## 3 Result
### Modeling and Comparison  
The model we used to predict the characteristics of the initial waves is exactly the same as that used in [29]; a very detailed description of the model can be found in that paper. Briefly, the model is based on the Orr-Sommerfeld equation, which is solved in both phases using the divided-attack approach. At the first stage, the liquid is considered rigid and the perturbations in the gas phase induced by a wavy wall are calculated under the assumption of linear response to the perturbations. Then, the tangential and normal stresses at the interface, calculated at the previous stage, are used as boundary conditions for solving the equation in the liquid phase. The solution is sought in the same form as Equation (2), which results in dispersion relationships, \(\alpha\) (\(k\)) and \(c(k)\). An example of a such calculation is shown in Figure 8 for water and WGS2 for the same combination of liquid and gas flow rates. As in [29], we assume that the initial waves are so-called waves of maximum growth, i.e., they evolve from perturbations with a wavenumber corresponding to the position of the maximum of the increment, \(k_{\rm m}\). Thus, the increment can be calculated as \(\alpha\) (\(k_{\rm m}\)), phase velocity-- as \(c(k_{\rm m})\), and frequency as \(c(k_{\rm m})\times k_{\rm m}/2\pi\). An additional complication is that in the model the thickness of an unperturbed film describes the stabilized film flow, which is usually thinner (hence, faster) than the film leaving the slot, where the waves appear. Thus, additional correction for the unperturbed film thickness is required. It is made in accordance with Equation (69) in [29]. Unfortunately, in its present state, the model is not adapted to the case of upward flow. For that reason, the comparison will only be made for downward flow. Figure 6: Phase velocity of the initial waves. (**a**,**b**) The effect of liquid viscosity at a fixed liquid flow rate for downward (**a**) and upward (**b**) flows. (**c**) The effect of flow orientation. (**d**) The effect of a liquid flow rate for a fixed gas velocity, \(V_{\rm G}=29\) m/s. The comparison between the model and the experiment is shown in Figures 9-11. The comparison is shown for the two liquid flow rates (5.9 and 9.3 mL/s) for the two working liquids (water and WGS2)  
Figure 8: Calculated dispersion relationships. (**a**) Spatial increment of the amplitude growth; (**b**) Phase velocity of the waves. \(Q=5.9\) mL/s; \(V_{\rm G}=22\) m/s, downward flow. Figure 7: Increment of the amplitude growth of the initial waves. (**a,b**) The effect of liquid viscosity at a fixed liquid flow rate for the downward (**a**) and upward (**b**) flows. (**c**) The effect of flow orientation. (**d**) The effect of the liquid flow rate for a fixed gas velocity, \(V_{\rm G}=29\) m/s. and the whole range of gas velocities. Qualitatively, the model reproduces the behavior of all three parameters of the waves (frequency, velocity, and increment) with gas velocity, namely, linear growth of \(f\) and \(\alpha\) with \(V_{\rm G}\) and the weak dependence of \(c\) on \(V_{\rm G}\). The growth of frequency and velocity with the liquid flow rate is reproduced correctly, as well as the increment decrease. The effect of liquid viscosity on modeled increment and velocity is in qualitative agreement with the experiments. However, the effect of viscosity on frequency is not reproduced properly. In the experiments, the frequency barely changes with \(v\) in downward flow, but it decreases drastically in the model. On the quantitative level, the modeled and experimental values are of the same order of magnitude, but the discrepancy is significant. Modeled frequency shows good agreement to the measured one for the lowest liquid flow rate for water; for the larger flow rates, the model overestimates the measurement results (Figure 9a). Based on this observation, it was supposed in [29] that the model works well for low liquid flow rates.