3.1 Convergence of the numerical solutions

The convergence of the numerical solutions was examined for different meshes. The defining characteristics, associated numerical error and solution results are included in table 3 and illustrated in figure 4. Simulations were begun from a hydrostatic solution, with vibrations initiated during the microgravity portion of the parabola and maintained for 1.25 s. In this regime, a two-scale wave pattern develops due to a secondary Faraday instability, which occurs after the formation of a (primary) frozen wave structure.

Figure 4.

Mesh convergence test for numerical simulations. Results for meshes nos. 1–4 are shown in blue, red, green and black, respectively. (a) Volume average of the level-set function $\langle \unicode[STIX]{x1D719}\rangle _{V}$ during the vibrated part of the simulation. (b) Horizontal velocity $v_{x}(y)$ at $x=4~\text{mm}$ . (c) Vertical velocity $v_{y}(x)$ at $y=3.75~\text{mm}$ .

Table 3.

Results of a mesh convergence test for numerical simulations of the FC-40 and 20 cSt silicone oil cell with $A=2~\text{mm}$ and $f=20~\text{Hz}$ (see § 6.1 for the relevance of this experiment). The selected mesh no. 3 is in bold, with deviations measured with respect to the maximum velocity for mesh no. 4 over the selected profiles: $v_{x}(y)$ at $x=4~\text{mm}$ and $v_{y}(x)$ at $y=3.75~\text{mm}$ . The interface thickness is $\unicode[STIX]{x1D700}=1.5(H/12.5)$ .

The interface thickness, which controls the scale of density and viscosity gradients across the interface ( $\unicode[STIX]{x1D719}=0.5$ ), was first varied with a fixed mesh size. The results were examined after the development of the two-scale wave pattern, and an interface thickness of $\unicode[STIX]{x1D700}=1.5(H/12.5)$ was selected.

The mesh was then modified by varying the maximum element size, defined as a fraction of $H/12.5$ , and different features were recorded in table 3: the number of mesh elements, the degrees of freedom (DoF), the simulation cost (in time) and the solution accuracy. We selected the maximum velocity of the profiles $v_{x}(y)$ at $x=4~\text{mm}$ and $v_{y}(x)$ at $y=3.75~\text{mm}$ as representative solution values, shown in figure 4(b,c), and measured the deviation from the most accurate solution. Mass conservation is checked in figure 4(a) by showing the level-set function $\unicode[STIX]{x1D719}$ , averaged over the fluid volume. We found mesh no. 3 to be a good compromise between computational cost and numerical accuracy. The level of numerical error with this choice should be well below that of the primary experimental uncertainties discussed in § 2.

The simulations presented throughout this paper are thus performed using a maximum element size of $H/12.5$ and $\unicode[STIX]{x1D700}=1.5(H/12.5)$ , which allows acceptable accuracy over a wide range of frequencies and amplitudes without excessive computational cost – an extended parameter study would otherwise be unfeasible. A generalised- $\unicode[STIX]{x1D6FC}$ (Erlicher, Bonaventura & Bursi Reference Erlicher, Bonaventura and Bursi2002) integration scheme is used, with a maximum time step manually set at $1/(10f)$ ; this step size may be reduced (automatically), as required for stability. Both streamline (Harari & Hughes Reference Harari and Hughes1992) and crosswind (Codina Reference Codina1993) stabilisation techniques are applied.

4.1 Quasi-equilibria driven by residual gravity

When forcing is initiated after the start of the microgravity phase of the parabolic flight, the fluids prior response to residual accelerations in the aeroplane is also captured. This allows some results concerning the (isolated) effect of flight disturbances on the shape and location of the immiscible interfaces. These states are referred to as quasi-equilibria, since the characteristic time scale of their evolution (of the order of the residual gravity variation) is significantly slower than the time scale of vibrational phenomena, especially for the medium to high frequencies used. In figures 1(b) and 6(c), some quasi-equilibria captured during the first part of a generic parabola (i.e. during the initial seconds of microgravity) are shown. The results of simulations are qualitatively consistent with the observed evolution of the interface in experiments due to varying residual gravity – illustrated in figure 6(a–c).

Figure 6.

Interface between FC-40 and 20 cSt silicone oil (the lower experiment cell) when subjected to residual gravity. (a–c) Snapshots showing the temporal evolution of the interface when subjected solely to residual gravity during an experiment. (d–f) Simulations when subjected to the gravity profiles of figure 5 prior to initialising forcing in microgravity: (d) initial ( $t=0.5~\text{s}$ , dashed-dotted), intermediate ( $t=1.5~\text{s}$ , dashed) and final ( $t=2.25~\text{s}$ , solid) interface locations for profile III (see 5 c); and (e,f) final quasi-steady equilibria selected with gravity profiles I and III, respectively.

Some computed quasi-equilibria states using gravity profiles I and III are shown in figure 6(d–f). The evolution of the interface for profile III after 0.5 s, 1.5 s and 2.25 s (see figure 5 c) is shown in (d), while (e) and (f) show the final (at $t=2.25~\text{s}$ ) unforced quasi-equilibria for profiles I and III, respectively. These computations are qualitatively consistent with experiments. During the hypergravity period, interfaces are nearly flat except near the endwalls, where meniscus effects are dominant. The average interface (figure 6 a and dashed-dotted line in panel d) is slightly tilted with respect to the vibrational axis, with an angle of the order of a few degrees, due to the combined effect of the $x$ and $y$ components of the aeroplane acceleration. At the initiation of the microgravity phase, interface curvature develops in both cells (figure 6 b and dashed line in panel d). The heavier FC-40 fluid (nearly transparent in the image) eventually forms an asymmetric drop on the left side of the container (figure 6 c and solid line in panel d), forced by the residual acceleration in the (negative) $x$ -direction. The lighter silicone oil (darker shading in the image) surrounds the heavier one. Thus, the interface position is selected by a combination of residual acceleration and interfacial tension, which are comparable in effect over most of the fluid domain. The final quasi-steady states are used as initial conditions for further numerical simulations with applied vibrations. Subsequent results should thus reflect the real experiments, which are performed in this manner.

The vibrational phenomena described in the remainder of the paper are associated with noticeable differences with respect to these unforced (quasi-steady) states. Although the full picture is complex with many potential interactions, we find it useful to separate the various interfacial phenomena into those tied to fast modes, like surface waves, which oscillate on the time scale of the forcing frequency, and slow modes, like vibroequilibria and frozen waves, which are related to mean, time-averaged effects.

5.1 Interfacial waves: harmonic waves and subharmonic Faraday mode

Both harmonic and subharmonic surface waves are excited by vibrations under weightless conditions. Depending on the vibrational velocity, these waves may appear superposed on top of a frozen wave pattern, as observed in miscible fluids by Shevtsova et al. (Reference Shevtsova, Gaponenko, Yasnou, Mialdun and Nepomnyashchy2016), or a drop-like configuration evolving on a slower time scale. In figure 7, a few different interfacial modes are shown; in each frame two different out-of-phase snapshots are superimposed and the interfaces in each snapshot are outlined for clarity.

Harmonic waves are not investigated in detail here, since, except at low vibrational amplitude, they are masked by more energetic phenomena like subharmonic waves. Many of the conclusions are thus based on this higher-amplitude regime. The dynamical modes observed here typically show some degree of symmetry (see e.g. figure 7) with respect to midplane reflection, although asymmetry due to finite-size effects and residual vertical acceleration, for example, is often clearly present as well, leading to a rotation of the (average) interface position. In addition, numerical and experimental results suggest that, in many cases, surface waves form on nearly vertical portions of the interface and are of nearly constant amplitude, so that the forcing mechanism can be considered roughly homogeneous and experiments with subharmonic waves, for instance, can be compared with the ideal case of a vertically vibrated container (Kumar & Tuckerman Reference Kumar and Tuckerman1994).

Figure 7.

Superimposed out-of-phase images for different surface wave modes with FC-40 and 2 cSt silicone oil: (a) harmonic mode at $f=40~\text{Hz}$ and $A=1.06~\text{mm}$ ; (b) subharmonic mode at $f=5~\text{Hz}$ and $A=4.1~\text{mm}$ (approximately odd with respect to midplane reflection); and (c) subharmonic mode at $f=18~\text{Hz}$ and $A=1.43~\text{mm}$ (approximately even). The time between frames is labelled under each picture, while solid and dashed lines outline the interfaces in each image.

Subharmonic Faraday waves exist over a wide range of parameters and can be seen experimentally in figure 7(b,c) and, below, in figures 8 and 15(d,e). Just past onset, both experiments and simulations show supercritical small-amplitude waves whose threshold may be compared with the case of a vertically vibrated container (Faraday Reference Faraday1831; Miles & Henderson Reference Miles and Henderson1990; Kumar & Tuckerman Reference Kumar and Tuckerman1994).

In Shevtsova et al. (Reference Shevtsova, Gaponenko, Yasnou, Mialdun and Nepomnyashchy2016), the Faraday wave instability in a miscible binary mixture was analysed. The threshold of subharmonic waves was estimated based on the damped Mathieu equation (Kumar & Tuckerman Reference Kumar and Tuckerman1994) by minimising

(5.1) $$\begin{eqnarray}\displaystyle A\unicode[STIX]{x1D714}=\frac{2}{\unicode[STIX]{x1D714}k}\left(\frac{\unicode[STIX]{x1D70C}_{1}+\unicode[STIX]{x1D70C}_{2}}{\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2}}\right)\sqrt{\left(\frac{\unicode[STIX]{x1D714}^{2}}{4}-\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}_{1}+\unicode[STIX]{x1D70C}_{2}}k^{3}\right)^{2}+4k^{4}\unicode[STIX]{x1D714}^{2}\left(\frac{\unicode[STIX]{x1D707}_{1}+\unicode[STIX]{x1D707}_{2}}{\unicode[STIX]{x1D70C}_{1}+\unicode[STIX]{x1D70C}_{2}}\right)^{2}}, & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D70E}=0$ for the miscible fluid case.

The threshold obtained by minimising (5.1) over wavenumber provides a reference value that is later shown in figure 18 (dashed blue line) of § 7 for the experiments of § 2. There is qualitative, albeit limited, agreement of this theoretical prediction with experiment, which both display $f^{-1/2}$ scaling at high frequencies, although separated by a clear gap. This discrepancy is not surprising in light of the differences between theoretical assumptions and the experimental set-up, including the inhomogeneous base state (the columnar frozen wave structure) and finite-size effects. The latter become increasingly important at low frequencies, which delays onset and modifies the scaling to  $f^{-1}$ .

The numerical simulations (dark blue solid line and blue open markers in figure 18 of § 7), on the other hand, match somewhat better with the experimental results. Numerical thresholds have been located using a similar approach as in Gandikota et al. (Reference Gandikota, Chatain, Amiroudine, Lyubimova and Beysens2014b ), using more precision in frequency and amplitude near threshold, so that the estimated uncertainty is of the order of $\pm 1.25~\text{Hz}$ and $\pm 0.0125~\text{mm}$ . For low-frequency vibrations, residual gravity can couple with the forcing to slightly modify the instability onset. This effect is observed in both cells.

Figure 8 shows a series of experimental snapshots separated by approximately one half-period of the $f=22~\text{Hz}$ driving force. Depending on the phase, standing waves can be seen (mainly) on the left or right side of the underlying columnar structure. The explanation for this phase shift is the sign change in the effective vibrational acceleration at each interface due to the sign of the density jump. Since the Faraday waves are subharmonic, the $\unicode[STIX]{x03C0}$ shift in forcing on opposite sides of the columnar structure implies a $\unicode[STIX]{x03C0}/2$ phase shift for the waves. The same type of behaviour was previously observed in miscible liquids (Shevtsova et al. Reference Shevtsova, Gaponenko, Yasnou, Mialdun and Nepomnyashchy2016).

Figure 8.

Successive snapshots showing the evolution of subharmonic Faraday waves on opposite sides of the frozen wave columnar structure for $f=22~\text{Hz}$ and $A=1.76~\text{mm}$ . Times are selected to best show the $\unicode[STIX]{x03C0}/2$ phase difference between these two subharmonic wave fields ( $1/4$ period of these 11 Hz subharmonic waves is approximately 0.0227 s).

The importance of the phase difference for weakly coupled subharmonic wave fields, like these two regions of interfacial Faraday waves, was investigated in Salgado Sánchez et al. (Reference Salgado Sánchez, Porter, Tinao and Laverón-Simavilla2016). It was found that the character of the instability not only changes from supercritical to subcritical depending on detuning (most relevant here at low frequencies where the container dimensions most constrain the wavenumber), but also can change from a steady-state pitchfork bifurcation to a Hopf bifurcation due to coupling. Hopf bifurcations have not been observed here, possibly due to the coupling being too weak, and thus the modulation frequency too small, to be noticed on the time scale of the experiments and simulations.

Figure 9.

Instability diagram of the subharmonic mode for FC-40 and 20 cSt silicone oil under different gravity profiles: (a) profile I and (b) profile III. Experimental (filled circles) and numerical (open circles) results exhibit various dynamical modes, marked with alternating blue tones. The solid curves are drawn for visual reference to distinguish different average vertical wavenumbers. The dashed curve indicates that all these modes belong to the main subharmonic instability tongue. Illustrative snapshots in the upper left corner show the initial condition, while the smaller images show the interface after $t\approx 2~\text{s}$ of applied vibration in microgravity.

By combining numerical with experimental results, we are able to construct the instability diagram for the subharmonic Faraday wave mode in each experiment cell. In figure 9, results for FC-40 and 20 cSt silicone oil are organised by the wavenumber $k$ measured in the centre of the pattern and vibrational velocity under gravity profiles I (panel a) and III (panel b). Four modes are distinguished according to the average number of wavelengths in the vertical direction, and corresponding curves are drawn for visual reference, suggesting boundaries within the main subharmonic tongue. Since the simulations of figure 9(a) are performed using the ideal gravity profile (no $g_{x}$ component), they are not compared directly with experiments, but provide a reference for evaluating the effect of residual gravity seen in figure 9(b), which shows results obtained using a more generic gravity profile that may be better compared to experiments. We note that numerical simulations near $k=1~\text{mm}^{-1}$ were not able to reproduce subharmonic waves as seen in experiments, perhaps due to the increased importance of residual gravity at those low forcing amplitudes in the experiments or to increased damping from the boundaries in the simulations. Over the limited region of overlap, the experimental results are consistent with the simulations, in terms of both wavenumber selection and the qualitative structure of the surface.

As discussed in Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013), gravity plays a dual role in shaping the instability diagram, modifying not only the instability threshold but also the location and profile of the mode boundaries. The effect of gravity can be seen in figure 9 by comparing the response of the interface to the two different microgravity profiles. When a small disturbance is introduced along the $x$ -axis (figure 9 b, profile III), the mode boundaries shift a bit towards lower wavenumbers (frequencies), while the threshold is modified only slightly. Note that the conditions differ only in the gravity level along the $x$ -direction, which is $0.01g_{0}$ for profile III. Some differences with respect to the predictions of Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013) are expected. Also note that, compared to the analysis of Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013) and Kumar & Tuckerman (Reference Kumar and Tuckerman1994), we only locate the subharmonic instability tongue, which is normally dominant in Faraday wave systems, and this is shown with the (dashed black) curve in figure 9. Other instability tongues (harmonic, for example) are masked by the primary subharmonic mode and difficult to observe.

Figure 10.

Instability diagram for the subharmonic mode with FC-40 and 2 cSt silicone oil under gravity profile I. Experimental data (filled circles) and numerical simulations (open circles) reveal various dynamical modes, marked with alternating blue tones. The solid curves are drawn for visual reference to distinguish different average vertical wavenumbers. The dashed curve indicates that all these modes belong to the main subharmonic instability tongue. Illustrative snapshots in the upper left corner show the initial condition, while the smaller images show the interface after $t\approx 2~\text{s}$ of applied vibration in microgravity.

Viscous and interfacial tension effects are revealed in a comparison of figures 9(a) and 10. Lower viscosity clearly reduces the threshold for Faraday waves. At the same time, the reduction of interfacial tension with 2 cSt silicone oil generally increases the observed wavenumber of the subharmonic mode (for a given forcing frequency). This is consistent with the dispersion relation, and is investigated experimentally, for example, in Gandikota et al. (Reference Gandikota, Chatain, Amiroudine, Lyubimova and Beysens2014a ), where the wavenumber of the Faraday instability in near-critical hydrogen is observed to increase with approach to the critical point where interfacial tension vanishes.

As vibrational velocity increases, the subharmonic Faraday waves become increasingly nonlinear and, at sufficient velocity, interfacial integrity breaks down, initiating a process of drop ejection. This process is important, since it places an upper bound on the wave amplitudes that can be investigated experimentally and numerically.

5.2 Drop ejection instability of the subharmonic mode

At large enough amplitude, the increasingly nonlinear Faraday waves may become unstable to drop ejection (Goodridge, Shi & Lathrop Reference Goodridge, Shi and Lathrop1996; Goodridge et al. Reference Goodridge, Shi, Hentschel and Lathrop1997; James et al. Reference James, Vukasinovic, Smith and Glezer2003b ). Visual observations indicate that the resulting interfacial pattern often has a three-dimensional structure, which makes the two-dimensional simulations less applicable. Despite the complexity of this process, especially when many drops are present, the experimental results allow us to distinguish two qualitatively different instability mechanisms that are associated with distinct velocity thresholds:

1. (i) A – heavier-to-lighter liquid ejection when a drop of FC-40 is ejected into the bulk of the silicone oil.

2. (ii) B – lighter-to-heavier liquid ejection when a drop of silicone oil is ejected into the bulk of the FC-40 fluid.

Figure 11.

Representative snapshots in time (values provided) showing the different drop ejection mechanisms observed in the FC-40 and 2 cSt silicone oil cell during microgravity: (a–c) mechanism A (near onset), (d–f) mechanism A (far from onset), and (g–i) mechanism B.

The dynamics of mechanism A is shown in figure 11(a–c) for $A=1.57~\text{mm}$ and $f=20~\text{Hz}$ . In this process, drops are created in the lighter fluid and ejected away from the interface. A variation of mechanism A can occur with larger vibrational velocities, as in figure 11(d–f) with $A=4.29~\text{mm}$ and $f=8~\text{Hz}$ , where drops are still created in the lighter fluid but, instead of moving away, tend to remain trapped near the interface.

Mechanism B occurs for larger amplitudes, sometimes concurrently with mechanism A. The dynamics of the process are as shown in figure 11(g–i) for $A=4.37~\text{mm}$ and $f=9~\text{Hz}$ .

Drop ejection is associated with a jet of fluid emerging from a large-amplitude surface wave that then collapses to create a drop of fluid and, often, other smaller drops, which are known as satellites (James et al. Reference James, Vukasinovic, Smith and Glezer2003b ; Castrejón-Pita et al. Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015). As forcing amplitude is increased, especially if it becomes comparable to (or larger than) the thickness of the liquid layer, $A>H$ , the scenario can be very complex. The dynamics is often fully three-dimensional and disordered in this regime (James et al. Reference James, Smith and Glezer2003a ) and the drops interact with the strongly nonlinear subharmonic waves and the associated mean flow, which can lead to the drop being recaptured by the interface. We note that the highly nonlinear behaviour of the system for $A>H$ was also observed in the case of miscible liquids, which led either to secondary Faraday waves over frozen waves (Shevtsova et al. Reference Shevtsova, Gaponenko, Yasnou, Mialdun and Nepomnyashchy2016) or to the formation of a fish-spine pattern (Shevtsova et al. Reference Shevtsova, Gaponenko, Yasnou, Mialdun and Nepomnyashchy2015b ).

Figure 12.

Drop ejection instability thresholds for FC-40 and 2 cSt silicone oil as a function of frequency $f$ and (a) amplitude $A$ and (b) vibrational velocity $A\unicode[STIX]{x1D714}$ . Circles mark experimental data points, coloured in red if drop ejection is found. Fits using different scalings for the drop ejection threshold (mechanism A, dashed black line) are given in colour: $A\propto f^{-1}$ (red), $A\propto f^{-2/3}$ (green) and $A\propto f^{-1/2}$ (blue). The threshold curve for mechanism B (solid black line) is estimated from experiments (no scaling assumptions).

The thresholds of the two droplet ejection mechanisms identified above are shown in figure 12 for FC-40 and 2 cSt silicone oil in terms of amplitude $A$ (panel a) and vibrational velocity $A\unicode[STIX]{x1D714}$ (panel b) versus frequency $f$ . The estimated thresholds shown are based on the available experimental data points (filled circles). The dashed black curve indicates the lower bound of the droplet ejection regime, while the solid black curve indicates the onset of mechanism B. For the majority of low-gravity experiments considered here, performed at $f<15~\text{Hz}$ , wave breaking with drop ejection is observed at $A\unicode[STIX]{x1D714}|_{c}\gtrsim 0.17~\text{m}~\text{s}^{-1}$ , as illustrated in figure 12(b).

We can compare the experimental results giving the (estimated) onset of drop ejection with the scalings suggested by Goodridge et al. (Reference Goodridge, Shi, Hentschel and Lathrop1997) for the cases of a system dominated by surface tension (with, for example, low-viscosity fluids like ethanol), i.e. $A\unicode[STIX]{x1D714}\sim c_{1}F_{1}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C})\unicode[STIX]{x1D714}^{1/3}$ , and for more viscous fluids (like water or glycerine), i.e. $A\unicode[STIX]{x1D714}\sim c_{2}F_{2}(\unicode[STIX]{x1D708})\unicode[STIX]{x1D714}^{1/2}$ . We also consider a constant velocity threshold, $A\unicode[STIX]{x1D714}\sim c_{3}~(=0.17~\text{m}~\text{s}^{-1})$ . Here $c_{j},~j=1,2,3$ , are constants while $F_{1}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C})=(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C})^{1/3}$ and $F_{2}(\unicode[STIX]{x1D708})=\sqrt{\unicode[STIX]{x1D708}}$ for a single-fluid system. The corresponding prefactors are obtained by fitting, and the results are shown in figure 12(b).

The results, although limited by the small number of experimental data points and the variable conditions onboard the parabolic flight (e.g. residual gravity effects), do suggest a transition from (nearly) constant critical velocity at low frequencies to an increase of critical velocity (most) consistent with the scaling proposed by Goodridge et al. (Reference Goodridge, Shi, Hentschel and Lathrop1997) for viscous fluids. Note that more experimental data points would be needed to more clearly determine the scaling definitively and that, in any case, there are substantial differences between the experiments of § 2 and the system considered by Goodridge et al. (Reference Goodridge, Shi, Hentschel and Lathrop1997), which would affect the $c_{i}$ and $F_{i}$ factors.

In the limit of high frequency, equation (5.1) gives a reference value for subharmonic Faraday waves, while experiments suggest a roughly constant gap between the onset of Faraday waves and the drop ejection instability (both scale approximately as $f^{-1/2}$ for higher frequencies). The regularity of this gap can be seen later in figure 18 of § 7 with the solid blue and red lines. Ignoring the details associated with both instabilities (see e.g. Goodridge et al. Reference Goodridge, Shi, Hentschel and Lathrop1997), the results suggest that the energy gap between Faraday waves and drop ejection depends only weakly on the frequency of the forcing, and mainly on the physical properties of the fluid. Since the less viscous combination of FC-40 and 2 cSt silicone oil allows more data points in the drop ejection regime, conclusions are drawn mainly from this case.

5.3 Energy estimation for drops and interfacial phenomena

For the experiments described in § 2, drop ejection generally occurs after the interface has evolved towards a columnar or bubble-like structure, and then undergone an instability to Faraday waves. This means that the average interface is constrained by the system size – reaching from top to bottom and oriented, to a large extent, perpendicular to the forcing. Both contact energy, which is relevant when $\unicode[STIX]{x1D6FD}\neq \unicode[STIX]{x03C0}/2$ , and the average interfacial energy are largely constrained (bounded) in this situation. An increase in interfacial energy must manifest through the dynamics of the interface via higher-amplitude waves or drop ejection.

We estimate the interfacial energy of the main interface and of the drops after ejection begins ( $A\unicode[STIX]{x1D714}>A\unicode[STIX]{x1D714}|_{c}\approx 0.17~\text{m}~\text{s}^{-1}$ for $f<15~\text{Hz}$ ) by measuring the visible interface length in selected frames of each experiment run. Consistent with the two-dimensional approximation (near onset, at least, the subharmonic waves do appear to be quasi-two-dimensional), we assume that the contribution from the perpendicular dimension $D$ is constant for different interface positions and for the initial drops. With this simplifying approximation, the interfacial energy depends on its visible perimeter, which can be measured in a given snapshot simply by summing pixels: $E_{S}\sim \unicode[STIX]{x1D70E}{\mathcal{P}}_{\unicode[STIX]{x1D70E}}$ , where ${\mathcal{P}}_{\unicode[STIX]{x1D70E}}$ is the pixel count of the perimeter. In figure 13, the measured interfacial energies (sum of visible lengths) of the main average interface (green squares), neglecting surface waves, and the drops (red circles) are given as a function of

(5.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{c}=A\unicode[STIX]{x1D714}-A\unicode[STIX]{x1D714}|_{c}, & & \displaystyle\end{eqnarray}$$

where $A\unicode[STIX]{x1D714}|_{c}$ is the drop ejection threshold.

Figure 13.

Interfacial energies, measured by ${\mathcal{P}}_{\unicode[STIX]{x1D70E}}$ (in pixels), obtained for FC-40 and 2 cSt silicone oil, as a function of $\unicode[STIX]{x1D6FF}_{c}=A\unicode[STIX]{x1D714}-A\unicode[STIX]{x1D714}|_{c}$ , where $A\unicode[STIX]{x1D714}|_{c}$ is the drop ejection threshold found to be $0.1743~\text{m}~\text{s}^{-1}$ from the fit. The interfacial energies of the drops (red circles) and the main interface (green squares) are included. The curves are fitted to quadratic and constant functions, respectively. The number of observed drops is marked in parentheses, while outlier data points marked with crosses are excluded from the fit.

As expected, we find that the main interfacial energy term is (nearly) constant since either the frozen wave instability has led to columnar structures, or the vibroequilibria effect has flattened an initially bubble-like configuration. In both cases, the main interfacial energy is bounded by the system size. Drop energy, in contrast, increases (roughly) as $\unicode[STIX]{x1D6FF}_{c}^{2}$ , as shown by a quadratic fitting to the experimental points.

We note several uncertainties of the experiments, which were not designed to study drop ejection, and of the fit in figure 13. First, drop number and size varied even near onset, and the significant interfacial energy of a single drop means that ${\mathcal{P}}_{\unicode[STIX]{x1D70E}}$ does not go to zero at onset, but jumps to a finite value instead. Also, some runs were excluded as outliers (marked by crosses in figure 13). Upon checking the corresponding datasets, we attribute these exceptions to the lack of repeatability in the residual gravity profile between parabolas, and to the potential hysteresis that occurs if drops survive between parabolas. The former modifies the contribution of gravity (neglected in the analysis) and may eventually interact with the drops (James et al. Reference James, Smith and Glezer2003a ) if a change in gravity couples with the ejection process, for example. Hysteresis is problematic too, with the experiment starting from a metastable initial condition, which changes the evolution (and energies) of the phenomena involved.

One can try to understand the results of figure 13 by recalling the energy balance suggested by the theory of vibroequilibria (Lyubimov et al. Reference Lyubimov, Cherepanov, Lyubimova and Roux1997; Beyer et al. Reference Beyer, Gawriljuk, Günther, Lukovsky and Timokha2001; Gavrilyuk et al. Reference Gavrilyuk, Lukovsky and Timokha2004), where selection of the time-average (quasi-static) interface shape is formulated as a minimisation problem using a Lagrangian method that considers vibrational and surface energy contributions, among others (like gravitational); surface waves, which do not affect the average interface (to leading order), are not accounted for.

Assuming constant interfacial tension, the system energy $E$ may be expressed (see e.g. Fernández et al. Reference Fernández, Tinao, Porter and Laverón-Simavilla2017b ) in a co-moving frame as

(5.3) $$\begin{eqnarray}\displaystyle E=(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})\unicode[STIX]{x1D700}(S)(A\unicode[STIX]{x1D714})^{2}+\unicode[STIX]{x1D70E}\int _{S}\unicode[STIX]{x2202}S+\cdots \,, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D700}(S)$ represents a non-dimensional factor obtained by integrating over the fluid domain (determined by the walls and interface $S$ ) and suppressed terms (signified by dots) include contact energy and gravitational energy.

In the absence of gravity and other relevant forces, the principal balance is between vibrational energy, which increases as $(A\unicode[STIX]{x1D714})^{2}$ , and interfacial energy. At a minimum of the Lagrangian (the vibroequilibria solution), these two energy terms have equal and opposite derivatives with respect to any (small continuous) variation. A finite increase in forcing will directly increase the vibrational energy of the system and this will, in turn, cause the interface to reorient, increasing interfacial energy but decreasing vibrational energy until a new balance (minimum) is achieved. Neglecting the initial interfacial contribution (of a flat, unforced interface, for instance), the interfacial energy of a vibroequilibria solution is expected to scale as $(A\unicode[STIX]{x1D714})^{2}$ .

It is interesting to note that the balance suggested by vibroequilibria theory, with increases in vibrational energy being partly transferred into (average) interfacial energy, seems to be replaced by an analogous process for drop ejection. Following a nonlinear process in which the surface wave amplitude increases until breaking occurs, subsequent increases in vibrational energy are partly channelled into the creation of ejected drops, with the $(A\unicode[STIX]{x1D714})^{2}$ scaling carried over as well when the shifted value $\unicode[STIX]{x1D6FF}_{c}=A\unicode[STIX]{x1D714}-A\unicode[STIX]{x1D714}|_{c}$ is used.

6.1 Frozen waves and the two-scale wave pattern

The series of snapshots in figures 14 and 15 show the evolution of the interface between immiscible liquids subjected to horizontal vibrations under microgravity conditions. Each figure contains experimental observations for $f=22~\text{Hz}$ that confirm the results of simulations for $f=20~\text{Hz}$ . It is clear that the amplitude, which is below the threshold for frozen waves in figure 14 and above this threshold in figure 15, has a big effect. Increasing from $A=0.87~\text{mm}$ to $A=1.76~\text{mm}$ in the experiments (from $A=1.5~\text{mm}$ to $A=2.0~\text{mm}$ in simulations) pushes the system beyond the critical value for the frozen wave instability. Recall that this instability is caused by a shear-driven mechanism similar to the Kelvin–Helmholtz instability.

Figure 14.

Snapshots showing temporal evolution of the FC-40 and 20 cSt silicone oil interface below the frozen wave threshold for both experiments and simulations. (a–c) Experiment response at $f=22~\text{Hz}$ and $A=0.87~\text{mm}$ . (d–f) Simulation performed at $f=20~\text{Hz}$ and $A=1.5~\text{mm}$ under gravity profile III. A comparison of experiment and simulations in time cannot be done quantitatively due to the definition of $\boldsymbol{g}(t)$ (see § 4), so time values are not labelled.

Figure 15.

Snapshots showing temporal evolution of the FC-40 and 20 cSt silicone oil interface above the frozen wave threshold for both experiments and simulations. (a–e) Experiment response at $f=22~\text{Hz}$ and $A=1.76~\text{mm}$ (above frozen wave onset) showing the growth of the instability (a,c) and the secondary Faraday waves excited on the lateral boundaries of the columns. The snapshots (d,e) separated by $\unicode[STIX]{x0394}t=0.05~\text{s}\approx 1/(22~\text{Hz})$ show the (nearly) out-of-phase subharmonic behaviour; waves on the right column are shifted in phase by $\unicode[STIX]{x03C0}/2$ . (f–h) Simulation performed at $f=20~\text{Hz}$ and $A=2~\text{mm}$ under gravity profile II. A comparison of experiment and simulations in time cannot be done quantitatively due to the definition of $\boldsymbol{g}(t)$ (see § 4), so the time values are not labelled.

Below the frozen wave threshold (figure 14), residual gravity drives the formation of a squeezed drop-like structure located near the left side of the cell. Subsequently, either harmonic surface waves (figure 14 a,b,d,e) or Faraday waves (figure 14 f) may be excited over the interface; see § 5.1 for further details.

Above the frozen wave threshold (figure 15), the interface forms a pattern with $\unicode[STIX]{x1D706}\sim L$ , which then develops into a columnar-like structure. Subharmonic Faraday waves (see § 5.1) can appear on both sides of a column (figure 15 d,e). Analogous two-scale patterns have been observed in miscible fluid experiments (Shevtsova et al. Reference Shevtsova, Gaponenko, Yasnou, Mialdun and Nepomnyashchy2016).

The (numerical) onsets of Faraday (dark blue circles and dark blue solid line in figure 18) and frozen waves (green squares and green solid line in figure 18) cross at a frequency $f^{\ast }\sim 35~\text{Hz}$ for 2 cSt silicone oil and $f^{\ast }\sim 17.5~\text{Hz}$ for 20 cSt silicone oil. If we imagine an idealised scenario with $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/2$ and an initially flat interface in microgravity, then Faraday waves cannot immediately appear with horizontal excitation (although harmonic and cross-waves can), even if their threshold is surpassed. It is only after frozen waves have been excited that columnar structures develop and portions of the interface are sufficiently perpendicular to the forcing axis to permit the excitation of Faraday waves. Thus, for $f<f^{\ast }$ one expects, with increasing amplitude, to first see the frozen wave instability, quickly followed (at sufficient wave amplitude) by Faraday waves – the frozen wave instability acts here, among other things, to facilitate or ‘trigger’ Faraday waves by reorienting the interface. For $f>f^{\ast }$ , on the other hand, the frozen waves develop free of Faraday waves, whether forming columnar structures or not, until the Faraday wave threshold is crossed. In a real parabolic flight experiment, however, plane manoeuvres and residual gravity do not permit a purely horizontal flat interface, and the situation is more complex. Faraday waves may appear any time a portion of the interface moves towards a (nearly) perpendicular orientation with respect to the vibrational axis and the Faraday wave threshold is exceeded.

Although some experiments were performed above the frozen wave threshold (see figure 18), the irregular microgravity conditions in those runs do not permit a satisfactory analysis of the instability. In addition, with the 2 cSt silicone oil, the experiments performed above the frozen wave threshold are complicated by high-amplitude Faraday waves or drop ejection, making the results hard to process. Here, we perform a simulation-based analysis of frozen waves and compare it, when possible, with experimental results and theoretical predictions. Unlike the experiments of § 2, we analyse the instability under idealised microgravity conditions (mainly, profile I) in the numerical model.

6.2 The frozen wave instability in microgravity

Columnar-like structures developing from the frozen wave instability have been seen in previous experiments in microgravity either in ordinary miscible liquids (Gaponenko et al. Reference Gaponenko, Torregrosa, Yasnou, Mialdun and Shevtsova2015b ) or in near-critical fluids where interfacial tension vanishes (Gandikota et al. Reference Gandikota, Chatain, Amiroudine, Lyubimova and Beysens2014a ,Reference Gandikota, Chatain, Amiroudine, Lyubimova and Beysens b ). Here we report, for the first time, columnar-like structures in microgravity for a pair of liquids with sizeable interfacial tension.

Previous treatments and investigations of frozen waves (Lyubimov & Cherepanov Reference Lyubimov and Cherepanov1986; Talib et al. Reference Talib, Jalikop and Juel2007; Goldobin et al. Reference Goldobin, Pimenova, Kovalevskaya, Lyubimov and Lyubimova2015; Lyubimov et al. Reference Lyubimov, Ivantsov, Lyubimova and Khilko2016) considered a sufficiently large container or the ideal case of an infinite domain so that finite-size effects could be neglected. In this limit, a linear stability analysis provides the velocity threshold

(6.1) $$\begin{eqnarray}\displaystyle (A\unicode[STIX]{x1D714})^{2}=\frac{(\unicode[STIX]{x1D70C}_{1}+\unicode[STIX]{x1D70C}_{2})^{3}}{\unicode[STIX]{x1D70C}_{1}\unicode[STIX]{x1D70C}_{2}(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})}\sqrt{\frac{\unicode[STIX]{x1D70E}g_{y}}{\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2}}}, & & \displaystyle\end{eqnarray}$$

for the inviscid case. The most unstable wavenumber $k=\sqrt{(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})g_{y}/\unicode[STIX]{x1D70E}}$ is related to the capillary length. When vibrated beyond the threshold (6.1), which is very small in microgravity, wavenumber selection depends on the vibrational velocity, which excites a band of wavenumbers. In Burnysheva, Lyubimov & Lyubimova (Reference Burnysheva, Lyubimov and Lyubimova2011), it was assumed that the wavelength selection in this problem was defined by the perturbation with maximum growth rate. A continuous and monotonic relationship between vibrational velocity and selected wavenumber was found.

In order to compare experiments and simulations with this theoretical prediction, figure 18 includes (6.1) with $g_{y}=0.05g_{0}$ (green dashed lines) for each experimental cell. Theoretical predictions give

(6.2) $$\begin{eqnarray}\displaystyle A\unicode[STIX]{x1D714}|_{theory}=0.134~\text{m}~\text{s}^{-1}~(2~\text{cSt oil}),\quad 0.1581~\text{m}~\text{s}^{-1}~(20~\text{cSt oil}), & & \displaystyle\end{eqnarray}$$

while the numerical results (green solid lines) yield

(6.3) $$\begin{eqnarray}\displaystyle A\unicode[STIX]{x1D714}|_{experiment}=0.157~\text{m}~\text{s}^{-1}~(2~\text{cSt oil}),\quad 0.207~\text{m}~\text{s}^{-1}~(20~\text{cSt oil}). & & \displaystyle\end{eqnarray}$$

Our results are in line with findings by Talib et al. (Reference Talib, Jalikop and Juel2007) that inviscid predictions underestimate the true onset value, which is affected by viscosity and, due to the geometry of the experiment cells, finite-size effects. The results here are obtained in relatively small containers where finite-size effects are expected to be important. To investigate this, the wavenumber of the instability is shown in figure 16(a) alongside the vibrational velocity with gravity profile I ( $g_{x}=0$ ) and for each fluid combination.

For small amplitudes, a constant wavenumber $k_{1}\sim 0.4~\text{mm}^{-1}$ ( $\unicode[STIX]{x1D706}_{1}\sim 15.7~\text{mm}\approx L$ ) is observed in both cells. When compared to the inviscid theory, it is evident that the wavenumber selection is driven by the container length. This wavenumber is nearly constant with increasing velocities until a certain value $A\unicode[STIX]{x1D714}\sim 0.3~\text{m}~\text{s}^{-1}$ is reached, at which point the wavenumber doubles to $k_{2}\sim 0.8~\text{mm}^{-1}$ . For this second mode, the results exhibit a certain amount of dispersion in the less viscous combination (green squares), while the wavenumber is nearly constant for the more viscous case (red squares). This mode persists until a velocity of $A\sim 1~\text{m}~\text{s}^{-1}$ , where the wavenumber appears to double again.

Figure 16.

(a) Frozen wave instability diagram showing wavenumber for both combinations (less viscous, green open squares) under gravity profile I, when vibrations are applied in microgravity. (b) Frozen wave instability diagram for FC-40 and 20 cSt silicone oil under different gravity profiles and forcing initiation times: profile I initiation in microgravity (red open squares), profile I initiation in hypergravity (pink open squares), and profile III initiation in hypergravity (cyan open squares). An experimental measurement (black solid square) is also shown. Illustrations are included to help visualise the features of the instability.

6.3 Residual gravity effects

The effect of residual gravity is analysed here and compared to the available experimental data point. In figure 16(b), the wavenumber of the frozen wave instability under different gravity conditions (i.e. gravity profiles) and forcing initiation times is shown for FC-40 and 20 cSt silicone oil.

First, note the differences arising from the initiation time of the forcing. This is shown for gravity profile I, both for initiation during the hypergravity period (pink open squares) and during microgravity (red open squares). Despite the fact that the final gravity level is identical for both experiments, the wavenumber at onset is distinct. For hypergravity initialisation, when $g_{y}$ is reduced after the vibration begins, the frozen wave threshold, which depends on $g_{y}$ via equation (6.1), may be surpassed before the final microgravity level is reached. Since the vertical gravity at this instant is larger, an increased wavenumber is expected. This wavenumber is nearly constant before jumping at a certain velocity, as in the previous case.

With increased vibrational velocity in this hypergravity case, we observe greater dispersion in the wavenumber, which may be explained in the same way. The variation of the gravity level results in varying (most dangerous) wavenumbers and so, depending on the growth rate of each unstable mode and its interaction with the changing gravity level, different final states may be found. In particular, a new mode is observed with $k\sim 1.2~\text{mm}^{-1}$ .

In the case of gravity profile III (cyan open squares), when there is a perturbation in the $x$ -direction, a further increase in $k$ is observed. In addition to that, the critical velocities for transition to different modes are notably increased as well; this limits the maximum expected wavenumber in experiments. We were able to process just one experimental point (black solid square), where the microgravity conditions of the parabola allowed the appropriate development of frozen waves.

To the best of our knowledge, the results presented here are the first to examine the effect of residual gravity and strong finite-size constraints on wavenumber selection for the frozen wave instability.

6.4 The frozen wave instability in hypergravity

Only one experiment run considered high-frequency and high-amplitude vibrations. Despite the scarcity of experimental data points in this regime, which was further aggravated by the fact that the run was quickly aborted for safety considerations, there is clear evidence of the frozen wave instability in both experimental cells. In figure 17, three snapshots of each cell after initiation of the forcing in hypergravity are shown; the excitation is with $f=50~\text{Hz}$ and $A=3.9~\text{mm}$ .

As with lower vibrational velocities, there may be a harmonic surface wave mode, typical of horizontally vibrated containers (see e.g. Porter et al. Reference Porter, Tinao, Laverón-Simavilla and López2012; Salgado Sánchez et al. Reference Salgado Sánchez, Porter, Tinao and Laverón-Simavilla2016), directly excited on the interface during hypergravity (figure 17 a,d). These surface waves, which are strongly nonlinear here, seem to propagate inwards from the ends of the container, but quickly evolve towards a different fluid configuration over a characteristic time $\unicode[STIX]{x1D70F}<0.0125~\text{s}$ during the hypergravity period, as illustrated in figure 17(b,e). Here, the weightlessness condition is less significant since the vibrational amplitude is so large, effectively dominating over gravity and allowing the formation of a nearly columnar pattern. The ratio of the estimated frozen wave instability wavenumber for the two experiments (2 cSt oil, figure 17 b, denoted by subscript 1; 20 cSt oil, figure 17 e, denoted by subscript 2) is measured as

(6.4) $$\begin{eqnarray}\displaystyle \left.\frac{k_{1}}{k_{2}}\right|_{experiment}=\frac{\unicode[STIX]{x1D706}_{2}}{\unicode[STIX]{x1D706}_{1}}\approx 1.26, & & \displaystyle\end{eqnarray}$$

which matches well with the predictions of Lyubimov & Cherepanov (Reference Lyubimov and Cherepanov1986):

(6.5) $$\begin{eqnarray}\displaystyle \left.\frac{k_{1}}{k_{2}}\right|_{theory}=\sqrt{\frac{\unicode[STIX]{x1D70E}_{2}(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})_{1}}{\unicode[STIX]{x1D70E}_{1}(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})_{2}}}\approx 1.29. & & \displaystyle\end{eqnarray}$$

With FC-40 and 2 cSt silicone oil, the frozen wave pattern coexists with numerous drops near the interface. In the case of the higher-viscosity 20 cSt silicone oil, however, the Faraday waves do not reach high enough amplitude to cause drop ejection in less than 0.0125 s (one could speculate that, had the experiment lasted longer, they would have). In figure 18(b), this experimental point is included for the sake of completeness, but marked in blue to emphasise the discrepancy with expected phenomena in this regime.

Figure 17.

Three snapshots, separated by 0.0125 s in time for FC-40 and 2 cSt silicone oil (a–c) and FC-40 and 20 cSt silicone oil (d–f) subjected to vibrations at $f=50~\text{Hz}$ and $A=3.9~\text{mm}$ . (a,d) Initial high-amplitude surface waves in hypergravity, (b,e) transient frozen wave pattern (in hypergravity), and (c,f) transition from vibrated to residual gravity equilibria.

Figure 18.

Response of the fluid interface to vibrations in microgravity as a function of frequency $f$ and amplitude $A$ : (a) FC-40 and 2 cSt silicone oil, and (b) FC-40 and 20 cSt silicone oil. Solid circles denote experimental points in three regimes: harmonic waves (yellow), subharmonic Faraday waves (blue) and drop ejection (red). Open markers denote numerical results for the Faraday wave instability (blue open circles) and the frozen wave instability (green open squares). These thresholds are fitted to obtain the dark blue and green solid lines, respectively. The Faraday instability scales as $f^{-1/2}$ for high frequencies ( ${\gtrsim}$ 20 Hz) and as $f^{-1}$ at lower frequencies where finite-size effects are important. Theoretical onset curves are shown for Faraday waves (equation (5.1), dashed blue lines) and frozen waves (equation (6.1) using g $_{y}=0.05g_{0}$ , dashed green line). The drop ejection threshold discussed in § 5.2 is shown in panel (a), and is extended (dashed curve) for visual reference to higher frequencies. Panel (b) includes a presumed curve (dashed red line) for drop ejection, although supported by just one experimental data point.