2.1 Small values of $m$ regime (nearly circular drop)

For small values of positive parameter $m$, we have the following expansions:

(2.25)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}Z_{max}(m)\underset{m\rightarrow 0^{+}}{=}\ell _{\mathit{c}}\sqrt{m}\left[1+{\textstyle \frac{1}{4}}m+\mathit{O}(m^{2})\right], & \displaystyle\end{eqnarray}$$

(2.26)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}Y_{max}(m)\underset{m\rightarrow 0^{+}}{=}\ell _{\mathit{c}}\sqrt{m}\left[1+{\displaystyle \frac{1}{4}}\left(1+{\displaystyle \frac{\unicode[STIX]{x03C0}}{4}}\right)m+\mathit{O}(m^{2})\right], & \displaystyle\end{eqnarray}$$

(2.27)$$\begin{eqnarray}\displaystyle & \displaystyle \mathscr{A}(m)\underset{m\rightarrow 0^{+}}{=}{\displaystyle \frac{\unicode[STIX]{x03C0}}{4}}m\ell _{\mathit{c}}^{2}+\mathit{O}(m^{2}). & \displaystyle\end{eqnarray}$$

Therefore, the cross-section of the drop is nearly that of a circle of radius $R_{0}$ that reads at leading order

(2.28)$$\begin{eqnarray}R_{0}(m)={\textstyle \frac{1}{2}}\sqrt{m}\ell _{\mathit{c}}\underset{m\rightarrow 0^{+}}{=}{\textstyle \frac{1}{2}}\unicode[STIX]{x0394}Z_{max}\underset{m\rightarrow 0^{+}}{=}{\textstyle \frac{1}{2}}\unicode[STIX]{x0394}Y_{max}\end{eqnarray}$$

and a contact length to the substrate $\unicode[STIX]{x1D6EC}_{\mathit{c}}(m)$ that satisfies

(2.29)$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathit{c}}\underset{m\rightarrow 0^{+}}{=}{\displaystyle \frac{\unicode[STIX]{x03C0}}{8}}m^{3/2}\ell _{\mathit{c}}\end{eqnarray}$$

which is then negligible compared to the radius $R_{0}(m)$. As the volume per unit length (in the $x$-direction) increases, we depart from the circular shape because of gravity that flattens the drop.

2.2 Parameter $m$ close to $1^{-}$ (flattened drop)

Let the parameter $\unicode[STIX]{x1D707}$ be such that $m=1-\unicode[STIX]{x1D707}$. We can compute the following asymptotic expressions for $\unicode[STIX]{x1D707}\rightarrow 0^{+}$ that read:

(2.30)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}Z_{max}(m)\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{=}2\ell _{\mathit{c}}\left[1-\sqrt{\unicode[STIX]{x1D707}}+o(\sqrt{\unicode[STIX]{x1D707}})\right], & \displaystyle\end{eqnarray}$$

(2.31)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}Y_{max}(m)\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{=}\ell _{\mathit{c}}\left[-\log \unicode[STIX]{x1D707}+\left(-4+2\sqrt{2}+2\log \tan {\displaystyle \frac{\unicode[STIX]{x03C0}}{8}}+4\log 2\right)\right]+\mathit{O}(\unicode[STIX]{x1D707}\log \unicode[STIX]{x1D707}), & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.32)$$\begin{eqnarray}\displaystyle & \displaystyle \mathscr{A}(m)\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{=}2\ell _{\mathit{c}}^{2}[-\log \unicode[STIX]{x1D707}+4(\log 2-1)]+\mathit{O}(\unicode[STIX]{x1D707}\log \unicode[STIX]{x1D707}), & \displaystyle\end{eqnarray}$$

(2.33)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6EC}_{\mathit{c}}(m)={\displaystyle \frac{\sqrt{m}}{2\ell _{\mathit{c}}}}\,\mathscr{A}(m)\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{=}\ell _{\mathit{c}}[-\log \unicode[STIX]{x1D707}+4(\log 2-1)]+\mathit{O}(\unicode[STIX]{x1D707}\log \unicode[STIX]{x1D707}), & \displaystyle\end{eqnarray}$$

with the following approximate numerical values of the additive constants to the logarithmic expressions, found in (2.31) and (2.33),

(2.34)$$\begin{eqnarray}\displaystyle & \displaystyle \mathscr{C}_{1}=-4+2\sqrt{2}+2\log \tan {\displaystyle \frac{\unicode[STIX]{x03C0}}{8}}+4\log 2\simeq -0.16173, & \displaystyle\end{eqnarray}$$

(2.35)$$\begin{eqnarray}\displaystyle & \displaystyle \mathscr{C}_{2}=4(\log 2-1)\simeq -1.2274. & \displaystyle\end{eqnarray}$$

Last, note that we have the following relationship between the value of the cross-sectional area and those of the vertical thickness and the contact extension

(2.36)$$\begin{eqnarray}\mathscr{A}(m)\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{=}\unicode[STIX]{x1D6EC}_{\mathit{c}}(m)\times \unicode[STIX]{x0394}Z_{max}(m)\left[1-\sqrt{\unicode[STIX]{x1D707}}+\mathit{O}(\unicode[STIX]{x1D707})\right],\end{eqnarray}$$

and we have $\ell _{\mathit{c}}/2(\mathscr{C}_{1}-\mathscr{C}_{2})$, corresponding roughly to half the capillary length, which is the length of the tip of the drop protruding above the wetted area.

In figure 3, we plot $\unicode[STIX]{x0394}Z_{max}$, $\unicode[STIX]{x0394}Y_{max}$ together with the aspect ratio $\unicode[STIX]{x1D71A}_{\mathit{A}}=\unicode[STIX]{x0394}Y_{max}/\unicode[STIX]{x0394}Z_{max}$. The latter is always larger than $1$ and diverges in the limit $m\rightarrow 1^{-}$ as $\unicode[STIX]{x0394}Y_{max}$ does. The plots are given with respect to the coordinate $m$ or $\unicode[STIX]{x1D707}=1-m$, in lin–lin or log–lin scales.

Figure 3.

(a) Plot of the horizontal extension $\unicode[STIX]{x0394}Y_{max}$ and the thickness $\unicode[STIX]{x0394}Z_{max}$ with respect to the variables $m$ and $\unicode[STIX]{x1D707}=1-m$ (inset); the $y$-coordinates are expressed in $\ell _{\mathit{c}}$-units. (b) Plot of the aspect ratio $\unicode[STIX]{x1D71A}_{\mathit{A}}(m)=\unicode[STIX]{x0394}Y_{max}/\unicode[STIX]{x0394}Z_{max}$ with respect to $\unicode[STIX]{x1D707}=1-m$. For $m\rightarrow 0$ (or $\unicode[STIX]{x1D707}\rightarrow 1$), the cross-section of the cylinder is nearly a circle, whereas for $m\rightarrow 1$ (or $\unicode[STIX]{x1D707}\rightarrow 0$), the horizontal extension diverges and the thickness tends to $2\ell _{\mathit{c}}$.

3.1 Saint-Venant governing equations

The Saint-Venant equations are a set of hyperbolic equations that were proposed by the eponymous Saint-Venant (Reference Saint-Venant1871) in order to study flows in open-channels along the $x$-direction. Their derivation is based on a long-wavelength (shallow water limit $k\rightarrow 0$) expansion and the basic hypotheses are the following: (i) the flow is considered quasi-parallel in the $x$-direction; and (ii) the pressure is hydrostatic. Under these hypotheses, if one sets the following quantities:

(3.1)$$\begin{eqnarray}\displaystyle & \displaystyle S(x,t),\text{ the transverse cross-sectional area of liquid,} & \displaystyle\end{eqnarray}$$

(3.2)$$\begin{eqnarray}\displaystyle & \displaystyle \bar{u}(x,t)={\displaystyle \frac{1}{S(x,t)}}\int _{S(x,t)}u(x,y,z,t)\,\text{d}y\,\text{d}z,\text{ the section-averaged velocity,} & \displaystyle\end{eqnarray}$$

(3.3)$$\begin{eqnarray}\displaystyle & \displaystyle P(x,y,z,t),\text{ the hydrostatic three-dimensional pressure field,} & \displaystyle\end{eqnarray}$$

the inviscid Saint-Venant equations read

(3.4)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}S+\unicode[STIX]{x2202}_{x}(S\bar{u})=0, & \displaystyle\end{eqnarray}$$

(3.5)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\bar{u}+\bar{u}\unicode[STIX]{x2202}_{x}\bar{u}+{\displaystyle \frac{1}{S(x,t)}}\int _{S(x,t)}{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}[P(x,y,z,t)]\,\text{d}y\,\text{d}z=0. & \displaystyle\end{eqnarray}$$

This set of equations is expressed in terms of section-averaged pressure gradient in the $x$-coordinate and turns into one-dimensional equations in space. For a complete derivation of these equations (including viscous effects), one can read the article of Decoene et al. (Reference Decoene, Bonaventura, Miglio and Saleri2009). As we are interested in the linear stability of a sessile drop, we will study a linearised version of these equations.

3.2 Special case of triangle-shaped cross-section

As a first elementary result given by the Saint-Venant equations, let us neglect the surface tension effects and consider a triangular cross-section of liquid, with both sides $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FD}}$, respectively, making an angle $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ with the horizontal. As no capillary effects are considered, we consider a flat horizontal interface. Let $S(x,t)$ be the cross-sectional area at coordinate $x$. We have $S(x,t)={\textstyle \frac{1}{2}}\unicode[STIX]{x1D706}H^{2}(x,t)$, with $H$, the liquid depth measured from the bottom of the substrate and $\unicode[STIX]{x1D706}={\textstyle \frac{1}{2}}(\sin (\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})/\sin \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FD})$. We denote as $S_{0}$, the liquid cross-sectional area at rest and as $H_{0}$, its corresponding depth. The pressure field is hydrostatic, hence it is independent of the $y$-coordinate and reads

(3.6)$$\begin{eqnarray}P(x,y,z,t)=\unicode[STIX]{x1D70C}g[H(x,t)-z].\end{eqnarray}$$

The Saint-Venant equations (3.4) and (3.5) after linearisation then read

(3.7)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}S+\unicode[STIX]{x2202}(S\bar{u})=2\unicode[STIX]{x1D706}H_{0}\unicode[STIX]{x2202}_{t}H+S_{0}\unicode[STIX]{x2202}_{x}\bar{u}=0, & \displaystyle\end{eqnarray}$$

(3.8)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\bar{u}+{\displaystyle \frac{1}{S}}\int _{S}{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}(\unicode[STIX]{x2202}_{x}P)\,\text{d}y\,\text{d}z=\unicode[STIX]{x2202}_{t}\bar{u}+g\unicode[STIX]{x2202}_{x}H=0, & \displaystyle\end{eqnarray}$$

so that we obtain the following wave equation on $H$

(3.9)$$\begin{eqnarray}\unicode[STIX]{x2202}_{tt}H-{\textstyle \frac{1}{2}}gH_{0}\unicode[STIX]{x2202}_{xx}H=0,\end{eqnarray}$$

hence the gravity wave speed

(3.10)$$\begin{eqnarray}c_{0}=\sqrt{{\displaystyle \frac{gH_{0}}{2}}}.\end{eqnarray}$$

3.3 Adiabatic perturbation hypothesis and discussion

In this article, we will make the following strong hypothesis: at any coordinate $x$, at any time $t$, we will assume that the cross-section shape (of area value $S(x,t)$) is adiabatically given by the equilibrium static shape calculated in § 2, at that very value $S(x,t)$, corresponding to a geometrical parameter $m(x,t)$. This assumption implies that the contact line is free to move on the superhydrophobic substrate, with contact angle at the substrate kept constant at value $\unicode[STIX]{x03C0}$. Moreover, we assume that the pressure field remains hydrostatic, hence independent of the $y$-coordinate.

In the current situation, translational invariance in the $x$-direction does not hold any longer so that the constant ${\mathcal{C}}_{0}$ given by (2.2) must depend on the coordinate $x$ and time $t$. In the framework of our assumption, the shape corresponds to that at equilibrium, provided it is given by some parameter $m(x,t)$ and we have

(3.11)$$\begin{eqnarray}{\mathcal{C}}_{0}(x,t)=\unicode[STIX]{x1D70C}g\unicode[STIX]{x1D701}(x,s,t)+\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}^{\mathit{t}}(x,s,t)\end{eqnarray}$$

at given $x$ and $t$, all along the interface, namely for all $s\in [0,s_{\mathit{tot}}(x,t)]$. It can be calculated anywhere along the interface. In the following, we will choose to calculate it at the top of the drop, i.e. at angle $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}$, at which we have $\unicode[STIX]{x1D701}(\unicode[STIX]{x03C0})=\unicode[STIX]{x0394}Z_{max}(m(x,t))$ and $\unicode[STIX]{x1D705}^{\mathit{t}}(x,s,t)=\unicode[STIX]{x1D705}_{\mathit{top}}(m(x,t))$, respectively, given by (2.16) and (2.17).

Let us now focus on the hydrostatic pressure field. A priori, for a point of coordinates $(x,y,z)$ inside the drop, its expression reads, because of the Young–Laplace law and (3.11)

(3.12)$$\begin{eqnarray}\displaystyle P(x,y,z,t) & = & \displaystyle \unicode[STIX]{x1D70C}g[\unicode[STIX]{x1D701}(x,s,t)-z]+\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}^{\mathit{t}}(x,s,t)+\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}^{\mathit{l}}(x,s,t),\end{eqnarray}$$

(3.13)$$\begin{eqnarray}\displaystyle & = & \displaystyle {\mathcal{C}}_{0}(x,t)-\unicode[STIX]{x1D70C}gz+\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}^{\mathit{l}}(x,s,t),\end{eqnarray}$$

with $s$ being an arclength coordinate satisfying $\unicode[STIX]{x1D702}(s,t)=y$ and the curvature term being the sum of the transverse curvature $\unicode[STIX]{x1D705}^{\mathit{t}}(x,s,t)$ (i.e. in the $(y,z)$-plane) and the longitudinal curvature $\unicode[STIX]{x1D705}^{\mathit{l}}(x,s,t)$ (i.e. in the $x$-direction) at a given point of coordinates $(x,s,t)$ located at the interface. At this stage, note that, because of the possible values of $s$, the pressure $P$ can be bivalued.

The adiabatic hypothesis regarding the equilibrium shape is valid in the limit $\unicode[STIX]{x1D705}^{\mathit{t}}\gg \unicode[STIX]{x1D705}^{\mathit{l}}$, which corresponds to the long-wave limit. Without entering into further technical details, at linear approximation, the longitudinal curvature term $\unicode[STIX]{x1D705}^{\mathit{l}}(x,s,t)$ will yield second-order contributions in $x$-derivatives of $S(x,t)$ in the expression of the pressure $P$, whereas $C_{0}(x,t)$ only involves zeroth-order contributions in $x$-derivatives of $S(x,t)$. Therefore, in the following, we will not account for the presence of $\unicode[STIX]{x1D705}^{\mathit{l}}$ and will then solely focus on the terms that will have non-dispersive contributions in the dispersion relation we are looking for. The section-averaged value of $\unicode[STIX]{x2202}_{x}P$ used in the Saint-Venant equations then reads

(3.14)$$\begin{eqnarray}{\displaystyle \frac{1}{S(x,t)}}\int _{S(x,t)}\{\unicode[STIX]{x2202}_{x}P(x,y,z,t)\}\,\text{d}y\,\text{d}z={\displaystyle \frac{1}{S(x,t)}}\int _{S(x,t)}\{\unicode[STIX]{x2202}_{x}{\mathcal{C}}_{0}(x,t)\}\,\text{d}y\,\text{d}z=\unicode[STIX]{x2202}_{x}{\mathcal{C}}_{0}(x,t).\end{eqnarray}$$

3.4 Plateau–Rayleigh instability in drops on flat hydrophobic substrate

We now turn to the stability study of the static sessile drops considered in § 2. As mentioned in § 3.3, we consider one particular class of perturbation by making the following assumption: the cross-section shape – of transverse area value $S(x,t)$ – is adiabatically given by the equilibrium static shape calculated in § 2, at that very value $S(x,t)$, corresponding to a geometrical parameter $m(x,t)$, at any given space coordinate $x$ and time $t$. The contact line is then free to move on the superhydrophobic substrate, keeping a constant contact angle equal to $\unicode[STIX]{x03C0}$. We focus on the varicose deformations; the liquid is supposed inviscid. Moreover, we choose the particular class of varicose perturbations with a symmetry about its $(x,z)$-mid-plane, as displayed in figure 2(c). The base state is supposed to correspond to the parameter $m_{0}$, with a cross-sectional area equal to $S_{0}=S(m_{0})$.

We set the perturbed cross-sectional area as being $S(x,t)=S_{0}+\unicode[STIX]{x1D70D}(x,t)$, with $\unicode[STIX]{x1D70D}\ll S_{0}$ and set $\bar{u}$ as the section-averaged velocity. Because of (3.14), the section-averaged value of $\unicode[STIX]{x2202}_{x}P$ can be expressed as

(3.15)$$\begin{eqnarray}\unicode[STIX]{x2202}_{x}\bar{p}=\unicode[STIX]{x1D70C}g\unicode[STIX]{x2202}_{x}[\unicode[STIX]{x0394}Z_{max}(x,t)]+\unicode[STIX]{x1D70E}\unicode[STIX]{x2202}_{x}[\unicode[STIX]{x1D705}_{\mathit{top}}(x,t)].\end{eqnarray}$$

The linearised Saint-Venant equations then read

(3.16)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D70D}+S_{0}\,\unicode[STIX]{x2202}_{x}\bar{u}=0, & \displaystyle\end{eqnarray}$$

(3.17)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\bar{u}+{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x2202}_{x}\bar{p}=0, & \displaystyle\end{eqnarray}$$

that is,

(3.18)$$\begin{eqnarray}\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D70D}={\displaystyle \frac{S_{0}}{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x2202}_{xx}\bar{p}.\end{eqnarray}$$

Knowing that

(3.19)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{x}[\unicode[STIX]{x0394}Z_{max}(x,t)]=\left[{\displaystyle \frac{\unicode[STIX]{x2202}[\unicode[STIX]{x0394}Z_{max}]}{\unicode[STIX]{x2202}m}}\right]\left[{\displaystyle \frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}m}}\right]^{-1}\left[{\displaystyle \frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}x}}\right], & & \displaystyle\end{eqnarray}$$

(3.20)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{x}[\unicode[STIX]{x1D705}_{\mathit{top}}(x,t)]=\left[{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{\mathit{top}}}{\unicode[STIX]{x2202}m}}\right]\left[{\displaystyle \frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}m}}\right]^{-1}\left[{\displaystyle \frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}x}}\right], & & \displaystyle\end{eqnarray}$$

we obtain the following wave equation, after only retaining the linear contributions of $\unicode[STIX]{x1D70D}$:

(3.21)$$\begin{eqnarray}\unicode[STIX]{x2202}_{tt}\unicode[STIX]{x1D70D}=g\left[{\displaystyle \frac{S_{0}}{{\displaystyle \frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}m}}(m_{0})}}\right]\left\{\!\left[{\displaystyle \frac{\unicode[STIX]{x2202}[\unicode[STIX]{x0394}Z_{max}]}{\unicode[STIX]{x2202}m}}(m_{0})\right]+{\displaystyle \frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}g}}\left[{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{\mathit{top}}}{\unicode[STIX]{x2202}m}}(m_{0})\right]\!\right\}\unicode[STIX]{x2202}_{xx}\unicode[STIX]{x1D70D}.\end{eqnarray}$$

Setting $\unicode[STIX]{x1D70D}(x,t)=\unicode[STIX]{x1D700}\exp [\text{i}(\unicode[STIX]{x1D714}t-kx)]$ eventually yields the following dispersion relation:

(3.22)$$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}(k)=gk^{2}\left[{\displaystyle \frac{S_{0}}{{\displaystyle \frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}m}}(m_{0})}}\right]\left\{\!\left[{\displaystyle \frac{\unicode[STIX]{x2202}[\unicode[STIX]{x0394}Z_{max}]}{\unicode[STIX]{x2202}m}}(m_{0})\right]+\ell _{\mathit{c}}^{2}\left[{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{\mathit{top}}}{\unicode[STIX]{x2202}m}}(m_{0})\right]\!\right\}.\end{eqnarray}$$

We have an instability whenever $\unicode[STIX]{x1D714}^{2}(k)<0$; conversely, if $\unicode[STIX]{x1D714}^{2}(k)>0$, the varicose mode is stable.

Let the following dimensionless quantities be

(3.23)$$\begin{eqnarray}\displaystyle & \displaystyle a(m_{0})=\left[{\displaystyle \frac{S_{0}}{{\displaystyle \frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}m}}(m_{0})}}\right]=2m_{0}(1-m_{0}){\displaystyle \frac{(m_{0}-2)\mathit{K}(m_{0})+2\mathit{E}(m_{0})}{(m_{0}-4)(m_{0}-1)\mathit{K}(m_{0})+(3m_{0}-4)\mathit{E}(m_{0})}},\qquad & \displaystyle\end{eqnarray}$$

(3.24)$$\begin{eqnarray}\displaystyle & \displaystyle b(m_{0})={\displaystyle \frac{1}{\ell _{\mathit{c}}}}\left[{\displaystyle \frac{\unicode[STIX]{x2202}[\unicode[STIX]{x0394}Z_{max}]}{\unicode[STIX]{x2202}m}}(m_{0})\right]+\ell _{\mathit{c}}\left[{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{\mathit{top}}}{\unicode[STIX]{x2202}m}}(m_{0})\right]=-{m_{0}}^{-3/2}, & \displaystyle\end{eqnarray}$$

we then obtain the dispersion relation, that reads

(3.25)$$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}(k)=g\ell _{\mathit{c}}k^{2}[a(m_{0})]\,[b(m_{0})],\end{eqnarray}$$

which yields the expression of the wave speed

(3.26)$$\begin{eqnarray}c_{\mathit{eff}}^{2}(m_{0})=g\ell _{\mathit{c}}[a(m_{0})]\,[b(m_{0})].\end{eqnarray}$$

This quantity is always negative because of the factor $b(m_{0})$ (see (3.24)), so that a cylindrical sessile drop on a flat superhydrophobic substrate is always unstable regardless of its size. As a visual illustration, in figure 4, we plot the quantities $a(m)$ and $c_{\mathit{eff}}^{2}/(g\ell _{\mathit{c}})=a(m)\,b(m)$ as functions of the parameter $m$.

Figure 4.

Plots of (a) the coefficients $a(m)$ together with (b) the renormalised $c_{\mathit{eff}}^{2}/(g\ell _{\mathit{c}})=a(m)\,b(m)$ with respect to the parameter $m$. The values of $c_{\mathit{eff}}^{2}$ are always negative, therefore, the drop is unstable. The larger $m$, the smaller the growth rate. In the limit $m\rightarrow 1^{-}$, we have $c_{\mathit{eff}}^{2}\rightarrow 0^{-}$; in this regime, the horizontal extension of the cylinder tends to infinity and the latter becomes marginally unstable.

Figure 5.

Geometry of a sessile cylindrical drop deposited upon a wedge-shaped substrate with angles to the horizontal equal to $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$. In dashed cyan is superimposed the shape of a drop of same horizontal extension $\unicode[STIX]{x0394}Y_{max}$, deposited upon a flat superhydrophobic substrate. Eight particular points are shown (see § 3.5 for their definition). Here, $\unicode[STIX]{x1D6E4}_{0}$ denotes the free surface, while $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FD}}$ correspond to the parts of the wedge wetted by the drop. They connect $M_{\unicode[STIX]{x1D6FC}}$ to $M_{\mathit{bott}}$ and $M_{\mathit{bott}}$ to $M_{\unicode[STIX]{x1D6FD}}$, respectively.

In the following, we will make more quantitative statements, by analysing these results in two distinct limits: small $m$ regime, i.e. small cross-section cylinder and large $m$ regime ($m$ close to $1^{-}$), i.e. large cross-section regime.

3.4.1 Small $m$ regime: nearly circle cross-section, Plateau–Rayleigh instabilityrecovered

Let us consider the asymptotic regime $m\rightarrow 0^{+}$. We have

(3.27)$$\begin{eqnarray}\displaystyle & \displaystyle a(m)\underset{m\rightarrow 0^{+}}{=}m-{\textstyle \frac{3}{4}}m^{2}+\mathit{O}(m^{3}), & \displaystyle\end{eqnarray}$$

(3.28)$$\begin{eqnarray}\displaystyle & \displaystyle b(m)=-m^{-3/2}, & \displaystyle\end{eqnarray}$$

and the dispersion relation then reads

(3.29)$$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}(k,m)\underset{m\rightarrow 0^{+}}{\simeq }g\ell _{\mathit{c}}k^{2}\left[-{\displaystyle \frac{1}{\sqrt{m}}}\right].\end{eqnarray}$$

We see that for small values of $m$, we have an instability corresponding to the classic Plateau–Rayleigh instability: rewriting this expression in terms of radius of the drop $R_{0}={\textstyle \frac{1}{2}}\ell _{\mathit{c}}\sqrt{m}$ (see (2.28)) indeed leads to

(3.30)$$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}(k,R_{0})\underset{m\rightarrow 0^{+}}{\simeq }=-{\displaystyle \frac{\unicode[STIX]{x1D70E}}{2\unicode[STIX]{x1D70C}R_{0}^{3}}}(kR_{0})^{2}.\end{eqnarray}$$

As a reminder, we recall that the growth rate $\unicode[STIX]{x1D6F4}$ of the classic Plateau–Rayleigh instability in the case of a free cylinder of radius $R_{0}$ reads

(3.31)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{2}(k,R_{0})={\displaystyle \frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}R_{0}^{3}}}{\displaystyle \frac{\text{I}_{1}(kR_{0})}{\text{I}_{0}(kR_{0})}}[1-(kR_{0})^{2}]kR_{0},\end{eqnarray}$$

with $\text{I}_{0}$ and $\text{I}_{1}$ the zeroth and first modified Bessel functions of the first kind, which satisfy for small $x$, $\text{I}_{0}(x)\underset{x\rightarrow 0}{{\sim}}1$ and $\text{I}_{1}(x)\underset{x\rightarrow 0}{{\sim}}x/2$. The angular frequency $\unicode[STIX]{x1D714}$ satisfies the exact same properties as the growth rate $\unicode[STIX]{x1D6F4}$ at small $k$, at leading quadratic order.

3.4.2 Large cross-section regime

For large cross-section drops, the Plateau–Rayleigh instability still holds, for $b(m)$ is always negative, for any $m\in \, ]0,1[$ (see (3.24)). Still, let us be more quantitative. For $m=1-\unicode[STIX]{x1D707}$ with $\unicode[STIX]{x1D707}\rightarrow 0^{+}$, we have

(3.32)$$\begin{eqnarray}\displaystyle & \displaystyle a(\unicode[STIX]{x1D707})=-\unicode[STIX]{x1D707}[\log (\unicode[STIX]{x1D707})+4(1-\log (2))]+\mathit{O}(\unicode[STIX]{x1D707}^{2}), & \displaystyle\end{eqnarray}$$

(3.33)$$\begin{eqnarray}\displaystyle & \displaystyle b(\unicode[STIX]{x1D707})\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{=}-1-{\displaystyle \frac{3\unicode[STIX]{x1D707}}{2}}-{\displaystyle \frac{15\unicode[STIX]{x1D707}^{2}}{8}}+\mathit{O}(\unicode[STIX]{x1D707}^{3}), & \displaystyle\end{eqnarray}$$

so that the wave celerity reads

(3.34)$$\begin{eqnarray}c_{\mathit{eff}}^{2}\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{\simeq }g\ell _{\mathit{c}}\unicode[STIX]{x1D707}[\log (\unicode[STIX]{x1D707})+4(1-\log 2)]\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{\longrightarrow }0^{-}.\end{eqnarray}$$

In the limit of very large cross-sectional areas, the growth rate vanishes and wide drops become marginally unstable. This result is reminiscent of the result evidenced by Diez et al. (Reference Diez, González and Kondic2009), in a different wetting situation (partial wetting), who stated that, because of gravity, a rivulet was unstable for a reduced interval of small wavenumbers.

As a general conclusion to this subsection, a sessile cylindrical drop placed upon a superhydrophobic flat substrate is always unstable whatever its volume by unit length may be.

3.5 Stabilisation due to substrate shape

In this section, we will study the stability of our previous sessile drop, this time deposited upon a non-flat superhydrophobic substrate, more specifically, a substrate that has a wedge-shaped geometry. We will start with the general case and then focus on two particular geometries used in our experimental set-ups, as a comparison with theoretical results. This study will greatly depart from the studies regarding wedge-shaped substrates that were cited in the introduction (Langbein Reference Langbein1990; Roy & Schwartz Reference Roy and Schwartz1999; Yang & Homsy Reference Yang and Homsy2007; Speth & Lauga Reference Speth and Lauga2009) because we account for gravity effects that will play a crucial role on stabilising sessile drops, as we will see.

The substrate that we consider is a wedge with angles to the horizontal equal to $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$. Without any loss of generality, we will assume that $0<\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D6FD}<\unicode[STIX]{x03C0}/2$. When $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$, the substrate is symmetric. The shape of the liquid domain will consist in a sessile liquid cap ($\unicode[STIX]{x1D6E4}_{0}$ will denote its boundary) connected to tangent straight lines of angle $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ with the horizontal, i.e. that of a (partial) sessile cap placed upon a triangular liquid cylinder, as depicted in figure 5. The sessile cap profile $\unicode[STIX]{x1D6E4}_{0}$ corresponds to a partial pendulum orbit starting at $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6FD}$, ending at $\unicode[STIX]{x1D719}=2\unicode[STIX]{x03C0}-\unicode[STIX]{x1D6FC}$ in the phase space. Such a cross-section shape can be characterised by the eight following characteristic points $M$, with the corresponding coordinates $(\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})$ given by (2.15). We first choose the reference point $M_{0}$ as the origin $(0,0)$ and the other points are

(3.35)$$\begin{eqnarray}\displaystyle & \displaystyle M_{2\unicode[STIX]{x03C0}},\quad \unicode[STIX]{x1D702}_{1}=\unicode[STIX]{x1D702}(2\unicode[STIX]{x03C0})=-\unicode[STIX]{x1D6EC}_{\mathit{c}},\quad \unicode[STIX]{x1D701}_{1}=0, & \displaystyle\end{eqnarray}$$

(3.36)$$\begin{eqnarray}\displaystyle & \displaystyle M_{\unicode[STIX]{x1D6FD}},\quad \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FD}}=\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6FD}),\quad \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FD}}=\unicode[STIX]{x1D701}(\unicode[STIX]{x1D6FD}), & \displaystyle\end{eqnarray}$$

(3.37)$$\begin{eqnarray}\displaystyle & \displaystyle M_{\unicode[STIX]{x03C0}/2},\quad \unicode[STIX]{x1D702}_{\mathit{tip}}^{+}=\unicode[STIX]{x1D702}\left({\displaystyle \frac{\unicode[STIX]{x03C0}}{2}}\right),\quad \unicode[STIX]{x1D701}_{\mathit{tip}}^{+}=\unicode[STIX]{x1D701}\left({\displaystyle \frac{\unicode[STIX]{x03C0}}{2}}\right), & \displaystyle\end{eqnarray}$$

(3.38)$$\begin{eqnarray}\displaystyle & \displaystyle M_{\unicode[STIX]{x03C0}},\quad \unicode[STIX]{x1D702}_{\mathit{top}}=\unicode[STIX]{x1D702}(\unicode[STIX]{x03C0}),\quad \unicode[STIX]{x1D701}_{\mathit{top}}=z(\unicode[STIX]{x03C0}), & \displaystyle\end{eqnarray}$$

(3.39)$$\begin{eqnarray}\displaystyle & \displaystyle M_{3\unicode[STIX]{x03C0}/2},\quad \unicode[STIX]{x1D702}_{\mathit{tip}}^{-}=\unicode[STIX]{x1D702}\left({\displaystyle \frac{3\unicode[STIX]{x03C0}}{2}}\right),\quad \unicode[STIX]{x1D701}_{\mathit{tip}}^{-}=\unicode[STIX]{x1D701}\left({\displaystyle \frac{3\unicode[STIX]{x03C0}}{2}}\right)=\unicode[STIX]{x1D701}_{\mathit{tip}}^{+}, & \displaystyle\end{eqnarray}$$

(3.40)$$\begin{eqnarray}\displaystyle & \displaystyle M_{\unicode[STIX]{x1D6FC}},\quad \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D702}(2\unicode[STIX]{x03C0}-\unicode[STIX]{x1D6FC}),\quad \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D701}(2\unicode[STIX]{x03C0}-\unicode[STIX]{x1D6FC}). & \displaystyle\end{eqnarray}$$

The last point $M_{\mathit{bott}}$ is located at the bottom of the wedge and has the following coordinates:

(3.41)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{\mathit{bott}}=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}+{\displaystyle \frac{\cos \unicode[STIX]{x1D6FC}}{\sin (\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})}}[(\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}})\sin \unicode[STIX]{x1D6FD}-(\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FC}})\cos \unicode[STIX]{x1D6FD}], & \displaystyle\end{eqnarray}$$

(3.42)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}_{\mathit{bott}}=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FC}}-{\displaystyle \frac{\sin \unicode[STIX]{x1D6FC}}{\sin (\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})}}[(\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}})\sin \unicode[STIX]{x1D6FD}-(\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FC}})\cos \unicode[STIX]{x1D6FD}]\equiv -H_{\mathit{bott}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},m).\qquad & \displaystyle\end{eqnarray}$$

Note that the points $M_{0}$ and $M_{2\unicode[STIX]{x03C0}}$ are located inside the cross-section, they do not belong to the boundary of the cross-section and serve as known references.

Using the Green–Riemann formula, we can exactly compute the cross-sectional area of the cylinder by an integration along $\unicode[STIX]{x1D6E4}$, the closed curve $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{0}\cup \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FC}}\cup \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FD}}$ with positive orientation, that is, using $S_{\mathit{tot}}=-\oint _{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D701}\,\text{d}\unicode[STIX]{x1D702}$,

(3.43)$$\begin{eqnarray}S_{\mathit{tot}}=-\int _{\unicode[STIX]{x1D6FD}}^{2\unicode[STIX]{x03C0}-\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D701}(\unicode[STIX]{x1D711}){\displaystyle \frac{\text{d}\unicode[STIX]{x1D702}}{\text{d}\unicode[STIX]{x1D711}}}(\unicode[STIX]{x1D711})\,\text{d}\unicode[STIX]{x1D711}-\int _{\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D701}^{(\unicode[STIX]{x1D6FC})}\,\text{d}\unicode[STIX]{x1D702}^{(\unicode[STIX]{x1D6FC})}-\int _{\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D701}^{(\unicode[STIX]{x1D6FD})}\,\text{d}\unicode[STIX]{x1D702}^{(\unicode[STIX]{x1D6FD})},\end{eqnarray}$$

with $(\unicode[STIX]{x1D702}^{(\unicode[STIX]{x1D6FC})},\unicode[STIX]{x1D701}^{(\unicode[STIX]{x1D6FC})})$ and $(\unicode[STIX]{x1D702}^{(\unicode[STIX]{x1D6FD})},\unicode[STIX]{x1D701}^{(\unicode[STIX]{x1D6FD})})$ straightforward parameterisations of the line segments $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FD}}$, respectively, connecting $M_{\unicode[STIX]{x1D6FC}}$ to $M_{\mathit{bott}}$ and $M_{\mathit{bott}}$ to $M_{\unicode[STIX]{x1D6FD}}$ (see figure 5). This yields the expression for the total cross-sectional area $S_{\mathit{tot}}(m,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$,

(3.44)$$\begin{eqnarray}\displaystyle S_{\mathit{tot}}(m,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}) & = & \displaystyle -{\displaystyle \frac{1}{m}}\left\{2(m-2)\left[\mathit{F}\left(\left.\unicode[STIX]{x03C0}-{\displaystyle \frac{\unicode[STIX]{x1D6FC}}{2}}\right|m\right)-\mathit{F}\left(\left.{\displaystyle \frac{\unicode[STIX]{x1D6FD}}{2}}\right|m\right)\right]\right.\nonumber\\ \displaystyle & & \displaystyle +\left.4\left[\mathit{E}\left(\left.\unicode[STIX]{x03C0}-{\displaystyle \frac{\unicode[STIX]{x1D6FC}}{2}}\right|m\right)-\mathit{E}\left(\left.{\displaystyle \frac{\unicode[STIX]{x1D6FD}}{2}}\right|m\right)\right]\!\right\}\nonumber\\ \displaystyle & & \displaystyle -\,{\displaystyle \frac{1}{2}}[(\unicode[STIX]{x1D702}_{\mathit{bott}}-\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}})(\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D701}_{\mathit{bott}})+(\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D702}_{\mathit{bott}})(\unicode[STIX]{x1D701}_{\mathit{bott}}+\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FD}})],\end{eqnarray}$$

with expressions of $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FC}}$, $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D6FD}}$, $\unicode[STIX]{x1D702}_{\mathit{bott}}$, $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FC}}$, $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FD}}$, $\unicode[STIX]{x1D701}_{\mathit{bott}}$ given by (2.14) and (3.35)–(3.42), which yield an explicit expression for the cross-sectional area, for given parameter $m$ and angles $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$. Note that the dependence of $S_{\mathit{tot}}$ on $m$ is monotonic, so that the relationship between both quantities is unequivocal. In the flat substrate case, for a given cross-sectional area value $S$, the interface has translational invariance in the $y$-direction if the starting point $M(s=0)$ is not specified. In contrast, in the wedge case, the relative position of the interface to $M_{\mathit{bott}}$ is unequivocal, once the cross-sectional area $S_{\mathit{tot}}$ is set.

In the following, we will focus on the symmetrical case, namely $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$. We will again study the dispersion relation of our system in the framework of the Saint-Venant equations (see (3.16), (3.17) and below), that is to say, in the non-dispersive limit. We will only retain the partial derivatives of order zero in $x$, i.e. only the quadratic terms in $k$, in the dispersion relation (3.25). Possible instabilities will appear at small wavenumber.

The presence of the symmetrical substrate adds a supplementary term to the expression of the hydrostatic pressure given by (3.15); due to the presence of the wedge underneath the liquid cap, the position of the interface now reads, on top of the drop

(3.45)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{\mathit{top}}(x,t)=\unicode[STIX]{x0394}Z_{max}(x,t)+H_{\mathit{bott}}(x,t),\end{eqnarray}$$

so that we can now write

(3.46)$$\begin{eqnarray}{\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x2202}_{x}\bar{p}=g\unicode[STIX]{x2202}_{x}[\unicode[STIX]{x0394}Z_{max}(x,t)+H_{\mathit{bott}}(x,t)]+{\displaystyle \frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}}}\unicode[STIX]{x2202}_{x}[\unicode[STIX]{x1D705}_{\mathit{top}}(x,t)].\end{eqnarray}$$

Equation (3.44) allows us to compute the derivative of the cross-sectional area with respect to the parameter $m$, so that the effective speed of propagation of waves along a sessile cylindrical drop above a wedge can be explicitly computed, using

(3.47)$$\begin{eqnarray}c_{\mathit{eff}}^{2}(m,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})=g\ell _{\mathit{c}}\left[{\displaystyle \frac{S_{\mathit{tot}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},m)}{{\displaystyle \frac{\unicode[STIX]{x2202}S_{\mathit{tot}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},m)}{\unicode[STIX]{x2202}m}}}}\right]\left[{\displaystyle \frac{1}{\ell _{\mathit{c}}}}{\displaystyle \frac{\unicode[STIX]{x2202}H_{\mathit{bott}}}{\unicode[STIX]{x2202}m}}-m^{-3/2}\right],\end{eqnarray}$$

for a parameter $m$ corresponding to that of the drop at rest, knowing that $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC}$. The full explicit expression of $c_{\mathit{eff}}^{2}(m,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$ is overly complicated to be written down. However, one important remark is that the negative factor $-m^{-3/2}$ yielding the Plateau–Rayleigh instability in the case of a flat substrate can be compensated by the additional positive term $\ell _{\mathit{c}}^{-1}\times [\unicode[STIX]{x2202}H_{\mathit{bott}}/\unicode[STIX]{x2202}m]$ stemming from the presence of the non-flat substrate underneath the sessile drop, provided it is sufficiently large.

At given angle $\unicode[STIX]{x1D6FC}$, the quantity $(1/\ell _{\mathit{c}})(\unicode[STIX]{x2202}H_{\mathit{bott}}/\unicode[STIX]{x2202}m)-m^{-3/2}$ is negative for small values of $m$. It becomes positive when $m$ is larger than a critical value $m_{\mathit{c}}=1-\unicode[STIX]{x1D707}_{\mathit{c}}$. This latter quantity can be numerically computed and its dependence on the angle $\unicode[STIX]{x1D6FC}$ is plotted in figure 6. For $\unicode[STIX]{x1D6FC}=0$, i.e. for a flat substrate, the cylinder is unstable whatever its cross-section may be (see § 3.4.2).

Figure 6.

Dependence of the critical parameter $m_{\mathit{c}}$ on the angle $\unicode[STIX]{x1D6FC}$ of a symmetrical V-shaped substrate ($\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$). For positive $\unicode[STIX]{x1D6FC}$, large drops ($m$ close to 1) are stable as long as $m>m_{\mathit{c}}(\unicode[STIX]{x1D6FC})$.

In the following, in order to illustrate all these calculations in a meaningful way, we perform some approximate computations and discuss their validity. They will come in handy for comparisons with the experimental results (see § 4).

Figure 7.

Geometry of the two different substrate shapes used in our experiments. (a) V-shaped symmetrical channel. (b) L-shaped asymmetrical channel. For a given parameter $m$, $W(m)$ is the width of the sessile cap, whereas $\overline{W}(m)$ denotes the width of the triangular cross-section, which can somehow be approximated by the wetted length $\unicode[STIX]{x1D6EC}_{\mathit{c}}(m)$ (see text).

As a reminder, we are dealing with the symmetrical case ($\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$) and we will make the following hypotheses: the angle $\unicode[STIX]{x1D6FC}$ is supposedly small and the parameter $m=1-\unicode[STIX]{x1D707}$ is close to 1, i.e. large flattened cross-sections are considered. We can approximate the geometrical properties of our system by considering a cap (given by the shape of a sessile drop on a flat surface) placed on top of a symmetrical triangle (see figure 7a). We have the following approximate quantities, using the asymptotic formulas of § 2.2,

1. (i) for the sessile cap:

   1. (a) width $W(m)\simeq \unicode[STIX]{x0394}Y_{max}\simeq \ell _{\mathit{c}}[-\log (1-m)+\mathscr{C}_{1}]$;

   2. (b) cross-sectional area $S_{\mathit{cap}}(m)\simeq W(m)\times \unicode[STIX]{x0394}Z_{max}(m)\simeq 2\ell _{\mathit{c}}^{2}[-\log (1-m)+\mathscr{C}_{1}]$;

2. (ii) for the triangle:

   1. (a) width $\overline{W}(m)\simeq \unicode[STIX]{x1D6EC}_{\mathit{c}}(m)\simeq \ell _{\mathit{c}}[-\log (1-m)+\mathscr{C}_{2}]$;

   2. (b) height $H_{\mathit{bott}}(m)\simeq {\textstyle \frac{1}{2}}\overline{W}(m)\tan \unicode[STIX]{x1D6FC}$;

   3. (c) cross-sectional area $S_{0}={\textstyle \frac{1}{4}}\overline{W}(m)^{2}\tan \unicode[STIX]{x1D6FC}$.

The total cross-sectional area is

(3.48)$$\begin{eqnarray}\displaystyle S_{\mathit{tot}}(m) & \simeq & \displaystyle S_{\mathit{cap}}(m)+S_{0}(m)\simeq 2\ell _{\mathit{c}}^{2}W(m)+{\displaystyle \frac{1}{4}}\overline{W}(m)^{2}\tan \unicode[STIX]{x1D6FC},\nonumber\\ \displaystyle & \simeq & \displaystyle \ell _{\mathit{c}}^{2}\left[2(-\log (1-m)+\mathscr{C}_{1})+{\displaystyle \frac{\tan \unicode[STIX]{x1D6FC}}{4}}[-\log (1-m)+\mathscr{C}_{2}]^{2}\right],\end{eqnarray}$$

so that

(3.49)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}S_{\mathit{tot}}}{\unicode[STIX]{x2202}m}}(m)=\ell _{\mathit{c}}^{2}{\displaystyle \frac{\tan \unicode[STIX]{x1D6FC}\,[\mathscr{C}_{2}-\log (1-m)]+4}{2(1-m)}}.\end{eqnarray}$$

The effective wave celerity then reads, using (3.47),

(3.50)$$\begin{eqnarray}c_{\mathit{eff}}^{2}(m)=g\ell _{\mathit{c}}\left[{\displaystyle \frac{S_{\mathit{tot}}(m)}{{\displaystyle \frac{\unicode[STIX]{x2202}S_{\mathit{tot}}}{\unicode[STIX]{x2202}m}}(m)}}\right]\left\{+{\displaystyle \frac{\tan \unicode[STIX]{x1D6FC}}{2\unicode[STIX]{x1D707}}}-1-{\displaystyle \frac{3\unicode[STIX]{x1D707}}{2}}+\mathit{O}(\unicode[STIX]{x1D707}^{2})\right\},\end{eqnarray}$$

with

(3.51)$$\begin{eqnarray}\left[{\displaystyle \frac{S_{\mathit{tot}}(m)}{{\displaystyle \frac{\unicode[STIX]{x2202}S_{\mathit{tot}}}{\unicode[STIX]{x2202}m}}(m)}}\right]=2(1-m){\displaystyle \frac{2[\mathscr{C}_{1}-\log (1-m)]+{\displaystyle \frac{\tan \unicode[STIX]{x1D6FC}}{4}}[\mathscr{C}_{2}-\log (1-m)]^{2}}{\tan \unicode[STIX]{x1D6FC}[\mathscr{C}_{2}-\log (1-m)]+4}}.\end{eqnarray}$$

If we have $\unicode[STIX]{x1D6FC}=0$ (flat substrate), we recover $c_{\mathit{eff}}^{2}<0$, corresponding to an unstable drop, whatever its size may be. For small values of $\unicode[STIX]{x1D707}$ and $\tan \unicode[STIX]{x1D6FC}\neq 0$, the drop is stable, provided $\tan \unicode[STIX]{x1D6FC}/2\unicode[STIX]{x1D707}-1-{\textstyle \frac{3}{2}}\unicode[STIX]{x1D707}>0$, that is

(3.52)$$\begin{eqnarray}\unicode[STIX]{x1D707}<\tilde{\unicode[STIX]{x1D707}}_{\mathit{c}}={\displaystyle \frac{\sqrt{1+3\tan \unicode[STIX]{x1D6FC}}-1}{3}}.\end{eqnarray}$$

As a comparison, in the experiments we present in the next section, we have typical values of width equal to $7.5\,\ell _{\mathit{c}}$, that corresponds to a parameter $\unicode[STIX]{x1D707}_{\mathit{exp}}\simeq 5\times 10^{-4}$, which satisfies the stability criterion.

Table 1.

Values of critical $\unicode[STIX]{x1D707}_{\mathit{c}}$ together with corresponding non-dimensional values of horizontal extension $\unicode[STIX]{x0394}Y_{max}(\unicode[STIX]{x1D707}_{\mathit{c}})/\ell _{\mathit{c}}$ and of cross-sectional area $S(\unicode[STIX]{x1D707}_{\mathit{c}})/\ell _{\mathit{c}}^{2}$, for a symmetrical V-shaped substrate. The values in the framework of the approximate computations (namely $\tilde{\unicode[STIX]{x1D707}}_{\mathit{c}}$, see § 3.5) are added for comparison.

Note that up to $\unicode[STIX]{x1D6FC}\simeq 10^{\circ }$, our approximations in terms of $\log \unicode[STIX]{x1D707}$ are valid for calculating the critical threshold. For large angles $\unicode[STIX]{x1D6FC}$, this approximate threshold value is arguable. One has to resort to the exact analytical computations presented previously. However, comparisons between these approximations and actual exact results remain surprisingly good, the relative errors being no greater than 30 % for $\unicode[STIX]{x1D6FC}$ up to $45^{\circ }$ (see table 1).

We can compute an ultimate term, that of (3.51), in the limit $\unicode[STIX]{x1D707}\rightarrow 0^{+}$, that reads

(3.53)$$\begin{eqnarray}\left[{\displaystyle \frac{S_{\mathit{tot}}}{{\displaystyle \frac{\unicode[STIX]{x2202}S_{\mathit{tot}}}{\unicode[STIX]{x2202}m}}}}\right]\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{{\sim}}2\unicode[STIX]{x1D707}{\displaystyle \frac{{\displaystyle \frac{\tan \unicode[STIX]{x1D6FC}}{4}}\log ^{2}\unicode[STIX]{x1D707}-2\log \unicode[STIX]{x1D707}}{-\tan (\unicode[STIX]{x1D6FC})\log \unicode[STIX]{x1D707}+4}}\underset{\unicode[STIX]{x1D707}\rightarrow 0^{+}}{{\sim}}-{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D707}\log \unicode[STIX]{x1D707}.\end{eqnarray}$$

We recover the expression of the wave celerity $c_{\mathit{eff}}$,

(3.54)$$\begin{eqnarray}c_{\mathit{eff}}^{2}=-g\ell _{\mathit{c}}{\displaystyle \frac{\tan \unicode[STIX]{x1D6FC}}{4}}\log \unicode[STIX]{x1D707}=g{\displaystyle \frac{H_{\mathit{bott}}}{2}},\end{eqnarray}$$

in the large cross-section limit as explained in § 3.2 (see (3.10)). It is the gravity wave speed directly given by the Saint-Venant equations, in the case of a symmetrical channel of angle $\unicode[STIX]{x1D6FC}$, in the absence of capillarity. The presence of a sessile cap is of no importance in this limit and it thus can be disregarded, as one would have expected it to be.

4.1 Experimental set-up

In this section, we present an experimental characterisation of the varicose waves propagating along cylindrical drops, when deposited upon superhydrophobic substrates. The critical volume at which breakup of these drops occurs is studied as well. In order to distinguish the theoretical liquid cylinder shape from its experimental realisation, we use the term ‘drop’ in this section to denote a slender volume of liquid deposited upon a non-flat substrate with translational invariance.

For this study, we use two different substrates. The first one is a symmetrical V-shaped Dural channel with angle $\unicode[STIX]{x1D6FC}=10^{\circ }$. The second one is an L-shaped Dural channel, placed upon a levelling table, so that it can be tilted by an angle $\unicode[STIX]{x1D6FC}$, set from $0^{\circ }$ up to $45^{\circ }$ (see the schematic of substrate cross-section in figure 7). The surface of these substrates is treated using Rust-Oleum NeverWet, so that the wetting conditions (contact angle around $160^{\circ }$) can be considered as superhydrophobic (see Gupta, Vaikuntanathan & Siakumar Reference Gupta, Vaikuntanathan and Siakumar2016). Both channels are 50 cm long. On either of them, a controlled volume of distilled water is deposited and stretched from one end to the other end of the channel. If the deposited volume is sufficiently large, we can pin such long drops at both ends of the channel without breakup.

In order to study the propagation of varicose waves along these drops, we force one end of the drop using a shaker at swept frequencies (1–25 Hz) (see figure 8a). Two photographs are shown as an illustration in figure 8(b,c). The plate connected to the shaker can be seen at the right-hand side of the photographs. The photographs correspond to excited drops of volume equal to 40 ml, deposited on the symmetric V-shaped substrate and the non-symmetrical L-shaped substrate, respectively, with a tilt angle $\unicode[STIX]{x1D6FC}=10^{\circ }$. In the symmetrical case, constriction modes are clearly visible near the shaker, whereas in the non-symmetrical case, only the deformations of one side of the drop can be captured, for deformations of the border of the drop at the steepest side of the substrate are too small to be visible. We then naturally focus on the interface located on the smallest slope of the substrate. The deformation of the border of the drop and consequently the position $\unicode[STIX]{x1D702}(x,t)$ of the border of the interface is recorded from above using a digital camera at 50 f.p.s. (AVT Manta G223). The interface position is numerically reconstructed using an algorithm of contour detection based on the change in the colour gradient, with a 0.03 mm accuracy: we record images of a region of interest of a border of the drop. Suppose the images have resolution $\mathtt{Nx}\times \mathtt{Ny}$, at all discrete $x$-coordinates $(x_{i})_{1\leqslant i\leqslant \mathtt{Nx}}$, we locate the minimum of intensity in the $y$-direction (for $1\leqslant j\leqslant \mathtt{Ny}$), using polynomial interpolations of the discrete values in this direction. The procedure is performed at each discrete time $t_{k}$ ($1\leqslant k\leqslant \mathtt{Nt}$), so that we obtain a discrete signal $\unicode[STIX]{x1D702}(x_{i},t_{k})$. Last, we compute the position at rest $\unicode[STIX]{x1D702}_{0}(x_{i})$ of the interface, using the same procedure.

Figure 8.

(a) Schematic of the experimental set-up. Top views of experiments of shaken drops of volume 40 ml (b) on a symmetrical V-shaped substrate (of angle $\unicode[STIX]{x1D6FC}=10^{\circ }$) and (c) on a non-symmetrical L-shaped substrate tilted by an angle $\unicode[STIX]{x1D6FC}=10^{\circ }$. They are lit so that the border of the drop is clearly visible for numerical treatments. Varicose modes, i.e. constriction modes, are clearly visible near the shaker. They are symmetrical in the V-shaped case. In contrast, due to the asymmetry of the L-shaped substrate, only the constrictions of the interface border located at the gentle slope side (upper side of the image) are clearly visible. They propagate and vanish far from the shaker because of dissipation and dispersion.

4.2 Experimental results

From the camera recordings, we compute the spatio-temporal spectrum $S(k,f)$ of the disturbed interface by performing fast Fourier transforms on the difference $\unicode[STIX]{x1D702}(x_{i},t_{k})-\unicode[STIX]{x1D702}_{0}(x_{i})$. The spatial resolution is $0.252~\text{mm}~\text{pixels}$, and we use a typical image resolution Nx equal to 1690 pixels. As for the temporal sampling, the typical recording duration is 2 minutes, which yields a typical number of time samples $\mathtt{Nt}\sim 6000$. Several branches can be identified from the spatio-temporal spectra (see figure 9). We focus here on the lowest branch, which goes to zero when $k\rightarrow 0$ (namely the varicose branch). Other branches (characterised by cutoff frequencies at $k=0$) are related to the other modes of wave propagation, the very first upper branch being related to sloshing motion (namely sinuous waves).

From the value of deposited volume, $V_{0}$, we can deduce the value at rest of the cross-sectional area, $S_{\mathit{tot}}=V_{0}/L$, with $L$, the length of the cylinder. Using the expression (3.44) of $S_{\mathit{tot}}(m)$, we can estimate the corresponding parameter $m$, hence the width $W(m)$. From these data, we can evaluate $c_{\mathit{eff}}$ and plot $f(k)=(1/2\unicode[STIX]{x03C0})c_{\mathit{eff}}k$ (the non-dispersive long-wavelength regime given by the Saint-Venant formalism) and compare them to the experimental measurements. The small-wavelength regime (dispersive regime) together with the sinuous modes will be discussed in a future article.

4.2.1 Case of the V-shaped symmetrical substrate

We now start with the spatio-temporal spectra of an elongated drop, placed upon a superhydrophobic symmetric V-shaped substrate, for four different volumes (23 ml, 40 ml, 60 ml, 80 ml) at fixed angle $\unicode[STIX]{x1D6FC}=10^{\circ }$. These spectra are plotted in figure 9. The smaller the volume, the smaller the slope at origin, which is the celerity of surface waves $c_{\mathit{eff}}$. We compare this experimental slope to its analytical value found using the Saint-Venant formalism, in the large drop limit (see (3.50) and (3.51)), for a parameter $m$ deduced from the value of deposited volume of water. The agreement is excellent (even better if we use the exact computations given by (3.47)). The larger the volume, the steeper the slope, that is the larger the value of $c_{\mathit{eff}}$.

We can clearly distinguish the existence of upper branches. The very first branch corresponds to sloshing modes. Its cutoff frequency depends on the volume of the drop. The larger the volume, the smaller the cutoff frequency. This point will be discussed in a future article.

Figure 9.

Spatio-temporal spectra of elongated drops, on a superhydrophobic symmetric V-shaped substrate with an angle $\unicode[STIX]{x1D6FC}=10^{\circ }$, for four different volumes $V$. We have superimposed the linear relation $f(k)=(1/2\unicode[STIX]{x03C0})c_{\mathit{eff}}k$ with $c_{\mathit{eff}}$ calculated using (3.50) and (3.51). (a) $V=23~\text{ml}$, $c_{\mathit{eff}}=7.47~\text{cm}~\text{s}^{-1}$; (b) $V=40~\text{ml}$, $c_{\mathit{eff}}=10.60~\text{cm}~\text{s}^{-1}$; (c) $V=60~\text{ml}$, $c_{\mathit{eff}}=12.86~\text{cm}~\text{s}^{-1}$; (d) $V=80~\text{ml}$, $c_{\mathit{eff}}=13.98~\text{cm}~\text{s}^{-1}$. The smaller the volume, the smaller the slope at origin, namely the effective celerity of long wavelength varicose modes. Below a critical volume $V_{\mathit{c}}\simeq 15~\text{ml}$, the drop becomes unstable. The agreements are excellent. The secondary branches correspond to other modes of propagation (in particular, sloshing modes).

4.2.2 Case of non-symmetrical substrate: L-shaped substrate or inclined V-shaped substrate

We now turn to the asymmetrical case which corresponds to a cap placed on top of an asymmetrical right-angled triangle (see figure 7b). The triangle properties will satisfy the following properties: small angle $\unicode[STIX]{x1D6FC}$ and steep angle $\unicode[STIX]{x1D6FD}=(\unicode[STIX]{x03C0}/2)-\unicode[STIX]{x1D6FC}$, width $W(m)$ supposedly large. The properties of the sessile cap are identical to those of the symmetrical case. As for the triangle, we can approximate the length of its hypotenuse $\overline{W}(m)$, with $\unicode[STIX]{x0394}Y_{max}$ minus the length of one of the tip of liquid protruding above the wetted area, which was approximated as being $\ell _{\mathit{c}}/2(\mathscr{C}_{1}-\mathscr{C}_{2})$. Here, $\overline{W}(m)$ turns out to be somehow equivalent to the mean value of $\unicode[STIX]{x0394}Y_{max}$ and $\unicode[STIX]{x1D6EC}_{\mathit{c}}$. We set $\mathscr{C}_{3}={\textstyle \frac{1}{2}}(\mathscr{C}_{1}+\mathscr{C}_{2})$, so that the triangle has the following properties:

1. (i) width $\overline{W}(m)\simeq {\textstyle \frac{1}{2}}[W(m)+\unicode[STIX]{x1D6EC}_{\mathit{c}}]=\ell _{\mathit{c}}[-\log (1-m)+\mathscr{C}_{3}]$;

2. (ii) height $H_{\mathit{bott}}(m)\simeq {\textstyle \frac{1}{2}}\overline{W}(m)\sin 2\unicode[STIX]{x1D6FC}$; and

3. (iii) cross-sectional area $S_{0}={\textstyle \frac{1}{4}}\overline{W}(m)^{2}\sin 2\unicode[STIX]{x1D6FC}$.

Therefore, we can compute the effective propagation speed

(4.1)$$\begin{eqnarray}c_{\mathit{eff}}^{2}(m)=g\ell _{\mathit{c}}\left[{\displaystyle \frac{S_{\mathit{tot}}(m)}{{\displaystyle \frac{\unicode[STIX]{x2202}S_{\mathit{tot}}}{\unicode[STIX]{x2202}m}}(m)}}\right]\left\{+{\displaystyle \frac{\sin 2\unicode[STIX]{x1D6FC}}{2\unicode[STIX]{x1D707}}}-1-{\displaystyle \frac{3\unicode[STIX]{x1D707}}{2}}+\mathit{O}\left(\unicode[STIX]{x1D707}^{2}\right)\right\},\end{eqnarray}$$

with

(4.2)$$\begin{eqnarray}\left[{\displaystyle \frac{S_{\mathit{tot}}(m)}{{\displaystyle \frac{\unicode[STIX]{x2202}S_{\mathit{tot}}}{\unicode[STIX]{x2202}m}}(m)}}\right]=2(1-m){\displaystyle \frac{2[\mathscr{C}_{1}-\log (1-m)]+{\displaystyle \frac{\sin 2\unicode[STIX]{x1D6FC}}{4}}[\mathscr{C}_{3}-\log (1-m)]^{2}}{\sin 2\unicode[STIX]{x1D6FC}[\mathscr{C}_{3}-\log (1-m)]+4}}.\end{eqnarray}$$

For a fixed volume of 40 ml, we show the spatio-temporal spectra for two different angles ($10^{\circ }$ and $30^{\circ }$) in figure 10. The values of the parameter $m$ used for the following comparisons between experiments and theory are set by the value of the deposited volume of water. For $\unicode[STIX]{x1D6FC}=10^{\circ }$, the agreement between the experimental slope at origin and $c_{\mathit{eff}}$ given by (4.1) is excellent, whereas in order to capture the correct theoretical value of $c_{\mathit{eff}}$ at angle $\unicode[STIX]{x1D6FC}=30^{\circ }$, we resort to exact calculations given by (3.47) and we can see that the agreement is excellent as well.

Figure 10.

Spatio-temporal spectra of elongated drops, on a superhydrophobic asymmetrical L-shaped substrate, for a volume equal to 40 ml and two different angles $\unicode[STIX]{x1D6FC}$. We have superimposed the linear relation $f(k)=(1/2\unicode[STIX]{x03C0})c_{\mathit{eff}}k$. (a) $\unicode[STIX]{x1D6FC}=10^{\circ }$, $c_{\mathit{eff}}=13.5~\text{cm}~\text{s}^{-1}$; (b) $\unicode[STIX]{x1D6FC}=30^{\circ }$, $c_{\mathit{eff}}=18.8~\text{cm}~\text{s}^{-1}$. The larger the angle, the larger the slope, namely the celerity of varicose modes. And last, (c) spectrum of an elongated drop, on a superhydrophobic asymmetrical tilted V-shaped substrate corresponding to the case $\unicode[STIX]{x1D6FC}=5^{\circ }$ and $\unicode[STIX]{x1D6FD}=15^{\circ }$, for volume $V=40~\text{ml}$. We have $c_{\mathit{eff}}=9.13~\text{cm}~\text{s}^{-1}$. In all the cases, the secondary branches correspond to other modes of propagation (for instance, sloshing modes).

Last, we have performed an experiment using our V-shaped substrate tilted by an angle $5^{\circ }$, that corresponds to the asymmetrical case $\unicode[STIX]{x1D6FC}=5^{\circ }$ and $\unicode[STIX]{x1D6FD}=15^{\circ }$. It is displayed in figure 10(c) together with the Saint-Venant dispersion relation $f(k)=(1/2\unicode[STIX]{x03C0})c_{\mathit{eff}}k$, where $c_{\mathit{eff}}$ is computed using (3.47), with an excellent agreement.

4.2.3 Experimental threshold of breakup

Section 3.5 was devoted to the critical value of the parameter $\unicode[STIX]{x1D707}$ (hence $m$) at which instability occurs, that is when $c_{\mathit{eff}}^{2}$ changes its sign, from positive to negative as volume decreases (that is when the parameter $\unicode[STIX]{x1D707}$ increases). In order to experimentally confirm these predictions, we pin both ends of a large controlled volume of water, deposited upon our superhydrophobic substrates (V-shaped substrate together with L-shaped substrate that we tilt at angle $\unicode[STIX]{x1D6FC}$). We progressively and carefully pump controlled volumes of water out from one end of the cylinder, of length $L$, using a syringe with a very sharp needle in order to minimise meniscus effects. We deduce the remaining deposited volume, down to a critical volume at which the drop breaks up. The (slow) pinch-off is located about the centre of the cylinder and retraction of both remaining halves eventually occurs, owing to surface tension (see supplementary movies, long shot and close-up, in the case of the symmetrical V-shaped substrate which are available at https://doi.org/10.1017/jfm.2020.114). The critical volume $V_{\mathit{c}}^{\mathit{exp}}$ allows us to obtain the critical cross-sectional area $S_{\mathit{c}}^{\mathit{exp}}$ of the drop ($S_{\mathit{c}}^{\mathit{exp}}=V_{\mathit{c}}^{\mathit{exp}}/L$), hence the critical value $\unicode[STIX]{x1D707}_{\mathit{c}}^{\mathit{exp}}$, using (3.44). We compare this experimental value to the theoretical one in table 2. The breakup of the drops systematically occurs at volumes approximately 1.3 times larger than the predicted theoretical volumes for a given substrate geometry.

At this limit regime, we have to stress that the system becomes very sensitive to any external perturbations such as the vibrations caused by the pumping procedure. Moreover, the translational invariance is broken because of the meniscus created by our pinning procedure. It induces a bulge at both tips of the drop, so that the effective volume per unit length, i.e. the effective cross-sectional area of the cylindrical part of our drop, turns out to be lower than that measured in our procedure. Using a longer channel would certainly help alleviate this discrepancy.

Table 2.

Values of the critical thresholds found in our experiments, in the cases of the symmetrical V-shaped substrate ($\unicode[STIX]{x1D6FC}=10^{\circ }$), and the non-symmetrical L-shaped substrates. Here, we have $\ell _{\mathit{c}}=2.5~\text{mm}$.