# Improved Calculation Method for Siphon Drainage with Extended Horizontal Sections
## 3 Analysis of the Formation Mechanism of Wet and Dry Slug Flow  
### Basis Modeling of Slug Flow  
In a cylindrical pipe formed by a gas-liquid two-phase fluid mixture in a slug flow, the bubble can be approximated as a cylinder with two nearly hemispherical caps. As shown in Figure 4, the middle section of the bubble can be approximated as consisting of a cylindrical cavity and a uniform thickness liquid film wrapped around the outside. The two ends of the bubble consist of a three-dimensional surface with a very complex shape and a nearly spherical surface at the top, as shown in Figure 5. The slug bubble that can exist stably in the circular tube has a length several times the diameter of the circular tube, while the gas flow rate is mainly calculated based on the main body of the bubble. Therefore, the actual shape of the bubble ends has little effect on the calculation of the total volume flow rate of the gas. Figure 4: Schematic diagram of a slug bubble moving with velocity \(U_{B}\) in a circular tube. Figure 3: Analytical solution curves for the front, the tail, and the total pressure drop of the bubble. Due to the random nature of the gas-liquid two-phase flow, the lengths of both bubbles and liquid slugs were not identical. It was assumed that all bubbles had the same length and were multiples of the diameter of the circular tube. Similarly, it was assumed that all liquid slugs were of the same length. The object of calculation and analysis was a cell consisting of a bubble and a slug. Based on the theory of multiphase media, a mathematical model of slug flow in a circular tube section in axisymmetric form was established. Firstly, it can be assumed that the computational domain of a single fluid based on the mass conservation equation can be obtained as the continuity equation in column coordinates (\(x\), \(r\), \(\theta\)) [28]  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{r}r)}{r\partial r}+ \frac{\partial(\rho u_{\theta})}{r\partial\theta}+\frac{\partial(\rho u_{x})}{ \partial x}=0 \tag{11}\]  
The Navier-Stokes equation projected onto the \(x\)-axis in the column coordinate system can be written in the following form  
\[\rho\bigg{(}\frac{\partial u_{x}}{\partial t}+u_{r}\frac{\partial u_{x}}{ \partial r}+\frac{u_{\theta}}{r}\frac{\partial u_{x}}{\partial\theta}+u_{x} \frac{\partial u_{x}}{\partial x}\bigg{)}=-\frac{\partial}{\partial x}(p- \rho g_{x}x)+\mu\
abla^{2}u_{x} \tag{12}\]  
Equations (11) and (12) describe the flow of bubbles and liquid films. Since the liquid was incompressible and the volume of the bubble remained almost constant, it can be assumed that the density \(\rho\) of each phase was constant. Since \(u_{\theta}=0\) and \(u_{r}=0\) characterized the axisymmetric model, \(\frac{\partial u_{x}}{\partial x}=0\) can be obtained from Equation (11), and further \(\frac{\partial^{2}u_{x}}{\partial x^{2}}=0\). Therefore, Equation (12) can be simplified to the following form  
\[v_{j}\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial u_{j}}{ \partial r}\bigg{)}=-g_{j}^{\prime}\,,\quad j=1,\,2 \tag{13}\]  
where \(u_{1}\) and \(u_{2}\) denote the velocity components of the continuous phase (liquid film or liquid) and the dispersed phase (bubble or droplet) along the \(x\)-axis, respectively, and \(g_{j}^{\prime}\) is the effective gravitational acceleration component of the gravity and pressure gradient acting on the dispersed or continuous phase along the \(x\)-axis. The conventional view is that the low viscosity properties of the gas result in minimal shear stress on the bubble surface, which can be neglected in calculations. However, the effect of shear stress on bubbles and droplets in slug flows can be substantial due to the  
Figure 5: The force balance diagram for the bubble’s head, transition, and smooth zone. small transverse dimensions of bubbles and liquid films in small-diameter circular tubes. At the axis of the circular tube (\(r=0\)), the shear stress in the bubble is zero.