5.1 Highlights of numerical simulations

Systematic numerical simulations of spatio-temporal evolution of the interface deflection have been carried out, starting from an initial instant when a particular perturbation, a Gaussian-like bump, is imposed on the interface. No initial disturbance on the lateral velocity perturbation $v$ has been considered. Due to the lack of experimental data in the subcritical regime, the numerical simulations have been utilized to validate the eigenvalues analysis, especially the treatment used to remove the singularity. A broad comparison with experimental results in the supercritical regime will be reported below. The temporal panels shown in figure 2 refer to the evolution computed for the subcritical case of $We=0.82$ with ${\mathcal{K}}=5\times 10^{4}$ , $\unicode[STIX]{x1D700}=6.7\times 10^{-4}$ , and to a bump disturbance initially located at $x=0.1$ , i.e. in the subcritical region $0<x<x_{s}$ , here resulting in $x_{s}=0.24$ (4.9). Note also that other simulations, not shown herein, revealed that the asymptotic deflection eventually reached by the sheet is not influenced by the initial position of the disturbance. The panels reported in figure 2 show an initial bump splitting into two parts, one travelling rightwards and being rapidly expelled, the other one travelling leftwards, where both have the expected velocities. The reflection of the left part on the inlet boundary at $x=0$ generates the temporary establishment of a low wavenumber mode which is rapidly characterized by the appearance of shorter wavelength ripples. At large times a travelling wave, exhibiting an almost regular structure with a wavenumber of approximately $k=15$ , is eventually established. Details about the transition from an initial impulse to the rise of capillary waves are presented in the movie, provided as supplementary material available at https://doi.org/10.1017/jfm.2017.241.

Figure 2.

Temporal sequence of panels of instantaneous nappe deflection extracted from the numerical simulation of (2.33)–(2.34), for $We=0.82$ , $Fr=1$ , ${\mathcal{K}}=5\times 10^{4}$ and $\unicode[STIX]{x1D700}=6.7\times 10^{-4}$ .

A comparison of large time sheet centreline deflection and lateral velocity between eigenmodes (a) and numerical simulations (b) is reported in figure 3, referring to the same parameters as the panels of the previous figure. The physical relevance of the result will be addressed later on, when discussing the shape of modes observed in actual experimental rigs.

Figure 3.

Comparison between leading eigenmode (a) and numerical simulation (b) of (2.33)–(2.34), for $We=0.82$ , $Fr=1$ , ${\mathcal{K}}=5\times 10^{4}$ and $\unicode[STIX]{x1D700}=6.7\times 10^{-4}$ .

The above comparisons appear quite satisfactory. In particular, note the perfect agreement of the two solutions near the singularity station $x_{s}=0.24$ , showing that the treatment of the singularity worked successfully in the spectral analysis. A further finding is the confirmation of the travelling wave feature of the solution; as was verified in a wide range of flow conditions, the wave velocity satisfies the following relationship to within good accuracy

(5.1) $$\begin{eqnarray}\bar{a}=\frac{\unicode[STIX]{x1D706}_{i}}{2\unicode[STIX]{x03C0}k},\end{eqnarray}$$

with $\unicode[STIX]{x1D706}_{i}$ indicating the disturbance circular frequency (i.e. the imaginary coefficient of eigenvalue) and $k$ the wavenumber, which can be evaluated by inspecting the eigenfunctions. The average $\bar{a}$ velocity is, on the other hand, directly evaluated as $\bar{a}=1/t_{cross}$ , where the crossing time is

(5.2) $$\begin{eqnarray}t_{cross}=\int _{0}^{1}{\displaystyle \frac{\text{d}x}{U\pm \displaystyle \sqrt{\frac{U}{We}}}}.\end{eqnarray}$$

5.2 Analysis of critical regime

Some significant results of the spectral analysis for $We$ around unity will be hereafter presented. Figure 4 compares spectra obtained in subcritical ( $We=0.98$ ) and supercritical ( $We=1.02$ ) regimes for $Fr=1$ and two typical values of the air stiffness, ${\mathcal{K}}=10^{5}$ and ${\mathcal{K}}=10^{6}$ . As predicted before, the subcritical regime (for $We=0.98$ it is $x_{s}=0.021$ ) is characterized by one branch only (associated with the faster travelling wave with velocity $U+\sqrt{U/We}$ ), constituted by two symmetrical parts exhibiting outer cusp-point maxima, as well as local maxima (independent of ${\mathcal{K}}$ ) represented by the inner pair of eigenvalues. The configurations reported in figure 4(a) are stable. Note also that, as already extensively discussed by De Rosa et al. (Reference De Rosa, Girfoglio and de Luca2014), according to the damped spring–mass model driven by the self-induced forcing $f_{i}$ of (3.6), the frequency of the less stable eigenvalue approximately scales with $\sqrt{{\mathcal{K}}}$ (of course in the driven regime it coincides with that of the forcing term). Another characteristic frequency is represented by the constant spacing $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i}$ of the imaginary parts of the eigenvalues which is related to the crossing time by $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i}\approx 2\unicode[STIX]{x03C0}/t_{cross}$ . The horizontal continuous line of figure 4(a) is the theoretical upper bound of inequality (4.23), which in the present case yields $\unicode[STIX]{x1D706}_{r}<-0.49$ and therefore it is satisfied. Figure 4(b) reports spectra for $We=1.02$ and the same values of ${\mathcal{K}}$ as the previous figure 4(a). Thus, the analysis of these two spectra sheds light on the critical region. The major feature is that, on the one the presence of a part of a branch representing the parametric continuation of the analogous branch found for $We=0.98$ is noted, on the other a new nucleus of eigenvalues having growth rate weakly less than zero clearly arises. The new nucleus of eigenvalues is to be ascribed to the travelling waves with velocity $U-\sqrt{U/We}$ and to their interference with low-frequency eigenvalues associated with the faster waves with velocity $U+\sqrt{U/We}$ . This last finding can be better appreciated by looking at figure 5(a,b), depicting the close-up views of the previous figures 4(a) and 4(b), respectively. It is very important to point out that in the above pictures the frequency scale is extended up to 1500 for the sake of completeness only, but values greater than approximately 300 are suspected not to have physical relevance because they are associated with high wavenumber modes, which are definitely damped by viscosity, while they also invalidate the assumption of thin sheet model. This issue will be raised again when discussing the comparison with experimental data in the supercritical regime. De Rosa et al. (Reference De Rosa, Girfoglio and de Luca2014) did not report this argument because their study was limited to a parametric investigation without further insights on the physical relevance of the higher-frequency modes.

Figure 4.

Spectra for $Fr=1$ , $\unicode[STIX]{x1D700}=6.7\times 10^{-4}$ and selected ${\mathcal{K}}$ values. (a)  $We=0.98$ , (b) $We=1.02$ .

Figure 5.

Inner eigenvalues of spectra for $Fr=1$ , $\unicode[STIX]{x1D700}=6.7\times 10^{-4}$ and selected ${\mathcal{K}}$ values. (a) $We=0.98$ , (b) $We=1.02$ .

A first estimate of the viscous dissipation in short wavelength capillary ripples can be made by following the approach of Lighthill (Reference Lighthill1978), in particular with reference to the so-called internal dissipation mechanism, i.e. under the assumption that the viscous damping is limited to the wave amplitude, with the main flow field being irrotational. This approach, well established more recently by Joseph, Funada & Wang (Reference Joseph, Funada and Wang2007), is the basis also of the work of Longuet-Higgins (Reference Longuet-Higgins1997) dealing with capillary–gravity waves. Lighthill (Reference Lighthill1978) showed that the time constant $\unicode[STIX]{x1D70F}^{\star }$ for the exponential decay of surface waves of wavelength $\unicode[STIX]{x1D706}^{\star }$ is

(5.3) $$\begin{eqnarray}\unicode[STIX]{x1D70F}^{\star }=\frac{\unicode[STIX]{x1D706}^{\star ^{2}}}{16\unicode[STIX]{x03C0}^{2}\unicode[STIX]{x1D708}},\end{eqnarray}$$

with $\unicode[STIX]{x1D708}$ being the fluid kinematic viscosity.

As asserted by Joseph et al. (Reference Joseph, Funada and Wang2007), the wave attenuation can be evaluated by the distance (namely, the number of wavelengths) a wave travels before it decays by a certain amount, or, in other terms, by the number of oscillation periods, $T^{\star }$ , corresponding to a given decay amount. Thus, the quantity $\unicode[STIX]{x1D70F}^{\star }/T^{\star }$ measures the number of periods for the energy of sinusoidal waves to be reduced by a factor $e$ . By assuming a constant sheet thickness, $T^{\star }$ is given by

(5.4) $$\begin{eqnarray}T^{\star }=\unicode[STIX]{x1D706}^{\star }\sqrt{\frac{\unicode[STIX]{x1D70C}_{l}H_{in}^{\star }}{2\unicode[STIX]{x1D70E}}}\end{eqnarray}$$

and the following is finally obtained

(5.5) $$\begin{eqnarray}\unicode[STIX]{x1D70F}^{\star }/T^{\star }=\frac{1}{16\unicode[STIX]{x03C0}^{2}}\frac{1}{Oh}\frac{\unicode[STIX]{x1D706}^{\star }}{H_{in}^{\star }},\end{eqnarray}$$

where the Ohnesorge number is introduced as

(5.6) $$\begin{eqnarray}Oh=\unicode[STIX]{x1D708}\sqrt{\unicode[STIX]{x1D70C}_{l}/(2\unicode[STIX]{x1D70E}H_{in}^{\star })}.\end{eqnarray}$$

For standard values of the thermodynamic properties of water, $H_{in}^{\star }=10^{-3}~\text{m}$ and $\unicode[STIX]{x1D706}^{\star }/H_{in}^{\star }=10$ , one obtains $Oh=2.62\times 10^{-3}$ and $\unicode[STIX]{x1D70F}^{\star }/T^{\star }\simeq 24$ , while in the case of very short capillary ripples with $\unicode[STIX]{x1D706}^{\star }/H_{in}^{\star }=2$ the wave energy is expected to be reduced by a factor of $e$ in approximately five periods. This last prediction is consistent with the experimental findings of de Luca & Costa (Reference de Luca and Costa1997b ) who visualized sinuous waves produced by an obstacle on a free-falling sheet. It is also important to note that, in the presence of surfactant agents in the liquid, another attenuation effect has to be taken into account, i.e. the surface dissipation, that can exceed the internal dissipation, as discussed by Lighthill (Reference Lighthill1978) and experimentally approached by de Luca & Costa (Reference de Luca and Costa1997b ), who took advantage of a previous work of de Luca & Meola (Reference de Luca and Meola1995).

Figure 5(a) shows a detail of the lowest frequency eigenvalues (inner conjugate couples) calculated for $We=0.98$ (but not less stable eigenvalues for ${\mathcal{K}}=10^{5}$ and ${\mathcal{K}}=10^{6}$ ), whilst figure 5(b) refers to the less stable supercritical ( $We=1.02$ ) eigenvalues (which are practically independent of ${\mathcal{K}}$ ) and it shows in great detail the interference of modes. Although for $We=0.98$ the less stable modes are the higher-frequency ones scaling with $\sqrt{{\mathcal{K}}}$ , the physical considerations above lead us to think that the sheet shapes more likely observable in experiments are represented by the inner modes (i.e. local maxima); the same should occur in the supercritical condition of $We=1.02$ , where the inner modes are the less unstable, too. The difference lies in the fact that in the subcritical regime, the prevailing modes are of fast type, whilst in the supercritical regime they are of slow type. In fact, in the former case the order of magnitude of the other characteristic frequency of the system, $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i}\simeq 16.1$ , is associated with the crossing time of the fast mode, whilst in the latter one $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i}\simeq 0.75$ corresponds to the crossing time of the slow mode. The supercritical spectrum of figure 5(b) exhibits a wavy trend whose period is just $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i}\simeq 16.1$ , which evidences the interference between the modes. Finally, it is found that for $We=0.98$ the ratio of the frequency local peak to $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i}$ is approximately 1.18 and for $We=1.02$ this ratio is approximately 1.27. This occurrence is reminiscent of the integer-plus-one-quarter resonance criterion of forced transverse vibrations of the nappe string, already uncovered by De Rosa et al. (Reference De Rosa, Girfoglio and de Luca2014) in the absence of surface tension. This will be discussed in more detail later on, in connection with comparisons with experimental data available in the literature for the supercritical regime and can also be verified by inspecting the relevant eigenmodes.

The inner leading modes are presented in figure 6 for ${\mathcal{K}}=10^{5}$ and $Fr=1$ . Panels (a,b) refer to subcritical ( $We=0.98$ ) and supercritical ( $We=1.02$ ) regimes, respectively. Their comparison gives a picture of the sheet shape variation encountered when traversing the critical regime. The nappe oscillation frequency accordingly changes from 19.13 for the subcritical regime to 0.99 for the supercritical regime. This finding has to be validated through experimental investigations because of the absence of experimental data as regards the critical regime, and will be further discussed in the subsequent section.

Figure 6.

Inner leading eigenfunctions for $Fr=1$ , ${\mathcal{K}}=10^{5}$ and $\unicode[STIX]{x1D700}=6.7\times 10^{-4}$ . (a)  $We=0.98$ , (b) $We=1.02$ .

5.3 Insight into the supercritical regime

A rather detailed comparison of the present theoretical predictions with experiments undertaken by Binnie (Reference Binnie1974) and by Sato et al. (Reference Sato, Miura, Nagamine, Morii and Ohkubo2007) in the supercritical regime will be made in this section. Such a comparison will also represent the validation of the interpretation of the results found in the critical regime. The apparatus arrangement of Binnie (Reference Binnie1974) was very similar to that sketched in figure 1, concerning the formation of a thin sheet of water falling in air, in the presence of an air enclosure behind the sheet. Tests were carried out for various geometrical quantities and water flow rate values, so as to vary the Weber number (basically by increasing the water flow rate) in the range from 3.77 to 287 (the corresponding Froude number ranged from 0.28 to 7.69) and the dimensionless air stiffness ${\mathcal{K}}$ in the range from $10^{5}$ to $10^{3}$ , respectively. Sheet oscillation frequency was measured by means a stroboscope and the associated deformed centreline shape was detected indirectly by means of an optical method able to individuate positions half a wavelength apart on the shape. Binnie (Reference Binnie1974) predicted the modes associated with the natural frequencies of the nappe through a simplified physical model which did not include surface tension effects, and reported such modes for various testing conditions. With reference to his flow configuration, named A, with $We=95.9$ and ${\mathcal{K}}=5.87\times 10^{3}$ , panels (a–d) of figure 7 show the first, second, third and fifth mode, respectively, corresponding to less stable inner eigenvalues, as discussed before in regard to the critical regime. Such modes agree strictly with the predictions arising from Binnie’s theory which, in turn, agrees with his own experiments. Note that the peak frequency of the most unstable mode, namely the spring–mass frequency scaling with $\sqrt{{\mathcal{K}}}$ , equal to approximately 80 (namely 154 Hz), was not observed in the experiment; on the contrary, the fifth inner mode was detected in this configuration.

Figure 7.

First, second, third and fifth eigenmodes for test A of Binnie (Reference Binnie1974) ( $We=95.9$ , $Fr=7.69$ , ${\mathcal{K}}=5.87\times 10^{3}$ , $\unicode[STIX]{x1D700}=7.6\times 10^{-4}$ ); the continuous lines refer to the present model, while the dashed lines refer to the Binnie’s model not including the surface tension.

A comparison between frequencies observed experimentally by Binnie (Reference Binnie1974) and calculated by the present physical model is shown in table 1. The tests from A to D refer to the same geometrical configuration, while in test F a wider nozzle width was used to work with a higher liquid velocity. Table 1 shows a quite good agreement between frequencies observed in experiments, $f_{e}^{\ast }$ , measured in Hertz, and frequencies evaluated numerically, $f_{n}^{\ast }$ , by the present analysis. Note that Binnie (Reference Binnie1974) too made a comparison between measured and calculated (via his simplified model) frequencies, $f_{b}^{\ast }$ , and attributed the discrepancy of the data to the difficulty in measuring correctly the liquid inlet velocity. In any case, the present comparison is better than Binnie’s. From Binnie’s experiments it is evident that the nappe always oscillates at lower frequencies corresponding to inner modes (which are all stable) of the spectrum, because higher-frequency modes are most likely damped by viscosity. The order of the observed mode cannot be predicted, of course, by the present theory and no explanation in this respect was furnished by the author of the experiments.

Table 1.

Comparison with experiments of Binnie (Reference Binnie1974);  $f_{e}^{\ast }$ is the observed frequency, $f_{n}^{\ast }$ is the frequency predicted by the present analysis, $f_{b}^{\ast }$ is the frequency computed using Binnie’s model.

Table 2.

Comparison with experiments of Sato et al. (Reference Sato, Miura, Nagamine, Morii and Ohkubo2007);  $f_{e}^{\ast }$ is the observed frequency, $f_{n}^{\ast }$ is the frequency predicted by the present analysis, $f_{b}^{\ast }$ is the frequency computed using Binnie’s model.

Another systematic comparison has been made with reference to experimental tests of Sato et al. (Reference Sato, Miura, Nagamine, Morii and Ohkubo2007), who carried out nappe shape visualizations and oscillation frequency measurements on an apparatus similar to that used by Binnie (Reference Binnie1974). However, Sato et al. (Reference Sato, Miura, Nagamine, Morii and Ohkubo2007) made direct visualizations by using a high-speed camera and compared observed frequencies with pressure fluctuations in the closed chamber behind the nappe. Table 2 refers to data points corresponding to the smallest slit width considered by the above authors. For this series of tests the agreement between measured frequencies $f_{e}^{\ast }$ and estimated frequencies $f_{n}^{\ast }$ is satisfactorily strict. Again one should note that the (stable) modes selected by the flow system do not correspond to frequencies of the order of $\sqrt{{\mathcal{K}}}$ , but to much lower values which are predicted by the inner eigenvalues. This point was outlined by Sato et al. (Reference Sato, Miura, Nagamine, Morii and Ohkubo2007) who, furthermore, pointed out that ‘the generated frequencies remain the same even when the chamber width (i.e.  $W^{\ast }$ ) was considerably changed’; in other words, they noted experimentally that the inner modes are independent of ${\mathcal{K}}$ , as confirmed by the present theoretical analysis.

Both Binnie (Reference Binnie1974) and Sato et al. (Reference Sato, Miura, Nagamine, Morii and Ohkubo2007) measurements gave evidence of the integer-plus-one-quarter resonance criterion, which is fulfilled by the present analysis. In the experimental investigation reported by Schmid & Henningson (Reference Schmid and Henningson2002), on the contrary, the nappe is seen to oscillate at the frequency corresponding to the base flow falling time, although from their theoretical analysis it is evident that the lowest frequency is approximately 1.28 times of that corresponding to the falling time.

It seems very interesting to compare spectra computed for a relatively high supercritical Weber number, at which the effects of the surface tension begin to be negligible, and for the critical values of $We=1.02$ and $We=0.98$ , hence just above and below the critical threshold, as illustrated in figure 8. The geometrical configuration of test A of Binnie (Reference Binnie1974) is assumed as the starting condition, for which $We=95.9$ , ${\mathcal{K}}=5.87\times 10^{3}$ , $Fr=7.69$ . Thus, due to the lack of experimental data in the subcritical regime, a virtual numerical experiment has been carried out by starting from the condition of test A of Binnie (Reference Binnie1974) and reducing the liquid flow rate (i.e. velocity) until obtaining $We=0.98$ , ${\mathcal{K}}=5.75\times 10^{5}$ , $Fr=0.076$ . Figure 8 uncovers the characteristic wavy trend of a supercritical regime and shows increasingly stable eigenvalues as the Weber number is reduced. Reducing the liquid flow rate from the value of test A of Binnie (Reference Binnie1974) down to the value of the subcritical regime leads to the frequency variations reported in table 3 for the first five inner modes. For each mode, the frequency is at first reduced until the critical threshold is reached, then it sharply increases below $We=1$ , due to the emergence of fast modes in the subcritical regime. Figures 9 and 10 report first, second, third, fifth modes associated with the corresponding inner eigenvalues for $We=0.98$ and $We=1.02$ , respectively, to be compared with the modes of figure 7 referring to $We=95.9$ .

Figure 8.

Spectra computed for various conditions of test A of Binnie (Reference Binnie1974).  $We=95.9$ , ${\mathcal{K}}=5.87\times 10^{3}$ , $Fr=7.69$ (*); $We=1.02$ , ${\mathcal{K}}=5.5\times 10^{5}$ , $Fr=0.08$ (○); $We=0.98$ , ${\mathcal{K}}=5.75\times 10^{5}$ , $Fr=0.076$ (▫).

Figure 9.

First, second, third and fifth eigenmodes for the test A of Binnie (Reference Binnie1974) and for $We=1.02$ , ${\mathcal{K}}=5.5\times 10^{5}$ , $Fr=0.08$ .

Figure 10.

First, second, third and fifth eigenmodes for test A of Binnie (Reference Binnie1974) and for $We=0.98$ , ${\mathcal{K}}=5.75\times 10^{5}$ , $Fr=0.076$ .

Table 3.

Computed inner modal frequencies of test A of Binnie (Reference Binnie1974) when reducing the liquid flow rate.

With reference to the first mode only, figure 11 illustrates graphically the sharp increase in the frequency of mode 1 when entering the subcritical regime as the Weber number is reduced. The arrow indicates the frequency route and $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i-}$ represent the frequencies related to the crossing time of the fast and slow waves, respectively, which are analytically evaluated as

(5.7) $$\begin{eqnarray}t_{cross\pm }=2Fr\left[\frac{1}{2}\left(U|_{1}-1\right)\mp \frac{1}{\sqrt{We}}\left(\sqrt{U|_{1}}-1\right)+\frac{1}{We}\log \frac{\sqrt{U|_{1}}\pm \displaystyle \frac{1}{\sqrt{We}}}{1\pm \displaystyle \frac{1}{\sqrt{We}}}\right].\end{eqnarray}$$

Figure 11.

Frequency route from supercritical to subcritical regime; mode 1; $Fr=0.077$ .

The result depicted in figure 11 requires an experimental validation. In addition, it should be used carefully because it refers to the inner first mode, and the present theory is not able to predict the actual selection mechanism of the frequency; hence, future experimental data should be compared with analogous routes referring to higher modes, and a jump from supercritical to subcritical modes of higher order cannot be excluded. Finally, to obtain the expected frequencies, one should multiply $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D706}_{i-}$ by a factor equal to approximately 1.25 according to the one-plus-one-quarter resonance criterion.

Figure 12.

Curves of marginal stability and routes towards the instability. (a) The thick blue lines refer to stable states, the dashed thick red lines to unstable states; the inset represents the dashed rectangular range; the open circle is the condition of test A of Binnie (Reference Binnie1974). (b) Trends for very low ${\mathcal{K}}$ values.

Starting again from the condition of test A of Binnie (Reference Binnie1974), and with reference to the first mode only, figure 12(a) depicts two routes towards the instability in the plane $Fr-{\mathcal{K}}$ , the vertical one referring to an increase in the chamber width $W^{\ast }$ , the hyperbolic one referring to an increase in the flow velocity $U_{in}^{\ast }$ . In both cases the thick blue line corresponds to stable states, the dashed thick red line to unstable states. Because of the one-to-one correspondence between physical quantities and dimensionless parameters observed before for the case of water sheet flowing in air, the Weber number of each state is unique (for instance, in the starting point, denoted by the open circle symbol, it is $Fr=7.69$ , ${\mathcal{K}}=5870$ , $We=95.9$ ) and the stable/unstable nature of a given status has to be inspected with reference to the neutral stability curve relative to that $We$ value. In particular, along the vertical path the $We$ number is constant, whilst along the other path $We$ increases towards the instability. Such routes are depicted together with various neutral stability curves evaluated for $We$ numbers equal to $95.9,300.98,10^{4}$ (with this last value closely simulating the case of absence of surface tension). Note that $We=95.9$ and $We=300.98$ coincide with the stable–unstable marginal values along the two routes here considered. According to a previous finding of De Rosa et al. (Reference De Rosa, Girfoglio and de Luca2014), for each $We$ number the unstable region is contained in between two marginal curves, with the lower ones being practically independent of the $We$ number, as is better evinced in figure 12(b) illustrating the range of very low ${\mathcal{K}}$ values. The asymptotic value of ${\mathcal{K}}=19.74$ attained for high $Fr$ numbers uncovers the analogous result of De Rosa et al. (Reference De Rosa, Girfoglio and de Luca2014) who neglected the surface tension and defined the Froude number as the reciprocal of the present one. Figure 12(a) clearly shows the stabilizing role of the surface tension, because the unstable region is reduced with decreasing the Weber number. The subcritical regimes are always stable, as predicted by the Frobenius analysis. For the sake of completeness the inset of figure 12(a), corresponding to the dashed rectangle in the main panel, magnifies the range of $Fr$ values less than 0.6 and gives an insight as regards the local minima of the upper marginal curves. Data reported in figure 12 have to be used very carefully because the system can exhibit higher modes in different flow conditions.