6.1. One-sided model: reference simulations

We first examine the $(2+1)$-D dynamics of a condensing layer subject to RTI by solving the one-sided model (5.7) with the random initialization (5.8a) in two cases: Case I for strong vapour thrust and Marangoni effect; and Case II for moderate vapour thrust and weak Marangoni effect. In addition, since the one-sided model is most relevant to experiments free of inert gases, we will turn to a quantitative comparison with GG's experiment in § 6.6 (Case III). As mentioned in § 5.2, we take $l=6{\rm \pi}$ as the side length of the domain, whose diagonal fits the disturbances with $\lambda =3\lambda _{m}^{nv}$, where $\lambda _{m}^{nv}=2\sqrt {2}{\rm \pi}$ is the fastest growth wavelength of a non-volatile (designated by the superscript ‘nv’) isothermal layer subject to pure RTI from linear theory (Yiantsios & Higgins Reference Yiantsios and Higgins1989; Fermigier et al. Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992). A uniform mesh of $171\times 171$ is employed, based on the grid-independency test.

The time series of interface evolution for Case I are presented in figure 5 with $K=0.1$, $\mathscr {E}^\ast =0.05$, $\mathscr {G}^\ast =0.2$, $\mathscr {M}^\ast =\mathscr {D}^\ast =-1$ and $Bi=0.5$. The interface starts with a general relaxation and the self-organization of random perturbations; several crests and troughs emerge as the initially linear instability is amplified (figure 5a). The local depressions and peaks experience the stages of deepening/elevating (due mainly to the RTI and strong vapour recoil) and of broadening and coalescing (due mainly to the strong thermocapillary and capillary effects). The dynamics unfolds two pairs of competitions between: (i) vapour recoil (tends to rupture the interface) and the capillary force (tends to coalesce bulges and smooth the interface) in the regions of thin film, where condensing fluxes and viscous forces are much larger than those in the thick regions; (ii) RTI (forming elongated drops as condensate drains) and stabilizing thermocapillarity (tending to broaden bulges) in the thick regions. The expansion of the drained regions results in an irregular polygonal network of liquid ridges (figure 5c). Further opening of the drained regions gives rise to breakup of the ridge network as well as the emergence of isolated droplets at the moment of rupture, as shown in figure 5(d,e). The interaction among drops and ridges (e.g. coalescence) during the late stage is responsible for the breakup of this polygonal structure. The primary and secondary droplets become gradually axisymmetric with the growth in amplitude (cf. figure 5f) due to the attenuating interaction with adjacent drops/ridges and the increasing capillary forces (see the transition from figures 5c to 5d).

Figure 5. Evolution of a Rayleigh–Taylor unstable Case I condensing layer, obtained by solving (5.7) with IC (5.8a) for $K=0.1$, $\mathscr {E}^\ast =0.05$, $\mathscr {G}^\ast =0.2$, $\mathscr {M}^\ast =-1$, $\mathscr {D}^\ast =-1$ and $Bi=0.5$ on a $171\times 171$ mesh. The side length of the periodic domain is $l=6{\rm \pi}$. The successive snapshots of interface contours with the global minimum and maximum thicknesses ($H_{Min}$, $H_{Max}$): (a) $t=2.0$, $(1.0452, 1.1316)$; (b) $t=5.0$, $(0.9169, 1.5832)$; (c) $t=7.0$, $(0.5170, 3.3498)$; and (d) $t_{r}=9.246$, ($0.20\times 10^{-4}$, $14.8632$). The bright (dark) shades correspond to thick (thin) regions. Each contour has its own brightness scale and thus different images cannot be compared directly (as well in figure 7). (e) Surface plot at rupture. (f,g) Evolution of representative profiles: $x=7.9$ and $x=10.5$.

With one order-of-magnitude weaker Marangoni effect of $\mathscr {M}^\ast =-0.1$ (along with $K=0.01$) and a significant but not dominant vapour recoil at $\mathscr {D}^\ast =-0.5$, Case II evolves into a more regular pattern of pendent droplets with more uniform amplitudes relative to Case I (comparing figures 5e and 6a). In this case, the free surface has the appearance of long-wave quasi-hexagons at $t_{r}=9.374$ (figure 6b), while an irregular pattern is seen in Case I. The representative $x$-cross-sections plotted in figure 6(d) show the occurrence of rupture on $x=0$ and the global maximum elevation $H_{Max}=6.5827$ along $x=2.107$. The heights of the droplets decrease significantly as compared with that of the primary drop in Case I. Furthermore, as suggested by figure 6(a), the interfacial temperature is close to the saturation value with small variations over the interface, consistent with the weak Marangoni effect. This is evidently because of the small degree of interface non-equilibrium ($K=0.01$), just as the experimental condition of Som et al. (Reference Som, Kimball, Hermanson and Allen2007). For that reason, we will present a comparison of the Case II with their experiment.

Figure 6. (a) Case II condensing layer subject to RTI at $t_{r}=9.374$, obtained by solving (5.7) with IC (5.8a) for $K=0.01$, $\mathscr {E}^\ast =0.05$, $\mathscr {G}^\ast =0.2$, $\mathscr {M}^\ast =-0.1$, $\mathscr {D}^\ast =-0.5$, $Bi=0.5$ and $l=6{\rm \pi}$ on a $171\times 171$ mesh. The surface is coloured by normalized $\varTheta _{I}^\ast$. (b) Upward view plotted on an extended domain of $[0, 3l/2)\times [0, 3l/2)$ to accommodate an evident pattern and compare with the film imaged by Som et al. (Reference Som, Kimball, Hermanson and Allen2007). (c) Shadowgraph image of the interface at the occurrence of complete coverage of pendent drops with the highest subcooling in an experiment. Copyright ©2006 Elsevier Ltd. Reprint of figure 8(c) with the permission of Elsevier from Som et al. (Reference Som, Kimball, Hermanson and Allen2007). (d) Rupture patterns of the representative profiles at $x=0$ and $x=2.107$ as highlighted in (a) along with the condensing fluxes at $0.98t_{r}$. The red dots in (b) and (d) denote the global minimum $(H_{Min}=0.45\times 10^{-4})$ and maximum thicknesses $(H_{Max}=6.5827)$ at the positions of $(0\pm ml, 9.267\pm ml)$ and $(2.107\pm ml, 11.975\pm ml)$, respectively, where $m=0,1,2,\ldots$.

In both Cases I and II, it is found that the finite-time rupture always occurs due to the vapour recoil and Rayleigh–Taylor instabilities. The rupture is nearby the ‘contact line’ of the highest droplet, where both the Marangoni stresses and localized vapour recoil are expected to be greatest. The one-sided model thus predicts that a moderate or strong vapour recoil together with RTI can prevail over the Marangoni and capillary stabilizations even if the bulk of mass gain occurs in the thin-film regions. For later reference, a significant Marangoni stabilization, however, can overcome a weak vapour thrust and leads to a non-rupture state despite the destabilization by gravity (see § 6.6).

6.2. One-sided model: comparison with experiments

It is instructive to compare the surface patterns of the one-sided simulations with those observed in the relevant RTI experiments: (i) the non-condensing silicone-oil layer by Fermigier et al. (Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992) to infer how the condensation effects and the ICs influence the interfacial stability, and (ii) the condensing $n$-pentane by Som et al. (Reference Som, Kimball, Hermanson and Allen2007) to demonstrate the applicability of the one-sided model.

First, we compare the dynamics of a Rayleigh–Taylor unstable Case II condensing layer with an experiment by Fermigier et al. (Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992), in which the instability was initiated by an axisymmetric disturbance of dust specks on the interface. Figure 7 displays the solution of (5.7) for the identical set of parameters as in figure 6, but now we use the axisymmetric IC (5.8b). As can be seen, the initial dimple deepens and expands into a ring (figure 7a), which then evolves into peaks with a fourfold symmetry (figure 7b), in the centre of which a small isolated drop appears; and four larger ones emerge from the corners. The evolution is almost in accord with the experiment in the early stage (see their figure 5), where the axisymmetric structure is revealed by the distortion of a ruled screen observed through the gas–liquid interface. Note that a small drop does form in the centre of the oil layer, whose image, however, dims out (figure 7c) due to large interface curvatures. This is a typical issue in the deflectometry technique (Fermigier et al. Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992).

Figure 7. Evolution of a Rayleigh–Taylor unstable Case II condensing layer, obtained by solving (5.7) with IC (5.8b) for the same values of parameters and mesh as in figure 6. The snapshots of interface contours with the global minimum and maximum thicknesses ($H_{Min}$, $H_{Max}$) at (a) $t=6.0$, $(0.8235, 1.4728)$ and (b) $t=9.0$, $(0.3461, 3.5234)$. (c) A photograph of a silicone-oil layer hanging on a horizontal substrate, growing from an axisymmetric perturbation in an experiment without phase change. Copyright ©1992 Cambridge University Press. Reprint of figure 5(e) with the permission of Cambridge University Press from Fermigier et al. (Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992). (d) Surface plot at $t_{r}=9.592$ with $H_{Min}=0.43\times 10^{-4}$ and $H_{Max}=4.4867$, coloured by normalized $\varTheta _{I}^\ast$. (e) Rupture patterns of the representative profiles, those highlighted in (d) are $x=5.332$ and $x=11.096$. The red dots denote the global minimum and maximum thicknesses at the positions of $(11.096, 7.275)$ and $(0, 0/6{\rm \pi} )$, respectively, and two local maxima, $H_{\textit{max}}=4.0388/4.0390$ at $(5.332,5.433/13.416)$. (f) The corresponding condensing fluxes at $0.98t_{r}$.

The simulation with the axisymmetric IC predicts a regular interface with enhanced condensing fluxes on the troughs (figure 7d–f), where the vapour thrust gives rise to local rupture due to the kinetic energy it produces. The rupture time $t_{r}=9.592$ is slightly larger than that of the Case II evolved from a random IC, while the heights of the primary droplets decrease at rupture, see figures 6(d) and 7(e). The distance between two adjacent peaks on the ring is $\lambda _{p}\approx 8.08$ at $t_{r}$, being close to the average distance of neighbouring peaks in the quasi-hexagon of the Case II from a random IC ($\lambda _{p}\approx 9.40$ in figure 6b). A comparison between the two Case II solutions demonstrates that the development of this nonlinear instability does depend on the choice of ICs. Here, the fourfold rotational symmetry of the interface is inherited from the axial symmetry of the initial Gaussian and the square symmetry of the periodic domain. However, a random disturbance breaks the symmetry of the solution since it does not possess any kind of symmetry.

Next, our Case II simulation with random perturbations yields a quasi-hexagonal array of drops (figure 6b), which qualitatively agrees with the experimental observations of the gravitational instability in a silicone-oil layer at a later stage (see Fermigier et al.'s figure 6 and the symmetric structures designated by ‘A6’ and ‘H’ in their figure 3 for similarity). However, there is a pronounced difference between the pathway of evolution in our simulation (cf. figure 5a–d, qualitatively similar in Case II but not shown) and that in the experiment, but this should not be of surprise because the Case II result has been obtained from a random perturbation. Moreover, it should be noted that, in addition to condensation, in their experimental and theoretical studies the Marangoni and buoyancy effects were also eliminated with an isothermal thin film (see their lubrication model (3.4)). Therefore, according to the experimental observations and the Case II simulation using a random IC, the tendency to a hexagonal symmetry from the distinct ICs, in turn, suggests that the preferred pattern to be selected should be the hexagon one, independent of condensation, Marangoni and buoyancy effects.

Another noticeable feature is that, with the common effects of negative gravity, capillarity and viscosity, there was no rupture observed within the time scale of RTI ($\tau _{M}\approx 350$ s from LSA, subscript ‘M’ representing the fastest growing mode) in their experiment and simulation (Fermigier et al. Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992). However, in Cases I and II with random and/or axisymmetric IC, finite-time rupture occurs at $t_{r}=9.246$, $9.374$ and $9.592$, respectively. All of the rupture times are conspicuously longer than the dimensionless time constant $t_{M}=\tau _{M}h_0^3 g^2 \rho ^2/(3\sigma _0\mu )=4$ and that of the instability in their experiment, e.g. $t_{exp}\approx 2.3\text {--}3.4$ for axisymmetric perturbations. The slower development of the instabilities in our simulations can be attributed to the stabilizing effect of thermocapillarity and the mass addition at troughs (Som et al. Reference Som, Kimball, Hermanson and Allen2007) by condensation. The finite-time rupture in our non-isothermal condensing layers indicates that the combined stabilizations of capillarity, thermocapillarity and mass gain together with viscous effects lose the competition with the localized intensifying vapour recoil during the RTI process. The physical explanation is that the local temperature gradient and thus momentum transfer associated with vapour thrust at an interfacial trough become greater as a trough gets closer to the cooled plate, where the heat and mass fluxes are locally highest (cf. condensing fluxes near $t_{r}$ in figures 6d and 7f).

Meanwhile, gravitational potential energy drives the condensate from the thin regions into the bulges where the pressure is reduced and the resulting pressure gradient further collects liquids into the lower lying droplets. It implies that the stabilization of thermocapillarity (to a weak or significant degree) and capillarity could not afford the continuous mass gain and the moderate-to-strong vapour recoil. Moreover, it is worth noticing that both the ratios of vertical to horizontal length scales in the two Case II results, $H_{Max}/\lambda _{p}\approx 0.7$ (figure 6) and $H_{max}/\lambda _{p}\approx 0.5$ (figure 7), are much larger than a corresponding ratio in the final stage of the experiment, $(e^\ast /h_0)/(\tilde {\lambda }_{p}/\hat {l}_{c})=0.155$ (Fermigier et al. Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992), where the typical thickness $h_0=0.2$ mm, the capillary length $\hat {l}_{c}=1.49$ mm, the critical thickness for dripping instability $e^\ast =0.31$ mm and $\tilde {\lambda }_{p}=14.9$ mm (for hexagonal patterns). Also, $\lambda _{p}\approx 9.40$ in the Case II quasi-hexagon is in good agreement with the experiment, $\lambda _{p}=10$. Therefore, if condensation occurs in a Rayleigh–Taylor unstable layer, it could be deduced that the Marangoni and condensation effects enhance the instability in amplitude, while its growth rate is greatly lowered by the Marangoni effect and mass gain in thin regions, which accords with LSA of the $1.5$-sided model (figure 4).

By comparing the Case II result in figure 6(b) with the observations of Som et al. (Reference Som, Kimball, Hermanson and Allen2007) (figure 6c), it is expected that for the particular Rayleigh–Taylor unstable film condensation (during the rise of pressure in an enclosed chamber), the one-sided model is applicable to reproduce the instability of the vapour–liquid interface with properly chosen control parameters. In their experiment, the gas is pure saturated vapour of $n$-pentane and condensation occurs near equilibrium condition. Particularly, they did not attribute the stability characteristics to thermocapillary flows and thus tacitly assumed a weak Marangoni effect. These are consistent with the parameter values of $K=0.01$ and $\mathscr {M}^\ast =-0.1$ in our Case II and the usual simplification (see § 1.1). Additionally, we shall present an illustrative computation for possible applications of this model in § 6.6.

6.3. Effects of VBL and initial concentration: vertical convection and diffusion

An understanding of the new mechanisms involved in the $1.5$-sided model can serve as a basis for discussing its results later and for comparing with the one-sided model. In this subsection, the effect of VBL thickness, quantified by $\mathscr {L}$, is probed. It is also instructive to look at the influence of the initial vapour concentration at interface on the vapour convection and diffusion in the gas phase. We briefly discuss the two issues using an unsteady basic state with an evaporating flat interface, in which the vapour transport in the vertical direction is separated from that along the interface and the resistance to phase change will be neglected ($K_{AB}=0$) for the sake of clarity. In terms of the rescaled variables in (5.1a,b) and (5.2a–f), the canonical form of (3.21a,b) reads

(6.1a,b)\begin{equation} \mathscr{N}\text{d}\bar{\tilde{x}}_{A,I}/\text{d}t= (1-\bar{\tilde{x}}_{A,I}) [\bar{J}_0^2-(\mathscr{L}^{-1}\ln\bar{\eta})^2 ], \quad \text{d} \bar{H}/\text{d}t=\mathscr{E} \bar{J}_0, \end{equation}

where the dynamics of $\bar {\tilde {x}}_{A,I}$ is governed by the vertical convection (via evaporation flow) and diffusion of vapour through VBL, described by the two right-hand side terms of (6.1a).

Figure 8 compares the temporal evolutions of $\bar {H}$, $\bar {\tilde {x}}_{A,I}$ and $\bar {J}_0$, obtained from a long time integration of (6.1a,b) with the imposition of either an initially unsaturated or saturated interfacial concentration (cf. (3.20)), shown in the upper and lower panels, respectively, for different $\mathscr {L}$. With an unsaturated initial concentration $\bar {\tilde {x}}_{A,I0}$, evaporation occurs immediately (see figure 8c), $\bar {\tilde {x}}_{A,I}$ first increases to a peak and then decreases to ambient value ($\tilde {x}_{{A}\infty }=0.2$) as a result of the slowly vertical diffusion of vapour through VBL, while $\bar {J}_0$ approaches zero monotonically. Moreover, $\bar {\tilde {x}}_{A,I}$ reaches a higher peak at a later time as $\mathscr {L}$ increases. In contrast, as can be seen in the lower panels, with a saturated $\bar {\tilde {x}}_{A,I0}$ the basic state sustains the initial equilibrium for $t={O}(1)$ by weak evaporation and then $\bar {\tilde {x}}_{A,I}$ monotonically decays to $\tilde {x}_{{A}\infty }$ without ‘overshooting’ under the effect of vapour diffusion. Meanwhile, significant evaporation starts only from $t={O}(1)$; $\lvert \bar {J}_0\rvert$ reaches its maximum during the transition of $\bar {\tilde {x}}_{A,I}$ to compensate for the vapour diffusion then vanishes as $\bar {\tilde {x}}_{A,I}\rightarrow \tilde {x}_{{A}\infty }$. Note also that the maxima of $\lvert \bar {J}_0\rvert$ are at least one order of magnitude less than the initial value (maximum) of $\lvert \bar {J}_0\rvert$ with the unsaturated $\bar {\tilde {x}}_{A,I0}$.

Figure 8. Temporal evolutions of the basic solutions for $\mathscr {L}=5$ (dotted), $10$ (dashed) and $20$ (solid), obtained by solving (6.1a,b) with ICs $\bar {H}(0)=\bar {H}_{st}=1$ and $\bar {\tilde {x}}_{A,I0}=0.1$ (a,b,c) or $\bar {\tilde {x}}_{A,I0}=\bar {\tilde {x}}_{A,I,st}=0.34$ (d,e,f) for $K_{AB}=0$, $\mathscr {E}=0.05$, $Bi=\varGamma =\mathscr {N}=1$, $\varTheta _\infty =0.5$, $\tilde {x}_{{A}\infty }=0.2$, $\tilde {x}_{A,w}=0.09$, $t_{Max}=10^6$. (a,d) Thickness $\bar {H}$; (b,e) interfacial concentration $\bar {\tilde {x}}_{A,I}$; (c,f) mass flux $\bar {J}_0$. In the inset of panel (b), the three curves are indistinguishable within line width.

The reasons for the ‘overshoot’ of $\bar {\tilde {x}}_{A,I}$ in the case of an unsaturated initial concentration are twofold: (i) in the beginning evaporative flux is strongest which is in favour of vapour build-up on the interface, and (ii) the speed of vertical diffusion of vapour is smaller than that of vertical convection by evaporation flow near the interface (Kanatani Reference Kanatani2013), which is appreciable in this result. It is also obvious that the transient processes of $\bar {H}$ and $\bar {\tilde {x}}_{A,I}$ are very slow with a transition time of $t_{s}={O}(10^4\text {--}10^5)$ for both values of $\bar {\tilde {x}}_{A,I0}$. It is found that the increase in $\mathscr {L}$ will reduce the instantaneous mass flux, slow down the vertical diffusion and consequently prolong the relaxation time to equilibrium. With the distinct $\bar {\tilde {x}}_{A,I0}$, the basic states reach the same equilibrium state for different values of $\mathscr {L}$. More precisely, $\bar {H}$ approaches the asymptotic solution (3.20) and $\bar {\tilde {x}}_{A,I}\rightarrow \tilde {x}_{{A}\infty }$ with ${\rm \Delta} \tilde {X}_{st}=\tilde {x}_{{A}\infty }-\tilde {x}_{A,w}$. For this set of parameters, $\bar {H}\rightarrow 11/39$ as $t\rightarrow \infty$. Meanwhile, both the diffusive and convective fluxes of the vapour vanish. This means that the composition of the vapour phase becomes uniform and the VBL itself disappears.

Finally, the initial ‘plateau’ in $\bar {\tilde {x}}_{A,I}$ suggests that for an initial concentration not too far from the saturated value the time derivative of $\bar {\tilde {x}}_{A,I}$ in (6.1a) or (3.21a) is indeed negligible for the prediction of initial growth rates of infinitesimal perturbations. In the slow-variation intervals of $\bar {H}$ and $\bar {\tilde {x}}_{A,I}$, the basic state can be regarded as pseudo-steady, and thus the linear analysis in § 4 with the frozen-time approach is warranted.

6.4. The $1.5$-sided model: $(2+1)$-D regular non-ruptured pattern

The investigations of Sultan et al. (Reference Sultan, Boudaoud and Amar2005) and Kanatani (Reference Kanatani2013) motivated the study of a more complicated situation when both non-equilibrium effect (resistance to advection of vapour molecules across the interface) and convection/diffusion of vapour are important in the nonlinear regime. We thus proceed to the fully nonlinear time-dependent simulations of Rayleigh–Taylor unstable condensing layers with the $1.5$-sided model to verify the LSA for various physical mechanisms based on the dispersion relations (4.2) and (F 1) and to provide quantitative results for finite-amplitude evolutions of the instability in the presence of vapour convection and diffusion. The results will be compared with those of the one-sided model. These are the important subjects in the current study. We consider the VBL thickness to be sufficiently large that the vertical diffusion of vapour and heat (recall (2.19), (3.18a) and § 6.3) are slow and consequently the mass transfer and ambient heating are expected to be weakened. The relevant parameters $K_{AB}=0.1$, $\mathscr {E}=0.05$, $\mathscr {D}=-0.1$, $Bi=0.1$ and $\mathscr {L}=10$, are appropriate for this situation. The other parameters (see the caption of figure 9) should be valid for the ethanol–nitrogen system (tables 1 and 2) under proper conditions, allowing direct comparison with experiments. In the $(2+1)$-D simulation, a uniform mesh of $131\times 131$ is employed for the $1.5$-sided model, which is close to exhausting the capabilities of the available computer.

Figure 9. Rayleigh–Taylor unstable condensing layer with the effects of convection and diffusion of vapour, which approaches a pseudo-steady state at $t_{s}\approx 26.5$, obtained by solving (5.3) with IC (5.8a) and $\tilde {x}_{A,I0}=0.1$ for $K_{AB}=0.1$, $\mathscr {E}=0.05$, $\mathscr {G}=0.2$, $\mathscr {M}=-1$, $\mathscr {D}=-0.1$, $Bi=0.1$, $\varGamma =3$, $\mathscr {N}=3$, $Bo=0.08$, $\mathscr {L}=10$, $\varTheta _\infty =0.5$, $\tilde {x}_{{A}\infty }=0.2$ and $\tilde {x}_{A,w}=0.09$ on a $131\times 131$ mesh. The side length of the periodic domain is $l=10{\rm \pi}$. (a) Surface plot showing the regular pattern of pendent droplets. (b) Interfacial height $H$ contours. The red dots indicate several local minimal/maximal thicknesses: $H_{min}$ at ${p}_1 (31.42, 5.86)$, ${p}_2 (25.43, 13.20)$, ${p}_3 (14.13, 26.18)$ and ${p}_4 (11.36, 11.58)$, and $H_{max}$ (not show labels) at ${P}_1 (1.41, 22.40)$, ${P}_2 (8.21, 14.23)$, ${P}_3 (17.58, 7.26)$ and ${P}_4 (23.71, 29.57)$, where ${p}_1$ (${P}_1$) is also the global minimum (maximum). (c) Evolution of $H_{min}$. (d) Interfacial molar fraction $\tilde {x}_{A,I}$. To expose features of the distribution, a small region containing the maximum $0.128$ at $(0.40, 6.13)$ beyond the plot range has been clipped. (e) Interfacial temperature $\varTheta _{I}$ contours. (f) Interfacial horizontal velocity $\boldsymbol {U}_{I}$, also superimposed on the contours in (b,d,e).

With the random interface disturbance (5.8a), nonlinearities rapidly take effect, as illustrated by the numerical results in figures 9 and 11. It is found that, in contrast to the Case I result of the one-sided model, there is no elongated droplet after a longer evolution and the pendent droplets are organized into a regular non-ruptured stable pattern in the form of LW quasi-hexagons (figure 9a–c). The surface pattern has a well-defined lateral length scale and the typical wavelength of the instability is comparable to the amplitudes of the droplets. These are similar to the experimental observations of GG for R-$113$ filmwise condensation (see their figure 4b,c). Figure 9(c) illustrates the transitions of $4$ local minimal thicknesses ($H_{min}$), from which the liquid layer appears to asymptotically approach a steady state. Actually, the system could achieve a dynamic equilibrium after a transition time, here $t_{s}\approx 26.5$, since then evaporation and condensation occur at nearly the same rate on the whole (further explained later), analogous to the chemical kinetics of reversible reaction. As shown in figure 9(d), in the intervening thin regions the interfacial concentration, $\tilde {x}_{A,I}$, remains closer to the equilibrium value $\tilde {x}_{A,w}=0.09$ of the wall temperature with $\tilde {x}_{A,I}-\tilde {x}_{A,w}={O}(0.01)$. In light of the quasi-equilibrium dependence of interfacial temperature on concentration (3.19) and the value of $\varGamma =3$, it is reasonable to infer that on those regions the gas immediately above the free surface is nearly saturated with vapour since a typical temperature of $\varTheta _{I}\approx 0.03$ indeed prevails in the intervening regions (figure 9e). Thus the asymptotic state is close to the diffusion-limited phase change with relatively weak mass fluxes (cf. figures 6d and 11b).

The interfacial liquid flows radially from the thin regions to the crests of the pendent droplets (figure 9b,f) due mainly to the local evaporation/condensation and the RTI (see figures 11b and 17c later) and secondarily to the local buoyancy in the droplets (Savino et al. Reference Savino, Paterna and Favaloro2002). The magnitudes of the horizontal velocity on the interface, $\boldsymbol {U}_{I}$, are relatively small on the whole, except for the very limited regions within two drops ($\|\boldsymbol {U}_{I}(t_{s})\|_{Max}\approx 0.8$). Note that the legend range in figure 9(f) is for $[0,t_{s}]$ to recognize the substantial decrease in $\|\boldsymbol {U}_{I}\|$ as it approaches a pseudo-steady state (see also figure 10). The interfacial velocities are relatively larger on the lateral surface of the droplets, consistent with the typical relation $\|\boldsymbol {U}_{I}\|\propto \|\boldsymbol {\nabla }_1 H \|$ of the lubrication approximation. Also, $\boldsymbol {U}_{I}$ essentially vanishes on the crests and the intervening films as $\boldsymbol {\nabla }_1 H\rightarrow \textbf {{0}}$ there, from where the stagnation points emerge, corresponding to the local maximal or minimal thicknesses (red dots in figure 9b). This interfacial convection allows the formation of pendent droplets and accumulates $\tilde {x}_{A,I}$ at the crests, toward where the interfacial flow entrains vapour, and thus $\varTheta _{I}$ increases around the crests (figure 9d,e). It should be mentioned that the radially outward arrow pattern in figure 3(b) of Burgess et al. (Reference Burgess, Juel, McCormick, Swift and Swinney2001) never represents the motion direction of fluid along the gas–liquid interface but manifests the variations of locally interfacial slopes (see also Fermigier et al. Reference Fermigier, Limat, Wesfreid, Boudinet and Quilliet1992) since it is an accumulated displacement field of a reference grid (imaged through an oil layer with a ruled screen) and a region with diverging arrows simply indicates a growing droplet.

Figure 10. (a) Evolution of horizontal interfacial velocity $\boldsymbol {U}_{I}(x,22,t)$ in the Rayleigh–Taylor unstable condensing layer. At $t=26.5$, $\lvert J \rvert$ peaks in figure 11(b) are marked with red/blue curly braces for evaporation/condensation (sink/source-like) in the dashed box. (b) Temporal evolution of $\| \boldsymbol {U}_{I} \|$ between consecutive moments at each $x$-position (red dots in (a) with ${\rm \Delta} x=1$). Accelerating and decelerating locations are marked as ‘$+$’ and ‘$-$’, respectively. Non-accelerated locations are grey. Separations between time levels are not to scale.

Figure 11. Evolution of the representative profiles of $y=22$ in the Rayleigh–Taylor unstable condensing layer: (a–d) $H$, $J$, $\varTheta _{I}$ and $\tilde {x}_{A,I}$. The $y$-cross-section is near to that involving $H_{Max}$ in figure 9(b). The consecutive curves are taken at $t=0$, $10$, $16$, $20$, $23.5$, $24$, $24.5$, $25$ and $t_{s}\approx 26.5$.

Figure 10 presents the evolution of the horizontal velocity on the interface $\boldsymbol {U}_{I}$ at $y=22$. At each instant the velocity field takes a different normalization in figure 10(a) so that the maximal norms (in orange) have the same length. Thus, different snapshots should not be compared directly. To illustrate the variation of the interfacial velocity magnitude, figure 10(b) compares $\| \boldsymbol {U}_{I}(x,22,t)\|$ between the consecutive moments for each $x$-position (see red dots in figure 10a). If the absolute value of $\| \boldsymbol {U}_{I}[t^{(n+1)}]\|-\| \boldsymbol {U}_{I}[t^{(n)}] \|$ is less than a small criterion of $0.002$, the velocity magnitude is considered unchanged and the $x$-position is marked as ‘$=$’ at $t^{(n+1)}$ level; if $\| \boldsymbol {U}_{I}[t^{(n+1)}]\|>\| \boldsymbol {U}_{I}[t^{(n)}] \|$, the $x$-position is marked as ‘$+$’ for acceleration; if $\| \boldsymbol {U}_{I}[t^{(n+1)}]\|<\| \boldsymbol {U}_{I}[t^{(n)}] \|$, marked as ‘$-$’ for deceleration. The most evident fact seen in figure 10 is that, following an initial acceleration due to the RTI, the interfacial velocity decreases significantly in most positions as it approaches a pseudo-steady state. This phenomenon can be ascribed to an induced Marangoni stabilization by interfacial convection in the presence of an inert component, besides the well-known stabilizing effects of viscous dissipation and surface tension.

The spatio-temporal evolutions of the representative profiles are presented for $y=22$ in figure 11, supplementing the results in figures 9 and 10. From figure 11(a), a coalescence of two bulges as a pendent drop can be observed at $t=24$. The mass flux shown in figure 11(b) suggests that the entire surface undergoes evaporation ($J<0$) in the early stage of the instability. It becomes understandable when recalling the initial mass flux $\bar {J}_0$ in figure 8(c). The essential reason of this phenomenon is that the initial concentration, $\tilde {x}_{A,I0}=0.1$, is less than the value, $\bar {\tilde {x}}_{A,I,st}\approx 0.1052$, for stationary equilibrium of the basic state ($\bar {H}_{st}=1$), as given by (3.20). Naturally, $\tilde {x}_{A,I}$ tends to increase initially via evaporation (figure 11d before $t\approx 16$) with the inhomogeneity owing its origin to the interfacial convection (figure 10a). The mass-flux profile at $t_{s}\approx 26.5$ is also consistent with the local interfacial velocity (see the dashed box in figure 10a). The evaporation ($J<0$) and condensation ($J>0$), respectively, at the crests and troughs contribute to suppressing the increase in interfacial deformation and to avoiding rupture.

The mechanism for the avoidance of interface rupture is as follows. On the one hand, the resulting concentration gradient, $\boldsymbol {\nabla }_1 \tilde {x}_{A,I}$, gives rise to the diffusion of vapour from warmer, higher-concentration regions towards cooler, lower-concentration regions along the interface over short scales compared to the LW interface instability (cf. § 4). Thus $\tilde {x}_{A,I}$ tends to diffuse from the relatively thicker regions to the thinner ones with SW variations, as will $p_{A,I}$ (see (B 2)) that allows differential mass fluxes. This is most clearly demonstrated in figure 11(c,d) after $t=20$. The SW variations in $\tilde {x}_{A,I}$ also justify the linear analysis of Kanatani (Reference Kanatani2013). As a result, the vapour diffusion along interface plays a stabilizing role through the differential mass fluxes. As seen in figure 11(a,b,d), two local maxima of $\tilde {x}_{A,I}$ arise in two local $H_{min}$ after $t=20$ due to the continuous diffusion, corresponding to the (annular) valleys (at $x\approx 11$ and $19$) around the central droplet, which results in gradual increase of the local condensation fluxes. In addition, the convection velocity associated with phase change is typically larger than those of vapour diffusion (cf. Kanatani (Reference Kanatani2013) and § 6.3) such that evaporation on the crests near $x\approx 1.5$ and $15$ suffices to compensate the ‘vapour dimples’ (see figure 11d), caused by the interfacial diffusion. On the other hand, the vapour convection along the interface, entrained by interfacial fluid under the initial instability, will enhance $\boldsymbol {\nabla }_1 \tilde {x}_{A,I}$ (figure 11d) until the interfacial convection is alleviated by the induced Marangoni effect that is reinforced just by the resulting concentration gradient (cf. § 4). Such a phenomenon manifests itself in figures 9(f) and 10(b). Thus the vapour convection offers a stabilization by enhancing the Marangoni effect, measured by the interface Schmidt number, $\mathscr {N}$.

Considering that the interface is initially unsaturated with vapour, there are two main scenarios to approach locally thermodynamic equilibrium with time. In respect to chemical equilibrium, during the initial evaporating stage ($t\lesssim 16$ in figure 11), in most regions $\tilde {x}_{A,I}$ increases as vapour is transported towards the interface by evaporating flow (convection) and by diffusion through the VBL (cf. figure 8b). Meanwhile, vapour will diffuse along the interface that $\tilde {x}_{A,I}$ naturally tends to local equilibrium, as described above. For thermal equilibrium, the temperature dependency (B 1) of coexistence pressure, $p_{s}(\theta _{I})$, indicates that any transient thermal variation in the interfacial liquid will alter the local $p_{s}$. As the local $\theta _{I}$ increases, for instance, due to the growth of a droplet under RTI into the hotter region below, the local $p_{s}$ will rise accordingly. Thus, if originally $p_{A,I}<p_{s}$ at a droplet apex, more evaporation can occur that evaporative cooling counteracts the temperature rise there (see ‘thermal dimples’ on the two apexes in figure 11c). A reverse situation occurs at a trough, i.e. enhanced condensation suppressing $\theta _{I}$ decrease. At the troughs around the central droplet, the mass and heat fluxes become locally highest, where the increase in $\varTheta _{I}$ can be identified in the late stage (figure 11c). Eventually, the RTI would be suppressed by the stabilization of differential mass fluxes as well as the mechanical equilibria of interfacial to viscous forces in thin regions and interfacial to gravitational forces in thick regions (see also § 7).

On suppression of the RTI of a volatile layer suspended from a cooled surface and heated from below, there are similarities between the $1.5$-sided result and those obtained from a one-sided and a simplified Cahn–Hilliard model by Bestehorn & Merkt (Reference Bestehorn and Merkt2006). Both the self-organized pattern formations result in regular surfaces without rupture, while coarsening and elongating do not occur. The stabilizing mechanism in Bestehorn & Merkt (Reference Bestehorn and Merkt2006) is a balance of local evaporation and condensation determined only by heat conduction in the liquid when an initial interface is in equilibrium with its vapour. However, it is intrinsically due to the convection and diffusion of vapour in our model. Without thermal Marangoni effect in their pure-component case, its stabilization can never be enhanced by the interfacial convection of vapour (see also figures 4a and 18a).

Furthermore, by making comparisons of figures 9(a–c) and 11(a) with figures 5(e,f) and 6(b,d), we found that, with a moderate-to-strong vapour recoil, the one-sided model predicts either an elongated solitary drop or a quasi-hexagonal pattern, but finite-time rupture appears to be inevitable under RTI. However, for the $1.5$-sided model the interfacial instability takes a longer duration to develop and can converge asymptotically to a non-ruptured quasi-hexagonal pattern, where the pendent droplets advance less into the hot vapour under the equivalent physical and initial conditions. The implication of this result is that when a pure liquid layer with phase change is subjected to RTI, even the solutal Marangoni effect being ignored, the effect of an inert component in the gas phase cannot be neglected owing to the local concentration variation of the volatile component, which has been taken into account in the $1.5$-sided model with the convection and diffusion of vapour near the interface.

Stabilization of the RTI has also been demonstrated in Kanatani & Oron (Reference Kanatani and Oron2011). It is appropriate at this point to briefly discuss the similarities and differences between the two models. A similar thermodynamic relation at the interface, derived previously by Kanatani (Reference Kanatani2010), was used in their model, where both the fluctuations of interfacial temperature and vapour pressure were connected to the mass flux (cf. (2.15)). On the other hand, in our $1.5$-sided model, the effect of the viscous stress of vapour is neglected (see (2.11)). However, they considered the interfacial shear stresses imparted by lateral vapour flow, arising from the confined geometry. Thus, they established a two-sided model, which incorporated viscous coupling between the liquid and vapour, but in the absence of an inert gas. Furthermore, in our model, the essential factor that influences the vapour convection near the interface is the phase-change-induced gas bulk flow, similar to that in Pillai & Narayanan (Reference Pillai and Narayanan2018), instead of the lateral variations of vapour pressure near the interface in their model. In the confined, thin, bilayer system, the vapour pressure variation served as a direct stabilizing mechanism for RTI: the vapour pressure becomes higher/lower where liquid evaporates/vapour condenses; the higher pressure pushes the interface where approaching the heated vapour-side wall, while the lower one pulls it where approaching the cooled liquid-side wall. The other major distinction is that they neglected the mass loss/gain term for the liquid layer so that the liquid mass was conserved (contrasting their (1a) with (2.44)). A corollary of the assumption is that the total interfacial flux over a period must always vanish (see (17) there). As indicated in their conclusion, the solutions for Rayleigh–Taylor unstable cases could be modified as the steady states were approached by including the mass-loss/gain effect.

6.5. The $1.5$-sided model: $(1+1)$-D dynamics with equilibrium initial concentration

In this section we present the numerical results for a $(1+1)$-D case by integrating the $1.5$-sided model (5.3) on $[0,l)$ with a random perturbation superimposed on the basic state $\bar {H}_{st}=1$ and an initial equilibrium concentration stipulated by (3.20), i.e. $\tilde {x}_{A,I0}=\bar {\tilde {x}}_{A,I,st}\approx 0.1052$. To quantify the intensity of phase change, it is useful to define a global mass flux, $J_{g}=\mathscr {E}\int _0^l J(x,t)\,\text {d}x$, which has been scaled by $\mathscr {E}$ and thus is independent of the temperature scale (see the last expression in (2.21)). The results are presented in figures 12 and 13 for $\mathscr {G}=0.2$ (those with $\mathscr {G}=0$ are similar), in which the values of the other parameters are the same as those of § 6.4. In order to shed light on the influence of buoyancy on internal convection, we switch it by setting $\mathscr {G}=0$ or $0.2$ in figure 14.

Figure 12. Evolution of the Rayleigh–Taylor unstable condensing layer, obtained by solving the $(1+1)$-D version of (5.3) with $l=10\sqrt {2}{\rm \pi}$, IC (5.8a) and $\tilde {x}_{A,I0}\approx 0.1052$. The other parameters are the same as those in figure 9. (a–d) $H$, $J$, $\varTheta _{I}$ and $\tilde {x}_{A,I}$. The consecutive curves are taken at $t=0$, $5$, $10$, $15$, $20$, $25$ and $t_{max}=28.2$. The insets show the early stages with ${\rm \Delta} t=5$.

Figure 13. Temporal evolutions of (a) the global mass flux $J_{g}$ and (b) the global maximal $\tilde {x}_{A,I}$, extracted from the $(1+1)$-D results in figure 12(b,d). The inset of (a) shows the transition from weak evaporation to the condensation-dominated regime at a critical time $t_{c}$.

Figure 14. Flow fields of the Rayleigh–Taylor unstable condensing layer at two representative moments, obtained by solving the $(1+1)$-D version of (5.3) with $l=10\sqrt {2}{\rm \pi}$, IC (5.8a) and $\tilde {x}_{A,I0}\approx 0.1052$. The other parameters are the same as those in figure 9 but $\mathscr {G}=0$ and $0.2$ in the upper and lower panels, respectively. (a,c) $t=15$ before which $J_{g}\approx 0$. Temperature contours (dashed) are separated by ${\rm \Delta} \varTheta =4.1\times 10^{-3}$, labelled with dashed frame. (b,d) $t=25$ in condensation-dominated regime, where ${\rm \Delta} \varTheta =4.8\times 10^{-3}$ and interfacial stagnation regions are indicated by curly braces. The streamlines (solid) are separated by ${\rm \Delta} {\varPsi }=3.2\times 10^{-4}$ and $4\times 10^{-3}$ in (a) and (b), and by ${\rm \Delta} {\varPsi }=3.4\times 10^{-4}$ and $5\times 10^{-3}$ in (c) and (d), respectively. Labels for streamlines at $t=15$ and $t=25$ have been multiplied by $10^4$ and $10^3$, respectively. In (b) the red dots highlight intersections of the connecting streamlines (${\varPsi }=8\times 10^{-3}$) and the free surface, although they appear to be tangent owing to the influences of RTI on the interfacial slope and velocity under weak mass fluxes.

Here, we do not follow the evolution of the film for a longer time but terminate the computation when the maximal concentration ($\tilde {x}_{A,I,Max}$) approaches the bulk value of $\tilde {x}_{{A}\infty }=0.2$, while $\partial _x\tilde {x}_{A,I}\ll 1$ still holds. As can be seen in figure 12(d), $\tilde {x}_{A,I,Max}\approx 0.1911$ at $x\approx 2.57$ and $max [\partial _x\tilde {x}_{A,I}]\approx 0.03$ when $t_{max}=28.2$, after which the $1.5$-sided model could be inappropriate (cf. § 5.2) with the increase in $\tilde {x}_{A,I,Max}$ (figure 13b). Therefore, only the results involving a transition from the early stage of weak evaporation to a condensation-dominated regime have been presented to focus on the effects of the interfacial convection and diffusion of vapour on the instability.

The transient interface in figure 12(a) shows that the film is Rayleigh–Taylor unstable. For $t\lesssim 15$, the troughs and crests basically develop at the same exponential rate with respect to the initially extended perturbations before nonlinear effects become important, after which the profile takes a regular LW form of $4$ pendent droplets. This can be attributed to the fact that with the initially equilibrium $\tilde {x}_{A,I0}$, evaporating and condensing fluxes nearly balance out up to $t\approx 15$ (see the inset of figures 12b and 13a), after which condensation dominates and the drops increase in amplitude under a balance of gravitational and surface forces (including the Marangoni and capillary stresses). As shown in figure 12(b,c), after $t=20$ the bulk of condensation and thus heat transfer mainly occur in the troughs, where $\varTheta _{I}$ starts rising, as also found in figure 11(c).

As can be seen in the inset of figure 12(d), even within the global weak-evaporation stage of $t\leqslant 15$ (precisely $0<t<t_{c}$, see figure 13a) the overall vapour concentration still decreases due to the vertical diffusion from the interface, according with the basic solution with a saturated initial concentration (see figure 8e,f); $\tilde {x}_{A,I}$ attains its local maxima in the crests of the interface at $t=15$ because of the vapour replenished by continuously local evaporation. Again, vapour diffusion along the interface is clearly reflected in the insets of figure 12(c,d). It is also important to recognize from the inset of figure 12(d) that the lateral diffusion of vapour causes concentration accumulation around the troughs (such as $x=21.25$ at $t=15$) with SW variations. The SW variation in $\tilde {x}_{A,I}$ is responsible for the enhanced condensation at the depressions, which plays a stabilizing role. In addition, the vapour convection along the interface, entrained by interfacial fluid under the initial Marangoni effect (figure 14a), also contributes to the $\tilde {x}_{A,I}$ variations. Although the local condensation contributes to decrease $\tilde {x}_{A,I}$ at a trough, it appears to be insufficient to consume the vapour diffused and convected from its two adjacent crests during the initially global evaporation stage. The interfacial temperature (figure 12c) decreases in the most regions in the weak evaporation stage due to the effects of evaporative and substrate cooling. It is also interesting to observe the transitions of $\varTheta _{I}$ and $\tilde {x}_{A,I}$ from $t=15$ to $t=25$: there is a delay in the increase of $\varTheta _{I}$ compared to that of $\tilde {x}_{A,I}$ at the depressions due to thermal inertia, while weak evaporative cooling at the elevations cannot override the heating effect in the condensation-dominated regime. The initial behaviours are in contrast to those of the case with an initially unsaturated concentration in § 6.4, where evaporation prevails over the entire profile of $y=22$ until $t\approx 16$, and both $\varTheta _{I}$ and $\tilde {x}_{A,I}$ increase in the most regions (figure 11). Both the initial dynamics of $\tilde {x}_{A,I}$ and $J$ (or $J_{g}$) in the two previously mentioned results with the different $\tilde {x}_{A,I0}$ are consistent with the pertinent basic solutions during the initial transition (see § 6.3), which owe their origins to the competition between vertical convection and diffusion through the VBL.

Figure 14 depicts the streamlines/isotherms for $\mathscr {G}=0$ and $0.2$ in the upper and lower panels, respectively, at two representative moments. With $\mathscr {G}=0$, at $t=15$ the convective cells are encompassed by the substrate $z=0$, the interface $z=H$, and the dividing streamlines ${\varPsi }=0$, which do not cross the free surface. Each bulge comprises $2$ symmetric cells of the same flow rate. As the troughs deepen, the increasing capillary pressure gradients along with the favourable Marangoni stresses drive liquid against the viscous resistance under the valley, where the capillary ridges emerge between the droplets at $t=25$. The convection cells become somewhat asymmetric in each drop, which are encompassed by $z=0$ and ${\varPsi }=0$ on the right and by $z=H$ and ‘connecting’ streamlines (${\varPsi }=8\times 10^{-3}$) on the left. The connecting streamlines do cross the free surface (see red dots on the interface in figure 14b). At $t=25$, the flow rates of the two recirculating cells are equal in the first and third drops, however, in the two slightly smaller drops the flow rates are higher on the left. In addition, the temperature field tends to be more distorted, especially near the lateral sides of the drops. The interfacial stagnation regions emerge from one side of the drops, reflected by the reversal of the connecting streamlines, which are close to the local minima of $\varTheta _{I}$ (see figure 12c).

Ordinarily, a film lying on a horizontal wall is very thin, so that it is a good approximation to treat the liquid density as constant. For typical liquids $\beta ={O}(10^{-3}\text {--}10^{-2})\ \textrm {K}^{-1}$; if $\theta -\theta _{w}\lesssim 10$ K, then $\vert {\rm \Delta} \rho /\rho \vert =\beta (\theta -\theta _{w})\ll 1$ with ${\rm \Delta} \rho$ for density variation within the Boussinesq approximation. Nevertheless, when pendent drops emerge from the surface of a Rayleigh–Taylor unstable liquid layer, there would be significant temperature differences across the locally thick regions in the presence of phase change, then buoyancy in the drops could be comparable to viscous stresses there, and so could not be negligible (Savino et al. Reference Savino, Paterna and Favaloro2002). Now, it would be of interest to look at the effect of buoyancy on the convection in the pendent drops. The lower panels of figure 14 for $\mathscr {G}=0.2$ are similar to the upper ones in terms of interfacial profile and temperature field. However, at $t=15$ the cell flow rate on the left/right of each drop is larger/smaller than its counterpart with $\mathscr {G}=0$ due to the presence of convection between the droplets. At $t=25$ the two recirculating cells inside each drop become highly asymmetric due to the distinct pressures on either side, in contrast to the case without buoyancy. The flow rates of the cells on both sides become smaller in comparison with those of $\mathscr {G}=0$, which is the source where the liquid for the stronger inter-drop convection comes from. The convection also contributes to avoid the possibility of rupture (Oron & Rosenau Reference Oron and Rosenau1992).

This convection is driven jointly by the gravity combined with buoyancy in the droplets, the surface forces and the interfacial mass fluxes. When the interface is deflected toward the cooled wall, the temperature in the liquid experiences a perturbation, which slightly increases the local density. Then a fluid element of volume $\delta V$ is driven by buoyancy ${\rm \Delta} \rho {\boldsymbol {g}}\delta V$. As shown in figure 14(d), a ‘plume’ of liquid, deflected by a background recirculation, descends around the centre of a drop to its crest, where a part of fluid crosses the interface due to evaporation loss. Under the continuous ambient heating, buoyant liquid then rises from the crest, due to slight expansion, with the help of favourable Marangoni stresses near the interface towards a trough, where the flow is dominated by viscous stresses and transverse pressure gradients. The pressure gradients, caused by the curvature variations of the secondary drops and the non-uniform mass fluxes (figure 12b), push the flow to the adjacent drop along the isotherms of the secondary drops.

6.6. Comparisons with experiments and simulations

As mentioned in the end of § 2 that the one-sided model is most relevant to experiments free of the effect of inert gases, it thus motivates us to simulate the instability observed by GG with this model and then compare with their experiment and simplified model. The parameters (see the caption of figure 15) are evaluated using the physical properties of R-$113$ (table 3) with $\hat {a}=0.01$ and an initial thickness $h_0=0.5$ mm. In figures 15(a) and 15(b), we present the pseudo-steady structure of the condensing surface after a transition. The global minimum thickness, $H_{Min}$, remains nearly constant after $t\approx 100$, which is two orders of magnitude less than the drop amplitudes (figure 15b,c). The drops are not stationary, but move slowly due to the disturbances of neighbouring ones. As vapour is primarily condensing on thin-film regions, where most of the heat transfer occurs, the condensate is flowing radially into the droplets, whose amplitudes increase slowly with time. We also evaluated the wavelengths of the drops by taking the geometric mean of those in the $x$- and $y$-cross-sections through the apex of a droplet (see illustration in figure 15a,b). For example, the wavelength of drop-$3$ is evaluated to be $\lambda =\sqrt {\lambda _x \lambda _y}\approx 8.09$. These are in reasonable agreement with the results of GG's drop model.

Figure 15. Rayleigh–Taylor unstable R-$113$ condensing layer (Case III), obtained by solving (5.7) with IC (5.8a) and $t_{max}=1000$ for $K=9.63\times 10^{-3}$, $\mathscr {E}^\ast =5.58\times 10^{-5}$, $\mathscr {G}^\ast =0.168$, $\mathscr {M}^\ast =-1.59$, $\mathscr {D}^\ast =-7.09\times 10^{-5}$, $Bi=0$ and $l=6{\rm \pi}$ on a $171\times 171$ mesh. (a) Surface plot and (b) the representative $y$-profiles through the apex of each drop at the ultimate moment. In (a) the red lines illustrate the $x$- and $y$-profiles through the apex of droplet-$3$, the red dots denote the apex and local minimum thicknesses around it. (c) $H_{Min}$ evolution.