## 2 Materials and Methods  
The annular flow was organized in a vertical acrylic resin pipe with an inner diameter \(D=11.7\) mm and a wall thickness of 10 mm. Prior to the experiments, the working liquid was pumped from the receiving tank into the pressure tank, located 3 m above the inlet. After that, the pump was switched off and the liquid flowed under the static pressure. The liquid flow rate was controlled by a float rotameter. The distributor vessel with a volume of about 1 L was filled with working liquid. From there the liquid was fed into the working section through a ring-shaped slot with a thickness of 0.5 mm between the inner wall of the pipe and the gas-feeding tube (the tube's wall thickness is 0.3 mm). The latter was mounted coaxially with the pipe; the gas was supplied from a centralized line of compressed air. The gas flow was controlled using an orifice meter. Thus, the liquid was introduced as a film and the annular flow started from the inlet. At the outlet, the gas-liquid mixture was separated so that the liquid returned to the receiving tank and the gas was thrown into the atmosphere. The scheme of the inlet is shown in Figure 1a for the case of a downward flow. Upward flow can be organized easily by literally turning the whole working section with the distributor and separator upside down. If the gas flow is strong enough, it will drag the liquid film upwards. Preliminary tests have shown that no liquid flows downwards in an upward flow configuration for superficial gas velocities over 15 m/s for the whole range of liquid flow rates employed. In both downward and upwards configurations, the range of superficial gas velocities, \(V_{\text{G}}\), is from 15 to 50 m/s with a step of 7 m/s. Three working liquids were used: water and two water-glycerol solutions with a mass concentration of glycerol of 20% and 33% (WGS1 and WGS2, respectively). The values of kinematic viscosity of the liquids, \(v\), are 1.15, 2 and 3 cSt (10\({}^{-6}\) m\({}^{2}\)/s), respectively. The viscosity was measured with a capillary viscosimeter at working temperature, which was about 16 \({}^{\circ}\)C for all cases. The error of the viscosity measurements did not exceed 2%. The surface tension of the glycerol solutions was only slightly lower than that of water, so its effect may be neglected. Liquid flow rates for water were selected to correspond to the liquid film Reynolds numbers of 140, 220 and 300, which were employed in [29]. The liquid Reynolds number is defined as:  
\[Re_{L}=\frac{Q}{\pi Dv}, \tag{1}\]  
where \(Q\) is the volumetric liquid flow rate. For the water-glycerol solutions, flow rates corresponding to the same \(Re_{L}\) were used. However, for WGS2, another set of liquid flow rates was used in addition to the same volumetric flow rates as for water. Figure 1: **(a)** Scheme of the inlet and layout of the camera and the illumination lamp for the shadow method; **(b)** instantaneous image of the film profile on one wall of the pipe and a fragment of a spatiotemporal matrix of film thickness. The measurements of local instantaneous film thickness were conducted using the shadow technique, used, e.g., in [21; 30]. In the present paper, we employed the same configuration as used in [29]. The detailed methodical analysis of the technique can be found in that paper; here, a brief description of its principles is given. The pipe was illuminated by a white lamp through a diffusing screen for more uniform illumination. The light passing through the pipe was registered by a high-speed camera located on the opposite side of the pipe. If a ray passes through the pipe wall and liquid film without \"touching\" the gas-liquid interface, it illuminates a particular pixel of the camera. If a ray crosses the interface, it is refracted/reflected (in the area we are looking at, it will be, as a rule, the total internal reflection) and the corresponding pixel will remain dark.