$ share identical frequencies, irrespective of $l$ . For any $k\geqslant 1$ , there are $k+1$ degenerate modes. Droplets on fibres with very small radii are expected to retain this property. However, as $R$ increases, this degeneracy breaks (see e.g. figure 5 $a{-}c$ ). A similar degeneracy breaking is observed in gravity-free sessile droplets with $\alpha =90^\circ$ and a free contact line (Bostwick & Steen Reference Bostwick and Steen2014; Steen et al. Reference Steen, Chang and Bostwick2019; Zhang et al. Reference Zhang, Zhou and Ding2023), where changes in the contact line, contact angle or the presence of gravity lead to diverse spectral orderings.

Figure 8.

Squared frequency $\lambda ^2$ against fibre radius $R$ for ( $a$ ) $k=1$ , ( $b$ ) $k=2$ , ( $c$ ) $k=3$ and ( $d$ ) $k=4$ in droplets on fibres with $\alpha =90^\circ$ and free contact lines. As $R$ tends to zero, these modes converge to the Rayleigh–Lamb spectrum (1.1) (dashed line). Each set of modes with the same $k$ contains $k + 1$ modes. In panel ( $a$ ), mode $\{1,1\}$ lies entirely within the instability region (shaded pink) where $\lambda ^2\lt 0$ , while in panel ( $b$ ), the $\{1,2\}$ mode becomes unstable when $R\gt 0.82$ (marked by a blue star). Insets in panels ( $a$ ) and ( $b$ ) illustrate the shapes of the unstable modes $\{1,1\}$ and $\{1,2\}$ at $R=1$ , respectively.

Figure 8 shows the squared frequencies $\lambda ^2$ as a function of $R$ for $k=1$ , 2, 3, 4 in droplets on fibres with $\alpha =90^\circ$ and free contact lines. At small $R$ , the modes align with the Rayleigh–Lamb spectrum (1.1), exhibiting frequency degeneracy ordered by increasing $k$ . As $R$ increases, this degeneracy breaks and frequencies of degenerate modes diverge. As a result, the spectral order does not strictly follow $k$ : for example, the zonal mode $\{4,0\}$ with $k=3$ has a lower frequency than the sectoral mode $\{1,4\}$ with $k=4$ at $R=0.1$ , but the reverse occurs at $R=1$ (figure 8 c,d). This arises because zonal mode frequencies typically increase with $R$ , while sectoral mode frequencies decrease, causing high- $k$ sectoral modes to drop below low- $k$ zonal modes. The $\{2,0\}$ mode with free contact lines maintains a zero frequency due to the droplet’s vertical translational invariance on the fibre (figure 8 a). Variations for $\alpha \neq 90^\circ$ and mixed (or pinned) contact line conditions are shown in figure 5. Generally, modes with small $l$ and large $n$ have frequencies that increase with $R$ , while those with large $l$ and small $n$ decrease, and intermediate modes show a non-monotonic variation.

The unstable modes of a droplet on a fibre include not only mode $\{1,1\}$ but also mode $\{1,2\}$ (figure 8 a,b). Mode $\{1,1\}$ , associated with the barrel-to-clamshell transition, remains unstable across all $R$ , with its growth rate $\textrm {Im}(\lambda )$ initially rising and then falling. In contrast, mode $\{1,2\}$ becomes unstable only when $R\gt 0.82$ , with its squared frequency decreasing consistently as $R$ increases. For relatively thick fibres ( $R\gt 0.82$ ), the droplet may destabilise via mode $\{1,2\}$ , leading to a symmetric split into two sub-droplets (see the inset in figure 8 b). At $R = 1$ , mode $\{1,2\}$ ( $\lambda ^2=-0.589$ ) dominates over mode $\{1,1\}$ ( $\lambda ^2=-0.442$ ) due to its higher growth rate. This instability is related to the Rayleigh–Plateau instability of static rivulets (Davis Reference Davis1980; Bostwick & Steen Reference Bostwick and Steen2018; Pham et al. Reference Pham, Perrard and Le2020; Wang & Tao Reference Wang and Tao2022), which will be further explored in § 4.5.

4.4. Effect of fibre radius on frequency spectra

This section examines how the fibre radius affects the frequency spectra, focusing on three representative cases: a fibre of intermediate thickness ( $R=0.2$ ), a thin fibre ( $R=0.05$ ) and a thick fibre ( $R=2$ ).

4.4.1. $R=0.2$

Figure 9 shows the frequency spectrum of droplets on a fibre with $R=0.2$ , comparing pinned, mixed and free contact line conditions. The natural frequencies are influenced by these conditions: free contact lines set the lower limit, pinned lines the upper, with the shaded grey region in figure 9(a,b) representing the range of dynamic contact lines. The spreading parameter in dynamic contact line condition governs this range, converging to the free contact line result at zero and the pinned result at infinity (Bostwick & Steen Reference Bostwick and Steen2014). For mixed conditions, the spectral line typically lies between the free and pinned cases, centred within the grey region.

For axisymmetric modes, the grey region is broad because modes with free contact lines undergo significant displacements at these boundaries, making them highly sensitive to contact line conditions (e.g. zonal mode $\{4,0\}$ in figure 9 e). In contrast, as $l$ increases, the displacements at the contact lines become smaller, reducing their sensitivity to different boundary conditions. For instance, mode $\{2,2\}$ exhibits nearly identical frequencies and shapes across contact line conditions, as minimal displacement at the contact line causes only slight differences (figure 9 f). For even higher $l$ , the spectral lines almost overlap (figure 9 d).

Figure 9 also highlights the influence of the contact angle on natural frequencies. For axisymmetric modes, the frequencies initially increase and then decrease as $\alpha$ increases, peaking around $110^\circ$ , except for the free mode $\{2,0\}$ , which always has zero frequency due to vertical translation invariance. For non-axisymmetric modes, those with $n$ much larger than $l$ show similar contact angle effects to axisymmetric modes. However, for modes with smaller $n$ , increasing $\alpha$ reduces the frequency, e.g. driving the frequency of mode $\{1,1\}$ to zero, then causing instability associated with the barrel-to-clamshell transition.

Figure 9.

Spectral lines for $R = 0.2$ : ( $a$ ) $l = 0$ , ( $b$ ) $l = 1$ , ( $c$ ) $l = 2$ and ( $d$ ) $l = 5$ , showing free (blue solid), mixed (black dotted) and pinned (red dashed) contact lines. The shaded grey region, enclosed by the lower limit of free and the upper limit of pinned contact lines, represents the possible range of dynamic contact line conditions. The mixed contact line result is positioned centrally within this range. Hollow dots indicate the Rayleigh–Lamb spectrum for reference. ( $e,f$ ) Mode shapes are displayed for free (left), mixed (centre) and pinned (right) contact lines: ( $e$ ) $\{4,0\}$ mode and ( $f$ ) $\{2,2\}$ mode.

4.4.2. $R=0.05$

We examine the frequency spectrum of droplets on thin fibres with $R = 0.05$ in figure 10. For axisymmetric modes, differences between contact line types persist, though slightly reduced compared with the $R = 0.2$ case. In contrast, for non-axisymmetric modes, the spectral lines nearly overlap, indicating that as fibre radius decreases, contact line effects weaken: distinctions remain in axisymmetric modes but vanish in non-axisymmetric ones. At very small radii, the influence of the contact angle also diminishes, with droplet modes approaching Rayleigh–Lamb modes.

Figure 10.

Spectral lines for $R=0.05$ , following the same notation as in figure 9(a–d).

4.4.3. $R=2$

The spectral lines of droplets on thick fibres show distinctive characteristics. Figure 11 presents the spectral lines for a fibre with $R=2$ , much thicker than the droplet itself. Here, the droplet resembles a liquid ring on a cylinder, similar to a static rivulet on a cylindrical surface, influenced by the fibre curvature, $1/R$ . As the curvature approaches zero ( $R\to \infty$ ), the fibre droplet becomes a static rivulet on a flat surface. In this case, the axisymmetric mode represents oscillations of a two-dimensional sessile droplet, while unstable non-axisymmetric modes correspond to instability of static rivulets. On thick fibres, the type of contact line and variations in contact angle have a greater effect compared with thin fibres. Except for sectoral modes ( $n = 1$ ), the frequencies of non-axisymmetric modes are largely insensitive to $l$ , increasing only slightly with $l$ (figure 11 b–d). This highlights the unique spectral characteristics for droplets on thick fibres.

Figure 11.

Spectral lines for $ R = 2$ , showing ( $a$ ) $l=0$ , ( $b$ ) $l=1$ , ( $c$ ) $l=5$ and ( $d$ ) $l=10$ , following the same notation as in figure 9(a–d).

4.5. Rayleigh–Plateau instability on cylindrical surfaces

Figure 12.

Shape of the sectoral mode $\{1,5\}$ for a droplet on a thick fibre with $\alpha =90^{\circ }$ , $R=5$ ( $R/H=22.3$ ) and free contact lines, illustrating an analogy to the Rayleigh–Lamb instability of a static rivulet on a cylindrical surface: ( $a$ ) three-dimensional view, ( $b$ ) top view and ( $c$ ) side view.

On thick fibres, we observe instabilities in sectoral modes $\{1,l\geq 1\}$ (figures 8 a,b and 11), which are closely related to the Rayleigh–Plateau instability seen in static rivulets on flat surfaces (Davis Reference Davis1980; Bostwick & Steen Reference Bostwick and Steen2018; Pham et al. Reference Pham, Perrard and Le2020; Wang & Tao Reference Wang and Tao2022). Specifically, a droplet on a fibre with a large radius resembles a static rivulet on a cylindrical surface (figure 12; see also figure 7( $a$ ) of Bostwick & Steen (Reference Bostwick and Steen2018)). In this case, the average liquid film thickness $H$ (defined as the cross-sectional area of the droplet divided by the wetting length) is much smaller than the fibre radius $R$ . In other words, only when $R\gg H$ does the droplet exhibit enough unstable sectoral modes to be analogous to the Rayleigh–Plateau instability. Thus, we use the ratio $R/H$ to quantify this analogy.

Figure 13.

Variation of $R/H$ with ( $a$ ) contact angle $\alpha$ for different fixed fibre radii $R$ (along the arrow $R=0.02$ , 0.2, 2, 5 and 10), and ( $b$ ) fibre radius $R$ for different fixed contact angles $\alpha$ (along the arrow $\alpha =30^\circ$ , $60^\circ$ , $90^\circ$ , $120^\circ$ and $150^\circ$ ).

Figure 13 shows how the ratio $R/H$ varies with contact angle $\alpha$ and fibre radius $R$ . As seen in figure 13(a), $R/H$ decreases with increasing $\alpha$ for all fibre radii considered, with larger $R$ yielding higher $R/H$ . For thicker fibres ( $R = 2, 5, 10$ ), $R/H$ is more sensitive to $\alpha$ as the droplet shape approaches that of a thin film, while for thinner fibres ( $R = 0.02, 0.2$ ), the influence of $\alpha$ is less significant. Figure 13( $b$ ) shows that $R/H$ increases monotonically with $R$ for all $\alpha$ , with smaller $\alpha$ resulting in higher $R/H$ . For small $R \lesssim 0.2$ , $H$ remains nearly constant at fixed droplet volume, leading to the scaling $R/H \sim R$ . Overall, fibres can be classified as thin when $R \lesssim 0.2$ ( $R \ll H$ ), intermediate when $0.2 \lesssim R \lesssim 2$ ( $R \sim H$ ) and thick when $R \gtrsim 2$ ( $R \gg H$ ).

In the above analysis, the droplet volume is fixed at $4\pi /3$ , whereas studies on static rivulets typically fix the liquid surface radius at 1, yielding a wetting length of $2\sin \alpha$ (distance between contact lines). To compare with static rivulet results, we introduce a characteristic length $l_{r}={h}/({2\sin \alpha })$ , scaling the wetting length of fibre droplets to match that of static rivulets. Here, $h$ represents the dimensionless wetting length of the droplet with volume $4\pi /3$ (see figure 1 a). Under this scaling, the eigenvalues $\tilde \lambda$ relate to the original eigenvalues $\lambda$ by

(4.2) \begin{equation} {\tilde \lambda }=l_r^{3/2}\lambda =\left (\frac {h}{2\sin \alpha }\right )^{3/2}\lambda . \end{equation}

Additionally, for the Rayleigh–Plateau instability, the axial wavenumber $\tilde {k}_m$ , which determines the wavelength $2\pi /\tilde { k}_m$ , is given by

(4.3) \begin{equation} \tilde {k}_m=\frac {l{l_r}}{R }=\frac {lh}{2R\sin \alpha }, \end{equation}

as the wavelength of the instability for the droplet on the fibre is ${2\pi R} / (l l_{r})$ , as shown in figure 12(b).

Figure 14.

Comparison of growth rate $\textrm {Im}(\tilde {\lambda })$ versus axial wavenumber $\tilde {k}_m$ for droplets on fibres with $R=5$ (circular dots) and for static rivulets on flat surfaces (solid lines) (adapted from Wang & Tao Reference Wang and Tao2022). Panels ( $a$ ) and ( $b$ ) show results for free and pinned contact lines, respectively. For unstable sectoral modes $\{1,l\geq 1\}$ , the mode number $l$ starting at 1 and increasing in steps of 1 relates to the axial wavenumber $\tilde {k}_m$ by (4.3).

Figure 14 compares the growth rates of the Rayleigh–Plateau instability for static rivulets on flat surfaces (Wang & Tao Reference Wang and Tao2022) and annular rivulets on cylindrical surfaces with $R=5$ (representing droplets on fibres). On cylindrical surfaces, the base state is toroidal and finite in length, leading to discrete unstable modes (circular dots in figure 14) characterised by the azimuthal wavenumber $l$ , corresponding to their $l$ -fold rotational symmetry. In contrast, for static rivulets on flat surfaces, the base state is infinite along the axial direction, allowing the axial wavenumber $\tilde {k}_m\geqslant 0$ to take any real value, resulting in a continuous growth rate curve as a function of wavenumber.

Figure 14 show that for both flat and cylindrical surfaces, the growth rate $\textrm {Im}(\tilde {\lambda })$ increases with wavenumber $\tilde {k}_m$ until reaching a maximum, then drops sharply to zero at the cutoff wavenumber. For cylindrical surfaces with $R=5$ , the discrete growth rates closely match flat surface results up to the maximum growth rate but shift slightly rightward (to higher $\tilde {k}_m$ ) beyond this point. This indicates that surface curvature for $R=5$ has little effect on the Rayleigh–Plateau instability up to the maximum growth rate, but slightly increases both the maximum growth rate and cutoff wavenumber at higher wavenumbers.

Figure 15.

Growth rates $\textrm {Im}(\tilde {\lambda })$ of sectoral modes as a function of $\tilde {k}_m$ for droplets on fibres (symbols) with different radii $R/l_r$ : ( $a$ ) free contact lines with $\alpha =90^\circ$ and ( $d$ ) pinned contact lines with $\alpha =120^\circ$ . The black dashed line represents growth rates for static rivulets on flat surfaces under the same contact angle and contact line conditions. In panel ( $a$ ), the maximum growth modes are $\{1,4\}$ , $\{1,13\}$ and $\{1,34\}$ , with cutoff modes $\{1,6\}$ , $\{1,19\}$ and $\{1,50\}$ . In panel ( $d$ ), the maximum growth modes are $\{1,4\}$ , $\{1,11\}$ and $\{1,29\}$ , with cutoff modes $\{1,6\}$ , $\{1,16\}$ and $\{1,42\}$ . Shapes of the maximum growth modes are shown in panels ( $b$ ),( $e$ ), and cutoff modes in panels ( $c$ ),( $f$ ), with their eigenvalues and wavenumbers listed in table 2.

To further investigate the effect of the cylindrical surface radius $R/l_r$ on the Rayleigh–Plateau instability, we plot the growth rates of unstable sectoral modes for droplets on fibres with varying radii $R/l_r$ . Figures 15( $a$ ) and 15( $d$ ) show cases with free contact lines ( $\alpha =90^\circ$ ) and pinned contact line ( $\alpha =120^\circ$ ), respectively. For comparison, growth rate curves for static rivulets on a flat surface (black dashed line) are included. For small cylindrical radii, only a few sectoral modes are unstable ( $\textrm {Im}(\tilde {\lambda })\gt 0$ ). These modes show a growth rate trend similar to that of static rivulets on flat surfaces. Before the peak, growth rates on cylindrical surfaces are slightly lower than on flat surfaces, but after the peak, they are slightly higher, indicating that curvature weakens growth rates for lower $l$ , but enhances them for higher $l$ .

Table 2.

Eigenvalues $\tilde {\lambda }$ and wavenumbers $\tilde {k}_m$ of the maximum growth rate modes and cutoff modes shown in figure 15 for droplets on thick fibres with varying radii $R/l_r$ (annular static rivulets) and static rivulets on flat surfaces ( $R/l_r\to +\infty$ ).

As the radius $R/l_r$ increases, more sectoral modes become unstable, with their growth rates closely matching those for flat surfaces (see figure 15 a,d). This suggests that fibre curvature has a diminishing effect as the radius grows, consistent with intuition. Since the cutoff wavenumbers for cylindrical surfaces are nearly identical to those for flat surfaces, the number of unstable sectoral modes for larger radii can be estimated. For free contact lines with $\alpha =90^\circ$ , where the cutoff wavenumber of static rivulets is $\tilde {k}_m=1$ (figure 15 a), the number of unstable sectoral modes can be approximately given by $\lceil R\tilde {k}_m/l_r \rceil$ , where $\lceil \cdot \rceil$ denotes the ceiling function. This estimate is consistent with table 2. Using a similar method, the sectoral mode with the maximum growth rate can also be approximately identified.

In summary, the instability of sectoral modes $\{1, l\}$ for droplets on thick fibres can be viewed as a Rayleigh–Plateau instability of annular static rivulets on cylindrical surfaces, where the mode number $l$ relates to the axial wavenumber $\tilde {k}_m$ by (4.3). These unstable sectoral modes exhibit discrete growth rates with respect to $\tilde {k}_m$ , yet they follow the same growth rate trend observed in static rivulets on flat surfaces (Wang & Tao Reference Wang and Tao2022). The curvature of the cylindrical surface affects the growth rate by increasing both the maximum growth rate and the cutoff wavenumber, while slightly reducing the growth rate at smaller wavenumbers (figures 14 and 15).