2. Numerical simulations

We carry out two-dimensional numerical simulations in the so-called closed-flow framework in which the Navier–Stokes equations are solved in a computational domain of length $L$ along the flow direction $x$ and the flow fields satisfy periodicity boundary conditions at $x=0$ and $x=L$; $y$ is the coordinate normal to the plate. The no-slip condition is applied at the plate surface located at $y=0$.

The calculations have been performed with the open-source code Basilisk (http://basilisk.fr), an improved version of the code Gerris used by Kofman et al. (Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018). The code adopts a second-order time- and space-adaptive finite volume methodology based on a projection method with a quadtree discretization (Scardovelli & Zaleski Reference Scardovelli and Zaleski1999; Popinet Reference Popinet2009). A conservative, non-diffusive, geometric volume-of-fluid method is used to simulate a gas–liquid two-phase flow. The interface between the phases is described by a volume fraction field which is governed by an advection equation. An improved version of the continuum surface force method (Brackbill, Kothe & Zemach Reference Brackbill, Kothe and Zemach1992; François et al. Reference François, Cummins, Dendy, Kothe, Sicilian and Williams2006) is used to account for surface-tension effects. We have assigned a density and viscosity 1000 times smaller than those of the liquid to the gas phase, thus approximating a situation in which the gas has negligible dynamical effects on the liquid. By varying these ratios we have confirmed that, with the choices made, the gas phase indeed has a negligible influence on the calculations. We have allowed a maximum grid refinement down to $2^{-11}L$ finding good numerical convergence. A comparison of the results of our simulations (solid lines) with the Gerris data taken from the published figures 2(a) and 2(b) of Kofman et al. (Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018) (circles), shown in figure 1, supports the reliability of our simulations.

Figure 1. Comparison of the present simulations (continuous lines) with the published results in figures 2(a) and 2(b) of Kofman et al. (Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018).

The cases simulated can be characterized in terms of four dimensionless parameters. We choose $V/h_0^2$, the volume of liquid $V$ in the computational domain (per unit spanwise length) non-dimensionalized by the mean film thickness $h_0$, the inclination angle $\beta$ (defined so that the plate is vertical for $\beta ={{\rm \pi} }/{2}$ and horizontal with the film on its underside for $\beta ={\rm \pi}$) and the Reynolds and Bond numbers. It has been customary in the literature on film flow on the upper side of inclined plates to incorporate the inclination angle into the definition of these numbers

(2.1a,b)\begin{equation} Re = \frac{h_0^3g\sin \beta}{3\nu^2} , \quad Bo = \frac{\rho h_0^2 g\sin\beta}{\sigma} , \end{equation}

with $g$ the acceleration of gravity, $\nu$ and $\rho$ the liquid kinematic viscosity and density and $\sigma$ the surface-tension coefficient. The same definitions are used by several authors also for film flow on the underside of inclined plates. However, when the effect of gravity normal to the plate becomes critical, as in this latter configuration, the effect of the angle cannot be completely captured by these definitions and this is the reason we consider the angle itself as a separate fourth parameter. Following other researchers (see e.g. Rohlfs et al. Reference Rohlfs, Pischke and Scheid2017; Charogiannis et al. Reference Charogiannis, Denner, van Wachem, Kalliadasis, Scheid and Markides2018), in order to characterize the simulations we use the so-called vertical version of the Reynolds and Bond numbers, $Re_\perp$ and $Bo_\perp$, defined as above by setting $\sin \beta =1$. Another dimensionless quantity encountered in the literature is the Kapitza number $Ka$ which can be expressed in terms of our primary dimensionless quantities as $Ka = (3Re)^{2/3}/Bo$, and $Ka_\perp = (3Re_\perp )^{2/3}/Bo_\perp$. The capillary length $\ell _c=\sqrt {\sigma /\rho g}$, not adjusted for the plate inclination, is related to the vertical Bond number by $h_0/\ell _c =\sqrt {Bo_\perp }$.

3. The quasi-equilibrium case

We carried out simulations with the slope of the plate very slowly reduced (i.e. the angle $\beta$ slowly increased towards $180^\circ$) starting from a computed steady non-dripping Kapitza wave. The rate of increase of the angle was $0.05^\circ$ per dimensionless time unit $\sqrt {g/h_0}\,t=1$ and was therefore so slow that the results can be considered as essentially progressing through a sequence of steady states until dripping sets in. By the way these simulations have been conducted, the gradual approach to the onset of dripping changes the liquid momentum of the growing wave at a very slow rate. The dripping that eventually sets in when the plate inclination is small enough is therefore little influenced by dynamic effects normal to the plate and essentially occurs when surface tension can no longer support the wave crest against gravity. For this reason, the dripping thresholds that we determine in this way can be considered upper limits. Indeed, we will show later examples in which, due to the rapid growth of the Rayleigh–Taylor instability, dripping takes place at smaller angles for the same values of $Re_\perp$, $Bo_\perp$ and $V/h_0^2$.

Figures 2$(a)$ and 2$(b)$ show some results for the dripping threshold as a function of the problem parameters $Re_\perp$, $Bo_\perp$ and $V/h_0^2$. The limit points for $\beta =180^\circ$ shown in the figures correspond to the minimum liquid volume necessary for dripping from a horizontal plate and are explained in § 6. Figure 2$(a)$ shows that the wave is increasingly stable as the Reynolds number $Re_\perp$ is increased from 20 (squares, purple) to 50 (circles, green) to 100 (diamonds, blue), so that the angle corresponding to the dripping threshold increases and the plate inclination decreases. As the Bond number decreases, the effect of surface tension increases, a smaller flow rate is sufficient to stabilize the wave and the dripping angle increases.

Figure 2. $(a)$ Dripping threshold vs the perpendicular Bond number $Bo_\perp$ for $V/h_0^2=40$ and, in descending order, $Re_\perp =100$, 50 and 20. $(b)$ Dripping threshold vs $Bo_\perp$ for $Re_\perp =50$ and, in descending order, $V/h_0^2=20$, 40, 60 and 80.

Figure 2$(b)$ shows that increasing the total liquid volume has a destabilizing effect, which causes the dripping angle to decrease and the plate inclination necessary for stability to increase. The primary reason is that, with more liquid available, the wave crest can grow fed by liquid that can flow into it remaining relatively far from the wall and, therefore, relatively unimpeded by the viscous shear caused by the no-slip condition. A heavier wave crest requires a stronger surface-tension effect to be supported against gravity and, therefore, a decreasing $Bo_\perp$ has a stabilizing effect. While increasing the liquid volume from 20 to 40 has a strong effect, further increases have a progressively weaker influence because for $V/h_0^2=40$ there is already enough liquid able to flow into the wave crest. Some of these considerations are amplified and better justified later in § 6.

Figure 3 shows three examples of the waveform at the instant at which $\beta =160^\circ$ with $Bo_\perp =0.4$, $V/h_0^2=40$ and $Re_\perp =20$, 50 and 100, all of which are stable cases. It is interesting to note the thickening of the wave tail as the Reynolds number is increased. This feature is reminiscent of the phenomenon encountered when a plate is pulled out of a liquid bath, in which process the thickness of the film left on the plate increases with the withdrawal velocity for a fixed Kapitza number (see e.g. Orsini & Tricoli Reference Orsini and Tricoli2017). By conservation of volume, the thickening of the wave tail decreases the wave amplitude, the volume of liquid in the crest and the crest curvature. We will return in § 5 on the importance of these factors in stabilizing the wave.

Figure 3. Three examples of the stable (non-dripping) wave shapes for $\beta =160^\circ$ for $Re_\perp =20$ (purple), 50 (green) and 100 (blue) with $Bo_\perp =0.4$ and $V/h_0^2=40$.

4. Curvature of the wave crest

An interesting feature of the simulations just described is illustrated in figure 4 for a few cases ($Re_\perp$ = 50 and 100, $Bo_\perp$ = 0.4, 0.6 and 0.8, and $V/h_0^2$ = 20, 40 and 80) but the same feature is also present in all the other cases used to generate the dripping thresholds of figure 2. Panel (a) of the figure shows the dimensionless growth rate $(\textrm {d}h_w/\textrm {d}t)/\sqrt {gh_0}$ of the amplitude $h_w$ of the wave under the plate while (b) shows $\kappa _w$, the curvature at the wave crest, non-dimensionalized by the capillary length $\ell _c$, both as functions of the dimensionless time $\sqrt {g/h_0}t$. In all cases, the growth rate of the waves starts to increase very rapidly once the dimensionless maximum curvature approaches a particular value $\kappa _{cr} \simeq 1.187/\ell _c$. We have found that, if the angle of inclination of the plate is kept constant just before this value is reached, the wave remains steady while, if it is stopped just above, the wave amplitude keeps increasing until a drop forms and detaches. This fact suggests that $\kappa _{cr}$ corresponds to or, at least, is correlated with, the dripping threshold in the condition of these simulations.

Figure 4. $(a)$ Growth rate of the wave amplitude. $(b)$ Curvature of the wave crest vs time as the slope of the plate is slowly reduced. The plate inclination angles at the instants marked by the vertical dashed lines are, from left to right, $171.4^\circ$, $165.2^\circ$, $168.2^\circ$, $164.3^\circ$, $160.5^\circ$ and $157.6^\circ$.

After reaching $\kappa _{cr}$, the tip curvature continues to increase rapidly approaching a maximum value close to $\kappa _\parallel =\sqrt {2}/\ell _c$ and then, equally rapidly, it starts to decrease until a drop forms and detaches at approximately the time the curvature has returned to $\kappa _{cr}$. The volumes (areas) of these drops are found to be very close to each other and approximately given by $4.6\ell _c^2$. A similar sequence can be observed in figure 10(b) of Yiantsios & Higgins (Reference Yiantsios and Higgins1989), in which the curvature of the drop tip is seen to increase up to the point at which the drop develops inflection points with vertical tangents, after which it starts to decrease.

The shapes of the wave as the angle of inclination gradually increases (i.e. the plate slope decreases) are shown in figure 5 for the case $Re_\perp =50$, $Bo_\perp =0.6$ and $V/h_0^2=40$; those for the other cases go through a similar progression. As the slope decreases the amplitude of the wave increases. The last profile shown is taken at the point at which the curvature reaches its maximum just before the base of the drop starts to shrink prior to detaching.

Figure 5. Wave shapes at different plate inclination angles for the case $Re=50$, $Bo=0.6$ and $V/h_0^2=40$ of the previous figure; the dimensionless times are $\sqrt {g/h_0}\,t=250$, 350, 450, 470 and 474.

In order to rationalize these results we refer to the observation of Pitts (Reference Pitts1973) who studied the static equilibrium of a given amount of liquid attached to the underside of a horizontal plate with a prescribed contact angle. He calculated the volume (per unit spanwise length) of liquid which remains in equilibrium under the action of gravity and surface tension as a function of $h_{drop}$, the elongation of the drop under the plate and, for each volume, found two values of $h_{drop}$, the larger one corresponding to unstable and the smaller one to stable equilibrium. For a zero contact angle, the two branches come together at a maximum volume $V_{cr} \simeq 5.211 \ell _c^2$, at which point the height of the drop is $h_{cr}\simeq 2.374 \, \ell _c$ with the tip curvature $\kappa _{cr}\simeq 1.187/\ell _c$. No equilibrium solution exists for liquid volumes greater than $V_{cr}$.

While Pitts used an energy argument, his results on the equilibrium drop shape are readily obtained from the relation which expresses the balance of surface tension and gravity for a pendant drop under a horizontal plate, namely (see Appendix A)

(4.1)\begin{equation} \sigma \kappa(z) = \sigma \kappa_0 -\rho g z . \end{equation}

Here, $z$ is the vertical distance from the tip of the pendant drop to the generic point on the drop surface, $\kappa (z)$ the surface curvature at that point and $\kappa _0$ the curvature at the drop tip. Figure 6$(a)$ shows the dimensionless pendant drop volume (per unit spanwise length) $V_{drop}/\ell _c^2$ vs the dimensionless elongation $h_{drop}/\ell _c$ for zero contact angle and agrees with figure 3 of Pitts (see also figure 12 of Yiantsios & Higgins Reference Yiantsios and Higgins1989).

Figure 6. $(a)$ Volume of a static pendant drop under a horizontal plate vs drop elongation under the plate for zero contact angle; the dashed lines mark the stability limit. $(b)$ Curvature vs drop elongation. $(c)$ Drop shape when $\ell _c\kappa _{cr}=1.187$. $(d)$ Drop shape when $\ell _c\kappa _\parallel =\sqrt {2}$.

Figure 4 implies that the balance argument of Pitts is relevant not only for the equilibrium drop under a horizontal plate, but also for the crest of a quasi-steady Kapitza wave. Kofman et al. (Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018) also referred to Pitts’ result using directly the limit volume $V_{cr} \simeq 5.211 \ell _c^2$, with $\ell _c$ adjusted for the inclination of the plate, as an indicator of the dripping threshold. Possibly because of fluid momentum effects in their waves, and even though they subtracted the volume of the liquid substrate (the definition of which may be somewhat uncertain, due to the fact that a significant amount of liquid accumulates in the tail of the wave above the substrate) their results were only moderately successful ranging from $\sim 4\ell _c^2$ to $\sim 8\ell _c^2$ about Pitts’ value $V_{cr}=5.211 \ell _c^2$. Some considerations on the scaling suggested by Kofman et al. (Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018) can be found in Appendix B.

The way in which we propose to use Pitts’ result is less direct but not dependent on the ambiguity associated with the drop volume or, because of the way the results of figure 4 were generated, on momentum effects. According to figure 6$(a)$, Pitts’ drop volume is in a one-to-one correspondence with the drop elongation. By establishing a connection between the drop elongation $h_{drop}$ and the tip curvature we can then relate the limit volume $V_{cr}$ to the curvature of the drop tip $\kappa _{cr}$. For this purpose we note that it is a simple consequence of (4.1) that, for a zero contact angle, the curvature at the tip of the drop satisfies the relation

(4.2)\begin{equation} \kappa_0 = \frac{h_{drop}}{2\ell_c^2} . \end{equation}

This result, together with other considerations related to (4.1), is proven in Appendix A. The relation (4.2) permits us to cast the drop elongation $h_{drop}=h_{cr}$ corresponding to the limit volume $V_{cr}$ into a critical value for the curvature at the drop tip, $\ell _c \kappa _{cr}\simeq 1.187$ marked in figure 4. The correspondence between the tip curvature and the drop elongation shows that the former is closely related to the ability of surface and potential energies to balance each other. The drop shape corresponding to the tip curvature $\kappa _{cr}$ is shown in figure 6$(c)$; this shape is the same as that shown in figure 12 of Yiantsios & Higgins. The significance of the conditions under which the tip curvature reaches $\kappa _{cr}$ is that, past this point, the release of potential energy associated with a further increase of the drop elongation can no longer be balanced by a corresponding increase of surface energy. Conservation of energy requires the appearance of another pathway to channel the excess potential energy released, and this is the kinetic energy of the forming and detaching drop, with the attendant viscous dissipation.

There is another pendant drop configuration that is worthy of specific attention. As the elongation of a pendant drop increases, its width near the base narrows and, at some point, a shape develops characterized by vertical tangents at the inflection points as shown in figure 6$(d)$. The special importance of this configuration resides in the fact that, when it occurs, the surface tension force acts vertically and is therefore most effective in counteracting gravity. As shown in Appendix A, the corresponding crest curvature equals $\kappa _\parallel =\sqrt {2}/\ell _c$ and is also marked in figure 4. In equilibrium conditions, the pressure along the line joining the two inflection points is evidently the constant external pressure and, if the volume of liquid under this line is $V_M$, equilibrium requires that $\rho g V_M=2\sigma$ or $V_M=2\ell _c^2$. Any increase beyond $V_M$ of the amount of liquid in the region below the line cannot be sustained by surface tension and a drop must eventually detach. The total liquid volume corresponding to the curvature $\kappa _\parallel$ is calculated in Appendix A for zero contact angle and it is found to be $V_\parallel \simeq 4.793 \ell _c^2$. It is therefore smaller than that corresponding to $\kappa _{cr}$, but the result $\kappa _\parallel =\sqrt {2}/\ell _c$ can be derived without reference to the total liquid volume, just focusing on the portion of the pendant drop between the tip and the points of vertical tangency. Thus, unlike $\kappa _{cr}$, $\kappa _\parallel$ is independent of the total amount of liquid under the plate and characterizes the largest liquid volume that can be sustained by surface tension under equilibrium conditions. This feature confers to this value of the curvature and the associated volume $V_M$ a special importance as we will see.

It is interesting to observe that, as can be seen in the examples of figure 7, the actual shape of the wave crest closely approximates the shapes shown in figures 6$(c)$ and 6$(d)$ at the instants when its curvature reaches $\kappa _{cr}$ and $\kappa _\parallel$. In the figure the two drop shapes of figures 6$(c)$ and 6$(d)$ are rotated and superposed on the wave crest at these times without any further adjustment. The axis of the drop shapes are nearly coincident with the direction of gravity, also shown by the arrows in the figure, with the slight misalignment a consequence of the wave momentum. This alignment justifies the use of $\ell _c$ as defined before without any adjustment for the plate inclination. The match between the drop and wave shapes is very nearly perfect for panels $(b)$ and $(d)$, for which $\kappa _w=\kappa _\parallel$. These results imply that the balance between surface and gravitational energies, that determines the dripping stability of a static drop under a horizontal plate, also determines, to a good approximation, the stability of quasi-equilibrium Kapitza waves. This connection between tip curvature and drop formation suggests that processes tending to stretch the base of the crest, thus preventing its curvature from approaching the critical value, will have a stabilizing effect against dripping. This appears to be a major mechanism by which flow stabilizes the wave as will be further argued below.

Figure 7. Drop shapes of figures 6$(c)$ and 6$(d)$ rotated and superimposed on the wave profiles at the moment the corresponding curvatures are reached for $Re=50$; $(a{,}b)$ $Bo=0.4$, $V/h_0^2=80$, $\beta \simeq 165^\circ$; $(c{,}d)$ $Bo=0.8$, $V/h_0^2=40$, $\beta \simeq 150^\circ$. The arrows indicate the direction of gravity.

This mechanism is inherently nonlinear and is therefore incompatible with an explanation based on the transition from absolute to convective instability of Brun et al. (Reference Brun, Damiano, Pierre, Balestra and Gallaire2015) and others. The mechanism proposed by Babchin et al. (Reference Babchin, Frenkel, Levich and Sivashinsky1983), who saw the convective steepening of the wave as stabilizing the Rayleigh–Taylor instability by increasing, rather than decreasing, the curvature of the wave crest, is only superficially incompatible with our proposal. What happens is that, indeed, the increasing curvature initially opposes the growth of the Rayleigh–Taylor instability but, if it were to be allowed to continue, it would shrink the base of the crest so much that a drop would form and detach. To stabilize the wave, therefore, it is necessary that the growth of the crest curvature be arrested before it becomes critical.

5. Normal momentum

It is evident from the wave shapes such as those shown in figure 5 that dripping may be influenced by the entire wave, but is ultimately a phenomenon localized at the wave crest. In the quasi-equilibrium situations of the previous section, the curvature of the crest is significant because it quantifies the amount of liquid in the crest and the crest length scale, which determine the propensity to instability thanks to the fact that the dominant forces at play are surface tension and gravity. The very fact that results obtained for the curvature of an equilibrium drop match well the results of quasi-equilibrium dynamical simulations gives a clear indication of the small importance of inertia and flow in the phenomena simulated in §§ 3 and 4. Of course there is recirculation in the wave (see e.g. Reck & Aksel Reference Reck and Aksel2015; Rohlfs & Scheid Reference Rohlfs and Scheid2015; Rohlfs et al. Reference Rohlfs, Pischke and Scheid2017), but the associated flow is so weak that its effect is not noticeable in the results of the previous two sections. As the flow is allowed to develop naturally rather than being guided, as it were, through a succession of quasi-equilibrium states, other channels for the potential energy released by the falling of the liquid near the crest become available beyond surface energy, namely kinetic energy and viscous dissipation. Furthermore, localized disturbances can also affect the wave and, in particular, its crest. It may well be expected, therefore, that at the stability limit the direct relation between curvature and stability valid for quasi-static equilibrium will be affected when dynamic effects become significant.

Figure 8 shows some typical examples of how the momentum of the developing wave normal to the plate affects the curvature of the wave crest when the calculation is started from a sinusoidal perturbation

(5.1)\begin{equation} \frac{h(x,t=0)}{h_0}=1+ A\sin \frac{2{\rm \pi} x}{L} , \end{equation}

with an initial amplitude $A=0.2$; here, $Re_\perp =50$, $Bo_\perp =0.6$ and $V/h_0^2=40$. With $h_0=1$ mm and typical properties of a silicone oil, $\sigma = 21\,\textrm {mN}\,\textrm {m}^{-1}$, $\rho = 971\,\textrm {kg}\,\textrm {m}^{-3}$, $\nu =8\times 10^{-6}\,\textrm {m}^2\,\textrm {s}^{-1}$, we find $Bo_\perp = 0.45$ and $Re_\perp \simeq 51$. The parameter choice can therefore be considered as representative of real situations. In the figure, the various lines correspond to different angles of inclination, kept fixed for each simulation. A prominent feature of these results is the evidence of finite-amplitude oscillations of the crest curvature which highlight the fundamental difference with the earlier simulations of figure 4. A closer inspection shows the presence of a smaller and faster oscillations super-imposed on a slower one. The computations show that the faster oscillation is caused by the absorption of a small disturbance at the front of the wave associated with the initial rapid phase of growth. For each image pair in figure 8 this disturbance is visible in the frame corresponding to the first peak and is shown more clearly for one example in movie 1 of the supplementary material associated with this paper available at https://doi.org/10.1017/jfm.2021.1032. Similar oscillations can also be seen in the supplemental material of Zhou & Prosperetti (Reference Zhou and Prosperetti2020). The slower oscillations reflect the oscillations of the entire wave and are reminiscent of those mentioned in Babchin et al. (Reference Babchin, Frenkel, Levich and Sivashinsky1983) who write ‘As a result of the combined effect of these three factors – destabilizing gravity, flow-induced scale contractions, and stabilizing surface tension – finite-amplitude oscillations set in’.

Figure 8. Curvature vs time as waves evolve for different fixed inclination angles with $Re_\perp =50$, $Bo_\perp =0.6$, $V/h_0^2=40$. The wave shapes are taken at the instant marked in the figure.

A sequence showing wave profiles over the course of the slow oscillation is shown in figure 9 for the case $\beta =145^\circ$ of figure 8. The figure shows successive images of the wave separated by equal time intervals $\sqrt {g/h_0}\,\Delta t = 2$ starting from $\sqrt {g/h_0}\,t=13$. The faster, small-amplitude oscillation is revealed by a careful look at the waveforms, but the slower, larger one is quite clear. As the oscillation amplitude decreases, liquid is pushed back toward the plate and tends to flow forward (possibly because of the low surface-tension-induced pressure at the foot of the wave), thus increasing the apparent speed of the wave front. In this process the tail grows ‘stretching’ the back of the wave, increasing the length scale of the crest and, with it, decreasing the crest curvature.

Figure 9. A sequence of wave shapes for the case $\beta =145^\circ$ of figure 8; the images are separated by equal time intervals $\sqrt {g/h_0}\,\Delta t = 2$ starting from $\sqrt {g/h_0}\,t=13$. Note how the growth of the tail ultimately ‘stretches’ the wave crest and diminishes its curvature.

A portion of the waveform at the maximum points of the curvature oscillations is shown in the small images surrounding the main panel of figure 8. Here the crest shape is not determined solely by a surface-tension/gravity equilibrium, but is significantly influenced by dynamic effects. This can be seen, for example, from the difference between the first crest shape for $\beta =125^\circ$ (middle inset on the left) and the shape shown in figure 6$(c)$ in spite of the fact that, in both cases, the tip curvature is close to $\kappa _{cr}$. A careful inspection of the insets for $\beta =135^\circ$, $145^\circ$ and $155^\circ$ shows that the wave tail thickens as the inclination increases; we will see later in figure 14 that, conversely, the tail is very strongly reduced by further decreases of the plate inclination. The same thickening is shown in figure 3 where it is produced by the increase of the Reynolds number. For the same substrate thickness, a thicker tail takes liquid away from the crest thus limiting the amount of liquid available to form a drop and ‘stretches’ the wave, thus increasing the length scale of the crest. Both factors increase with the inclination of the plate and exert a stabilizing effect on the wave.

In all cases of figure 8 excepting the last one for $155^\circ$, the crest curvature ultimately decays without drop formation. When the curvature exceeds $\kappa _\parallel = \sqrt {2}/\ell _c$ the wave shape closely resembles the one shown in figure 6$(d)$, both for the stable case $\beta =145^\circ$ and the unstable case $\beta =155^\circ$. In the former, the wave shape is close to that of figure 6$(d)$, but the tangents at the inflection points of the wave shape do not quite become parallel to gravity and no drop forms. For the unstable $\beta =155^\circ$ case, however, the wave shape has evidently just passed the point of vertical tangency and, indeed, the wave becomes unstable and a drop detaches. These results suggest that $\kappa _\parallel$ maintains its correlation with the maximum liquid volume that can be supported by surface tension even in the presence of moderate dynamic effects. We have encountered stable waves with maximum curvatures temporarily larger than $\kappa _\parallel$ only when the wave was excited by short-wavelength disturbances, stronger than the ones giving rise to the oscillations visible in figure 8 mentioned before. In these cases, however, the large curvature was not associated with a sufficient liquid volume to overcome surface tension and, in any case, the large curvature did not last long enough to allow for the formation and detachment of a drop. Generalizing, it might be said that, in most cases, the shortest length scales that can arise are those generated by the Rayleigh–Taylor instability for which the crest curvature is a good indicator.

It may also be noted that here we find dripping for $\beta =155^\circ$ while, for the same parameter values, in the quasi-equilibrium case of figure 4 dripping did not occur until the inclination reached an angle $\beta =160.5^\circ$. The decrease of the dripping angle that we encounter here arises from the fact that, evolving from the initial condition, the wave is off-equilibrium. As the liquid gets re-distributed and the velocity field develops, a sufficient and sufficiently persistent accumulation of liquid in the wave crest can occur to overcome the restraining power of surface tension. Through this process, dynamic effects can render unstable a situation that would be stable in quasi-equilibrium conditions. These considerations support the interpretation of the conditions of figure 4 as providing an upper limit for dripping. We will see further examples when we explore the effect of initial conditions in § 7.

6. Horizontal and nearly horizontal plate

Up to this point we have considered plate inclination angles lower than, or just above, the critical angle for dripping. In preparation for the study of phenomena occurring farther from criticality, it is useful to consider the largest possible angle, i.e. a horizontal plate with $\beta =180^\circ$. As we will see, away from criticality the evolution of the system can be quite different from the one outlined in the previous sections and conclusions based on the quasi-equilibrium picture become less and less applicable. Some qualitative features of the results that we find in this case will also be encountered in the next section where, in the study of the effect of initial conditions, we encounter situations far from criticality because of conditions other than the angle of inclination of the plate. As will be noted in the next section, this is a consequence of the large (actually, infinite) dimensionality of the parameter space of the problem at hand which can in no way be reduced to considerations based on linear theory.

A fundamental study of the Rayleigh–Taylor instability with a horizontal plate was conducted by Yiantsios & Higgins (Reference Yiantsios and Higgins1989) who considered two superposed fluids in the Stokes flow limit. The same configuration was studied by others (see e.g. Whitehead Reference Whitehead1988; Lister & Kerr Reference Lister and Kerr1989), but we refer to Yantsios & Higgins’ study as, although their configuration is different from ours, many of their results are relevant. For example, these authors demonstrated the strong effect of initial conditions, a feature that we have already encountered and which we will illustrate further in the next section. They pointed out the limitations of predictions based on the (linear) fastest growing wavelength which may or may not dominate the final outcome of the instability and, in addition, they demonstrated the importance of the total liquid volume. Another somewhat distantly related work is that of Elgowainy & Ashgriz (Reference Elgowainy and Ashgriz1997) who studied an unsupported liquid layer subjected to a pressure difference on its two sides. The absence of a supporting plate makes the results of this study of limited use for our intents, but the growth of the so-called Rayleigh–Taylor spikes has some similarities with the example shown later in figure 11.

We focus on the dependence of the crest curvature on the total liquid volume in the computational domain. Of particular interest is the maximum curvature which is reached just before a drop starts to form. We fix $L/\ell _c$, the domain length normalized by the capillary length, and study the effect of varying the normalized film thickness $h_0/\ell _c$. The initial interface configuration is prescribed as before in (5.1) with $A=0.1$. Several examples of the time dependence of the drop tip curvature are shown in figure 10 for $L/\ell _c=10$ and different normalized liquid volumes $V/V_{cr}=Lh_0/V_{cr}$. In the present situation in which there is no travelling wave, a suitable dimensionless quantity to characterize the effect of viscosity is the Ohnesorge number

(6.1)\begin{equation} Oh = \frac{\mu}{\sqrt{\sigma \rho h_0}} = Ka_\perp^{{-}3/4} Bo_\perp^{{-}1/4} , \end{equation}

which is a measure of the viscous damping rate to the frequency of capillary waves. For the examples of figure 10 $Oh$ ranges, from left to right, from 0.050 to 0.120. In all cases, at first the curvature increases with time. As the initial wave crest gradually forms into a pendant drop, the curvature attains a maximum value soon after the free surface has developed two inflection points with vertical tangents. The curvature then briefly relaxes as the drop forms and starts to fall away from the plate reaching the bottom of the computational domain. The simulations are stopped at this point. We have found very similar results for $L/\ell _c=8$, 12 and 15. The Ohnesorge number controls the speed of the process but also, not coincidentally, the maximum curvature, which is seen to increase as the process is slowed down, which causes an increase of the importance of viscosity and a concomitant decrease of dynamic effects.

Figure 10. Time dependence of the curvature at the tip of a pendant drop under a horizontal plate for $L/\ell _c=10$. From left to right, the curves are for Ohnesorge numbers, defined in (6.1), equal to 0.050, 0.071, 0.091, 0.105, 0.112, 0.115 and 0.120. The computations show that the maximum curvature is reached just after the free surface develops two inflection points with vertical tangents and then relaxes as the base of the drop shrinks leading to detachment. The dashed line is a guide to the eye.

Figure 11. An example of the shape of a pendant drop under a horizontal plate at the moment at which two inflection points with vertical tangents appear; here, $Oh=0.05$ and $L/\ell _c=10$. The dashed line permits a comparison with the equilibrium shape of Pitts shown in figure 6$(d)$ and drawn to the same scale.

Smaller values of $Oh$ facilitate the formation of drops that are larger than the volumes $V_{cr}$ and $V_\parallel$ of figures 6$(c)$ and 6$(d)$ and have a curvature smaller than $\kappa _{cr}$ and $\kappa _\parallel$. An example is shown in figure 11 which compares the pendant drop shape for $Oh=0.05$ and $L/\ell _c=10$, at the moment inflection points with vertical tangents appear, with the drop of figure 6$(d)$ drawn at the same scale. In this case, the volume of the drop that ultimately detaches is more than three times as large as the volume of the drops that detach when the waves of figure 4 become unstable. A similar finding was reported by Abdelall et al. (Reference Abdelall, Abdel-Khalik, Sadowski, Shin and Yoda2006) who experimented with drop detachment from the underside of a porous plate. The reason is that, unlike the drops found following Pitts’ analysis, the drop shapes that develop in these conditions are not equilibrium solutions of the gravity–surface-tension balance relation (4.1). An intuitive explanation of these results may be given noting that a liquid flow accelerating in the direction of gravity is, in a very rough sense, equivalent to a reduced gravity since, if the liquid was in a free fall, the apparent gravity would vanish. A smaller $g$ would increase the value of $\ell _c$ thus reducing $\kappa _\parallel$ and increasing the drop volume, for both of which $\ell _c$ is the characteristic length scale. The relative rapidity with which drops form as $Oh$ is reduced may be interpreted by noting that $Oh$ can be decreased by reducing the liquid viscosity, which permits a freer flow of liquid into the wave crest under the action of the gravity-induced pressure gradient. An equivalent interpretation may rely on reducing $Oh$ by increasing the mean thickness of the liquid layer $h_0$. As can be seen from (5.1), the initial wave amplitude is given by $Ah_0$, and therefore a larger $h_0$ places a proportionately larger amount of liquid below the average liquid depth and far from the wall with a similar effect.

Figure 10 shows a monotonic increase of the maximum curvature as the crest growth is slowed down. This effect is made clearer by figure 12, which shows the maximum curvature as a function of $Lh_0/V_{cr}$, with $V_{cr}\simeq 5.211 \ell _c^2$ Pitts’ minimum volume for drop formation. Each line connects points obtained with the same values of $L/\ell _c$ and $Ka_\perp =21.5$ or 54.3; the results seem to depend mostly on the total liquid volume with $Ka$ having a minor effect. As $V/V_{cr}\rightarrow 1$, the maximum curvature increases past Pitts’ $\kappa _{cr}$, although a limit value, if it exists, is not apparent from the figure. This effect is caused by a small residual amount of liquid which remains trapped on the plate near the main pendant drop. This liquid cannot readily flow into the main drop due to the strong viscous effect in the thin layer that separates it from the main drop. A similar effect was reported by Yiantsios & Higgins (Reference Yiantsios and Higgins1989) (see e.g. the last example in their figure 8) and, more recently, by Lister, Rallison & Rees (Reference Lister, Rallison and Rees2010) and Lerisson et al. (Reference Lerisson, Ledda, Balestra and Gallaire2020). The effect is negligible when $Lh_0/V_{cr}$ is large, but becomes noticeable when it prevents the volume ratio to get sufficiently close to 1. Because of this residual volume, the data in the figure should not really be plotted against the nominal value $Lh_0/V_{cr}$ used in the horizontal axis of the graph, but rather against an effective volume $(Lh_0-V_{residual})/V_{cr}$.

Figure 12. The maximum curvature reached by a pendant drop under a horizontal plate (i.e. the curvature corresponding to the maxima in figure 10) as a function of the normalized liquid volume $Lh_0/V_{cr}$. All the lines, except the second one from the left (upside down triangles, yellow) have $Ka_\perp =21.5$. From left to right, the lines are for $L/\ell _c=8$ (circles, blue), 10 (upside down triangles, yellow, with $Ka_\perp =54.3$), 10 (triangles, orange), 12 (squares, green) and 15 (diamonds, purple).

Figure 13(a) shows the data of figure 12 near $V/V_{cr}=1$ which are seen to lie approximately on straight lines. We have found that the effect of the residual volume is negligible for the case with $L/\ell _c=8$. Extrapolation of the line connecting the data for this case (open blue circles in the figure) toward $Lh_0/V_{cr} = 1$ leads to $\kappa _w\simeq \kappa _\parallel$. This fact suggests that this should be the limit for the other cases as well if the effect of the residual volume could be accounted for. In order to estimate $V_{residual}$ we extrapolate the straight lines of figure 13$(a)$ to $\kappa _w=\kappa _\parallel$ and run a simulation using this initial volume with the same initial condition as before. Eventually two pendant drops are left, a larger one and a smaller one. We identify the volume of the latter with $V_{residual}$ and re-plot the data of figure 13$(a)$ as functions of $(Lh_0-V_{residual})/V_{cr}$ as shown in figure 13$(b)$. All the lines now converge very closely to $\kappa _\parallel$ as $(V-V_{residual})/ V_{cr} \rightarrow 1$. This procedure therefore leads to the conclusion that $\kappa _\parallel$ is indeed associated with $V_{cr}$ when the liquid volume is just enough to cause dripping under a horizontal plate. It may be noted that the agreement with Pitts’ result is not obvious a priori given that the present simulations were conducted under the assumption of spatial periodicity, which implies a fixed contact line at the end points of the domain, rather than fixed contact angle, conditions. A similar conclusion as to the existence of a minimum amount of liquid necessary for drop formation under conditions similar to the present ones was reached by Yiantsios & Higgins (Reference Yiantsios and Higgins1989) (reiterated in Kofman et al. Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018) and, for the three-dimensional case, by Fermigier et al. (Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992).

Figure 13. Panel $(a)$ shows a subset of the data of figure 12 near $V/ V_{cr} = 1$. Panel $(b)$ shows the same data adjusted by the subtraction of $V_{residual}$ as explained in the text. See the caption to figure 12 for an explanation of the symbols.

Figure 14. Continuation of figure 8 showing the curvature vs time for fixed inclination angles with $Re_\perp =50$, $Bo_\perp =0.6$, $V/h_0^2=40$. The data for $\beta =155^\circ$ are the same as shown in figure 8 and permit a comparison with those for $\beta =165^\circ$ and $175^\circ$ in which cases the formation of the drop resembles that under a horizontal plate.

The association of $V_{cr}$ with $\kappa _\parallel$, rather than $\kappa _{cr}$, may appear somewhat unexpected. However, it is known from Pitts’ work that $V_{cr}$ is the largest liquid volume for which a drop on the underside of a horizontal plate can be stable and, as pointed out in § 4, the curvature $\kappa _\parallel$ corresponds to drop shapes for which surface tension is most effective in opposing gravity. We have seen the importance of $\kappa _\parallel$ already in figure 4 as the maximum curvature reached when drops start to form and detach, in figure 8 as being mostly close to the maximum curvature in the oscillations of Kapitza waves and in figure 10 as an upper limit for the curvature as drop formation under a horizontal plate is progressively retarded. In the first case $\kappa _{cr}$ was found to play a role at the threshold, but did not appear to be associated in particular with drop formation. In the other two, the crest curvature reached and surpassed $\kappa _{cr}$ without any particular feature or event marking the passage. The crucial difference between the two special values of the curvature, which explains the different importance of their roles, is that $\kappa _{cr}$ is dependent on the overall wave shape and total liquid volume while, as already noted, the association of maximum surface-tension effect and $\kappa _\parallel$ is based only on the local features of the wave shape near the crest.

In this light, the results reported in figure 4 are not surprising: even in a flowing film, the first drops that are possible when the equilibrium provided by surface tension becomes unstable are drops with the characteristics of the (quasi-)equilibrium situation. The wave first becomes unstable when $\kappa _{cr}$ is attained, but drops do not start to form and detach until the curvature reaches $\kappa _\parallel$ because it is at this point that surface tension becomes unable to support the volume of liquid in the crest. Since these are the first possible drops, one would expect that their features would be nearly independent of the total amount of liquid in the computational domain. Indeed, for example, the cases with $Re_\perp =50$ and $Bo_\perp =0.6$ in figure 4 go through essentially the same sequence irrespective of whether the liquid volume is $V/h_0^2=20$ or $V/h_0^2=80$. As figure 2$(b)$ shows, with a larger liquid volume, drops can form at a larger plate inclination (smaller $\beta$) but the process of their formation is otherwise similar.

The residual liquid left on the plate may have a curious effect when the total liquid volume is just above $V_{cr}$. We have found that, in this case, the volume of the pendant drop, deprived of the residual liquid, may be below the critical value. However, the very slow motion induced by tilting the plate by as little as $1^\circ$ – from $180^\circ$ to $179^\circ$ – is sufficient for the pendant drop to pick up the residual liquid growing sufficiently to start to drip. As pointed out by Yiantsios & Higgins (Reference Yiantsios and Higgins1989) and Lister et al. (Reference Lister, Rallison, King, Cummings and Jensen2006), given enough time the residual volume might flow into the main drop (unless the film evaporates or breaks under the action of the disjoining pressure caused by van der Waals forces) so that it is possible that even the pendant drop on the horizontal plate might ultimately become unstable. The practical consequences of this possibility – if any – are hard to assess.

It is evident that the physics determining the evolution of the interface for a horizontal plate will not appear abruptly when the inclination angle reaches $180^\circ$, but will set in gradually as this condition is approached. For this reason, the horizontal plate results shed light on what happens when the inclination of the plate studied in § 5 is decreased to $\beta =165^\circ$ and $175^\circ$ (figure 14). As we have already mentioned, the case with $\beta =155^\circ$, repeated in figure 14, is close to the critical angle and, accordingly, dripping is associated with a crest curvature close to $\kappa _\parallel$. The other two, however, are farther from the critical angle: the wave shape is markedly different, most noticeably in the tail, which is greatly diminished, and in the crest curvature, which is smaller. The amplitudes vs time of the waves for $\beta =180^\circ$, $175^\circ$, $165^\circ$, $148^\circ$ (all unstable) and $145^\circ$ (stable) are shown in panel $(a)$ of figure 15. Remarkably, the time dependence of the first three amplitudes is virtually indistinguishable. The corresponding wave shapes, taken at the same times at three successive instants, are shown in panels $(b)$ of the same figure, with the arrow indicating the direction of gravity. For these nearly horizontal plates the Rayleigh–Taylor instability develops very rapidly, much before the flow, which starts from rest, is able to generate a Kapitza wave. When the plate tilt is increased to $155^\circ$ (not shown) and $148^\circ$, the growth rate weakens and finally stabilizes for an inclination of $145^\circ$.

Figure 15. $(a)$ Time dependence of the amplitude of the wave crests for $\beta =180^\circ$, $175^\circ$, $165^\circ$, $148^\circ$ (all unstable) and $145^\circ$ (stable). The first three very nearly superpose and are hardly distinguishable. The corresponding maximum curvatures are shown in figure 16. $(b)$ Two-dimensional pendant drop shapes for the three largest inclination angles at $\sqrt {g/h_0}\,t=20$, 21 and 22; the arrow indicates the direction of gravity. The drops slowly translate with respect to their initial position in the top panel due to the inclination of the plate.

Figure 16 collects the maximum curvatures of figures 8 and 14. The filled and open symbols denote dripping and non-dripping cases, respectively. As the dripping angle for the particular initial condition used in these simulations is approached from below, the maximum curvature of the wave increases slightly overshooting the value $\kappa _\parallel$. This is an indication that the length scale of the wave crest diminishes, a tendency which clearly favours dripping. As the angle increases past the dripping threshold, the maximum curvature decreases similarly to what we have seen for the horizontal plate case as one moves farther away from the critical condition with increasing liquid volume and decreasing viscous effects.

Figure 16. The maximum curvatures reached by the waves of figures 8 and 14 (augmented by the results of three additional simulations with $\beta =180^\circ$, $\beta =148^\circ$ and $140^\circ$) vs the inclination angle of the plate. The filled and open symbols denote dripping and non-dripping cases, respectively.

7. Effect of initial conditions

A peculiarity of the present system already noted by Yiantsios & Higgins (Reference Yiantsios and Higgins1989), is that ‘the final state of the interface can be different depending on different initial conditions and always present disturbances’. A similar point was mentioned by Rohlfs et al. (Reference Rohlfs, Pischke and Scheid2017) who note that ‘droplet (or ligament) detachment is sensitive to the initial condition, so that dripping $\ldots$ can also occur for lower inclination numbers if a favorable initial condition is used’. Kofman et al. (Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018) modified their original initial conditions to avoid ‘a temporal development of the wave $\ldots$ characterized by an overshoot in amplitude $\ldots$ which accentuates the phenomenon of dripping’. While, as these quotes show, this aspect of the problem has been noted before, it does not seem to have received much attention in spite of its obvious importance. For example, it is obvious that this feature prevents the development of a phase map of possible outcomes dependent only on the problem parameters (Reynolds number etc.) such as, for example, the one developed by Rohlfs & Scheid (Reference Rohlfs and Scheid2015) for waves on the top side of an inclined plate. In order to be successful, any phase map must also include information on a suitable characterization of initial conditions.

The proper interpretation of these complex features must be based on concepts familiar from the theory of dynamical systems. It is well known that, when a dynamical system has more than one attractor, the phase space is divided into domains, or basins, of attraction of each one of them, with one or the other limiting behaviours ultimately prevailing depending on the particular domain of attraction in which the initial condition happens to be located (see e.g. Lichtenberg & Lieberman Reference Lichtenberg and Lieberman1983; Wiggins Reference Wiggins1988; Ott Reference Ott2003). In a system with an infinite number of degrees of freedom like the present one, the phase space is a Sobolev space but the notion of domains of attraction still applies. There will be a domain of attraction of the dripping solution, a domain of attraction of the Kapitza wave with one hump and, possibly, domains of attraction of Kapitza waves with two or more humps, an outcome that we have occasionally encountered with long computational domains but not studied in detail. A complete characterization of these domains of attraction requires a significant effort because it must be carried out iteratively accounting for the full nonlinearity of the problem. Tools for this sort of nonlinear non-modal analysis are currently being developed (see e.g. Gallino, Schneider & Gallaire Reference Gallino, Schneider and Gallaire2018; Kerswell Reference Kerswell2018) but we will not pursue the matter here, limiting ourselves to the presentation of results of some exploratory calculations that illustrate the effect and highlight the subtlety of the influence of initial conditions.

We begin by describing the results of two types of simulations, in both of which the initial condition is given by (5.1). In the first type we start the simulations with perturbations of the same shape but different amplitudes. In the second one we use the fact that the Rayleigh–Taylor instability is dependent on the component of gravity normal to the plate, $g\cos \beta$, while the Kapitza instability is primarily dependent on the component parallel to the plate, $g\sin \beta$. If a simulation is started without the $g\cos \beta$ component, which is then restored later, the Kapitza instability is allowed to develop for a while in the absence of the other instability. This would be equivalent to imposing an initial condition on the system at the instant at which the $g\cos \beta$ component is restored.

Figure 17$(a)$ shows the wave amplitude vs time when the amplitude of the initial sinusoidal perturbation (5.1) is varied from 0.1 to 0.8; the vertical Reynolds number is $Re_\perp = 50$, the Bond number $Bo_\perp = 0.6$, the liquid volume $V/h_0^2$ = 40 and the plate inclination angle $\beta$ = 140 $^\circ$. From figure 4, the critical inclination angle for these conditions is $160.5^\circ$ so that one would expect a stable non-dripping Kapitza wave in quasi-equilibrium conditions. For $A = 0.3$, 0.5 and 0.8, the vertical velocity component develops very quickly and the amplitude of the wave rapidly grows, leading to the formation and ultimate detachment of a drop. For $A = 0.1$ and 0.2, however, the wave initially grows, reaches a maximum, and then decreases settling down to a constant value. This asymptotic value is independent of the initial amplitude, showing that both initial conditions with $A=0.1$ and 0.2 are situated in the domain of attraction of the stable steady state. Even in these stable cases, the power of the Rayleigh–Taylor instability manifests itself in causing the overshoot above the eventual steady-state level, as noted by Kofman et al. (Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018); a similar phenomenon is responsible for the oscillation of figure 9. The smallest amplitude used here, while small, is not small enough to justify the application of linear theory. We have found that, with a truly small initial amplitude, the interface develops in a complex way with a combination of small waves with the wavelength dictated by the fastest growing Rayleigh–Taylor mode. A similar outcome is shown in figure 5 of Yiantsios & Higgins (Reference Yiantsios and Higgins1989).

Figure 17. $(a)$ Wave amplitude vs time starting from initial amplitudes $A=0.8$, 0.5, 0.3, 0.2 and 0.1 in descending order; $Re = 50$, $Bo = 0.6$, $V/h_0^2 = 40$, $\beta =140^\circ$. $(b)$ Wave amplitude with the gravity component normal to the plate restored at different times marked by arrows; $Re = 50$, $Bo = 0.6$, $V/h_0^2 = 40$, $\beta =152^\circ$. An animation of the wave growth when gravity is restored at $\sqrt {g/h_0}\,t=10$ and $\sqrt {g/h_0}\,t=15$ is shown in movie 2 of the supplementary material associated with this paper.

The detaching drops in the three unstable cases of figure 17$(a)$ are shown in figure 18 where it is evident that their volume increases with $A$. The reason is that, as anticipated in the previous section, there is a well-defined critical dripping amplitude associated with the initial condition, inclination angle and control parameters used for this simulation. Increasing $A$ increases the distance from this critical dripping amplitude which leads to larger detaching drops. Qualitatively, this is the same tendency found for a horizontal plate in § 6 for diminishing Ohnesorge numbers. A physical interpretation is that a larger initial amplitude places more liquid into the crest of the wave at some distance from the wall from which this liquid can flow into the forming drop relatively unhindered by the viscous shear caused by the no-slip condition at the distant wall. Conversely, with a smaller initial amplitude, it takes longer for a liquid mass sufficient for dripping to collect in the wave crest as liquid flow is hindered by viscosity. Therefore, the Kapitza instability has more time to establish itself and stabilize the dripping instability by stretching the wave crest and ‘opening up’ the base of the drop.

Figure 18. Drops about to detach for the three unstable cases of figure 17$(a)$. From left to right the images correspond to $A=0.8$, 0.5 and 0.3. The slight deviation of the trajectory away from the direction of gravity is due to the drag exerted by the second fluid in the computational domain as explained in § 2. Note that the structures shown here are actually two-dimensional, not real three-dimensional droplets.

Turning now to the second type of simulation, we show some results in figure 17$(b)$ with the same parameter values as for the previous figure except the inclination angle that here is $\beta = 152^\circ$, again smaller than the critical angle for quasi-equilibrium conditions. The initial condition is the same sinusoid (5.1) as before with $A$ = 0.2. The particular value of the inclination angle used here has been chosen because it is slightly above the stability threshold for the conditions of figure 8 and therefore, under normal conditions, it should produce an unstable wave. The curves correspond to restoring the $g\cos \beta$ gravity component at progressively later times. If the Kapitza instability is not allowed enough time to develop before the normal gravity component is restored, the wave grows and a drop eventually forms as expected. On the other hand, the film is stabilized if the action of the normal gravity component is delayed and, with it, the development of vertical inertia and the compression of the horizontal length scale toward $\ell _c$. However, it is also evident from the overshoot above the eventual steady state that, as soon as it is restored, the Rayleigh–Taylor instability injects a significant amount of energy into the system ‘trying’ to destabilize the wave. In this case as well the amplitude of the steady-state wave is independent of the time at which the vertical gravity component is restored and the same comment on the location of the initial conditions in the domains of attractions of the two asymptotic solutions is applicable. An animation of the wave growth when gravity is restored at $\sqrt {g/h_0}\,t=10$ and $\sqrt {g/h_0}\,t=15$ is shown in movie 2 of the supplementary material associated with this paper.

In both previous cases the initial condition was a sinusoid with a wavelength equal to that of the computational domain. We now show results obtained with different initial shapes. For the first example we use

(7.1)\begin{equation} \frac{h(x,t=0)}{h_0} = 1+ \frac{A}{\tau} \tan^{{-}1} \frac{\tau \sin 2{\rm \pi} x/L}{1-\tau \cos 2{\rm \pi} x/L} , \end{equation}

with $0\leq \tau \leq 1$ a dimensionless parameter. The shape of this curve which, depending on $\tau$, interpolates between a single sine ($\tau =0$) and a sawtooth function ($\tau =1$) (McCaughan Reference McCaughan2017), is shown in figure 19 for $\tau =0.5$ and $A =-0.3$ (forward leaning, solid line) and $A=0.3$ (backward leaning, dashed line). Figure 20, for $Re_\perp =50$, $Bo_\perp =0.6$ and $V/h_0^2=40$, shows that a significant difference between the two cases develops already for $A=\pm 0.1$, with the forward-leaning wave becoming stable and the backward-leaning one unstable. A close inspection shows that the oscillations affecting the forward-leaning case with $A=-0.1$ are not due to the Fourier modes that make up the initial condition, but to small capillary disturbances absorbed at the front of the wave as for the oscillations of figure 8. Thus, a Fourier decomposition is of little value in determining the dripping/non-dripping outcome of the wave development. The same difference between forward- and backward-leaning initial shapes persists all the way to $A=\pm 0.4$ while, for $A=\pm 0.5$, both initial shapes become unstable. A likely explanation of the different outcomes is that, in the forward-leaning case, the initial shape is already close to that of a Kapitza wave and such that surface tension starts out strong (cf. a similar argument in Babchin et al. Reference Babchin, Frenkel, Levich and Sivashinsky1983), while its effect has to gradually develop in the backward-leaning case leaving enough time for the liquid to accumulate into the wave crest and ultimately drip.

Figure 19. The initial condition (7.1) for a ‘forward-leaning sine’ ($A=-0.3$, solid line) and a backward-leaning sine ($A=0.3$, dashed line) with $\tau =0.5$; note the downward-oriented vertical axis.

Figure 20. Evolution of $h_w/h_0$, the normalized maximum height of the wave, for the initial conditions (7.1) with $\tau =0.5$. The solid lines are for a forward tilt and the dashed lines for a backward tilt. In descending order, the lines are for $A=\pm 0.5$ (blue), $\pm 0.3$ (green) and $\pm 0.1$ (purple).

As a fourth example we use the superposition of two sines

(7.2)\begin{equation} \frac{h(x,t=0)}{h_0} = 1+ A\left( \sin \frac{2{\rm \pi} x}{L} + 0.1 \sin \frac{4{\rm \pi} x}{L}\right) . \end{equation}

The maximum and minimum amplitudes of this shape differ by less than 2 % from $\pm A$ and, therefore, these simulations can legitimately be compared with the results found omitting the second term (dashed lines). Both initial conditions produce stable Kapitza waves for $A=0.1$ and unstable waves for $A=0.3$ and 0.5. However, for $A=0.2$, the one-term initial condition produces a stable wave while the two-term one produces an unstable solution. These results illustrate the sharpness and convoluted shape of the domains of attraction of the two asymptotic solutions, demonstrating the difficulties that must be overcome to understand their topologies and, even more, to develop a precise determination of their boundaries. It is also interesting to note that the solid lines give no straightforward evidence of the presence of two Fourier components in the initial condition. The nonlinear interaction is evidently quite strong already from the beginning.

8. Summary and conclusions

In this paper we have considered the dripping of a liquid film flowing along the underside of a titled plate. The Rayleigh–Taylor instability tends to cause dripping by reducing the length scale of the wave crest so much that the weight of the liquid that concentrates into it cannot be sustained by surface tension. Conversely, the Kapitza wave owes its stabilizing effect to its ability to prevent the accumulation of liquid in the crest by thickening the wave tail and stretching its length scale. We have found that, when the wave is allowed to evolve in quasi-equilibrium conditions, the balance of gravity and surface tension is broken in much the same way as when a drop forms on the underside of a horizontal plate when the liquid is just sufficient for this to happen (Pitts Reference Pitts1973). Key to the establishment of this correspondence has been our recasting of Pitts’ stability condition in terms of the curvature of the wave crest, which does not rely on the contact angles and liquid volumes used in his original formulation. In this way, the crest curvature has been turned into a sensitive indicator of the liquid volume in the crest and of the proximity of the crest to the dripping instability. As dynamic effects become important, the value of the crest curvature as an indicator of the weight of the liquid accumulated in it weakens, but remains fairly good provided conditions are not too far from the quasi-equilibrium dripping threshold. Increasing distance from the threshold, as in the case of a horizontal or nearly horizontal plate or a large-amplitude initial condition, produces progressively larger drops with decreasing maximum curvature.

The dependence of the outcome of the simulations on initial conditions, which has been noted, but not studied in depth, before (Yiantsios & Higgins Reference Yiantsios and Higgins1989; Rohlfs et al. Reference Rohlfs, Pischke and Scheid2017; Kofman et al. Reference Kofman, Rohlfs, Gallaire, Scheid and Ruyer-Quil2018), is a very striking feature of the process investigated in this work. The analysis of the horizontal plate case reveals an important factor that suggests an explanation for this effect, namely the need for the rapid flow of liquid into the wave crest. If the initial condition places enough liquid relatively far from the supporting plate, this liquid can rapidly flow into the crest because it is relatively unhindered by the viscous shear caused by the no-slip condition at the plate. In this case, then, a drop is more likely to form. By ‘stretching’ the wave, the flow pushes the liquid closer to the plate thus subjecting it to viscous shear and preventing it from flowing into the crest to form a drop. For this stabilizing process to take place, though, enough time must be allowed for the Kapitza wave to develop which, depending on conditions, may be too long to stop the dripping instability. In support of this explanation we have shown that, if the Kapitza instability is allowed sufficient time to establish itself before the Rayleigh–Taylor instability is turned on, dripping is prevented. As the flow Reynolds number is increased, the tail behind the crest of the wave becomes longer and thicker and, therefore, by conservation of mass, the wave amplitude decreases, both effects countering the development of a large curvature. Indeed, we have found that smaller Reynolds numbers are less stable than larger ones.

With the domain lengths on which we have focused (up to 80 times as long as the mean film thickness) and with the initial conditions that we have considered, we have not found a tendency for the appearance of a second wave and, therefore, we have not had to deal with wave–wave interactions. There are several reasons why these interactions are important for film flow above a plate (see e.g. Chang & Demekhin Reference Chang and Demekhin2002; Pradas, Tseluiko & Kalliadasis Reference Pradas, Tseluiko and Kalliadasis2011; Pradas, Kalliadasis & Tseluiko Reference Pradas, Kalliadasis and Tseluiko2012). An additional one for a film on the underside of a plate is that, as shown experimentally by Fermigier et al. (Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992), dripping can result from the coalescence of two waves which, individually, would be stable. Another important class of problems related to the one we have investigated is film flow on the underside of curved, rather than planar, surfaces which is important, for instance, for vapor condensation on tubes in heat exchangers and other applications (see e.g. Limat et al. Reference Limat, Jenffer, Dagens, Touron, Fermigier and Wesfreid1992; Takagi & Huppert Reference Takagi and Huppert2010; Balestra, Nguyen & Gallaire Reference Balestra, Nguyen and Gallaire2018). It may be expected that our focus on the crest curvature as an indicator of the formation of short length scales conducive to dripping as well as the quasi-equilibrium approximation would be relevant for such applications as well.

In conclusion it is appropriate to consider which of our findings would likely carry over to the three-dimensional situation. The quasi-equilibrium result of § 4 is only based on the fact that the balance of surface tension and gravity that governs dripping from a horizontal plate is also relevant in the case of waves on a flowing film provided the limit angle is approached very slowly. It would seem that this conclusion should apply to the three-dimensional problem as well. The marked effect of initial conditions is a qualitative feature that should be strong enough to carry over to the three-dimensional case. The mechanism leading to oscillations such as those shown in figures 8, 9 and 14 should affect also a three-dimensional process. No such oscillations have yet been observed in experiments, but this could be due to small Reynolds numbers, large viscosity, small surface tension or a combination of these factors. For example, in the experiments of Brun et al. (Reference Brun, Damiano, Pierre, Balestra and Gallaire2015) $Re \leq 0.1$, in those of Rietz et al. (Reference Rietz, Scheid, Gallaire, Kofman, Kneer and Rohlfs2017), $Re \leq 1.6$; in those of Charogiannis et al. (Reference Charogiannis, Denner, van Wachem, Kalliadasis, Scheid and Markides2018) $Re \leq 193$, but the smallest Kapitza number was 13.1; in those of Ledda et al. (Reference Ledda, Lerisson, Balestra and Gallaire2020) the Kapitza number was approximately 0.087. The role of the substrate and wave tail in ‘stretching’ the wave crest, limiting the curvature and thereby stabilizing the wave appears to be a robust qualitative feature which may also be expected to affect the three-dimensional flow.

Figure 21. Wave evolution from the two-term initial condition (7.2) with (solid lines) and without (dashed lines) the second Fourier component; $A =0.1$ (purple), $A=0.2$ (green), $A=0.3$ (blue) and $A=0.5$ (orange).