# Exchange Flows in Inclined Pipes with Different Viscosity Ratios
## 3 Results and Discussion
### Exchange Flow Experiments in Pipes
#### 3.1.1 Changes in the Flow Regime of Light Fluids
This is because, at this time, the \(F_{af}\) and \(F_{ar}\) of a dominant fluid movement are of a critical point so that this will lead to part of the light fluid flowing against the wall and part of the flow moving away from the wall; when the angle of inclination is greater than the emergence of the SBS of the smallest angle of inclination (\(\theta\)), the light fluid flow state completely turns into SBS because, at this time, the direction of the light fluid flow is dominated by the \(F_{ar}\), meaning the light fluid will be adhering to the wall and upward movement. With an increase in \(\beta\), the flow state into SBS of the required \(\theta\) is greater; this is because of the low \(\beta\), where viscous forces (\(f_{a}\)) dominate. When blocking the movement of light fluids and forcing the fluid to the wall of the pipe, where the role of \(F_{a}\) is small, resulting in a small \(F_{af}\), the light fluids do not flow easily along the pipe axis. With the increase in \(\beta\), the \(F_{a}\) gradually dominates; when the \(F_{af}\) is large, it will make it easier for the fluid to flow along the axis of the pipe, and so a greater angle of inclination (\(\theta\)) is needed in order to make the light fluid flow against the wall. Figure 5b illustrates the variation of the ratio of the width of the rising light fluid to the width of the pipe (\(\delta_{a}\)) with the tilt angle (\(\theta\)) for different viscosity ratios (\(\beta\)). In order to minimize the error, the \(\delta_{a}\) is measured by taking three different positions to measure the \(\delta_{a}\) and then taking the average value. As the \(\theta\) becomes larger, the \(\delta_{a}\) becomes larger and then smaller. This is because when the tilt angle is small, the \(F_{af}\) plays a dominant role; when the tilt angle is large, the \(F_{ar}\) plays a dominant role. The \(\delta_{a}\) at each \(\beta\) does not differ much when \(\theta\) is small, which is due to the fact that at a small \(\theta\), the variation in the individual forces is small and, therefore, does not have much effect on the width of the fluid flow. With the gradual increase in \(\theta\), the \(\delta_{a}\) at a high \(\beta\) is larger than that at a low \(\beta\), which is because the light fluid flow is dominated by \(F_{af}\) at a high \(\beta\). It is easier for the light fluid to flow towards the centre of the pipe, resulting in the widening of the width of the light fluid at this time. We found that as a light fluid rises, it does not occupy the centre of the pipe and flow upward. This causes the forces on both sides of the light fluid to be asymmetrical. This is very difficult to capture. It is currently not possible to set boundary conditions for this situation when considering the theoretical perspective. Thus, in our theoretical model, we have tried to simplify it. We assume that the rising fluid occupies the centre of the pipe at all tilt angles, making the forces on both sides of the light fluid symmetrical. This allowed us to derive a velocity model for the rising fluid. Furthermore, we collected the ratio of the width of the light fluid to the diameter of the pipe (\(\delta_{a}\)) and recorded it in Table 3, which will be used as a parameter to calculate the velocity for our later theoretical model. We found that the \(\delta_{a}\) increases and then decreases as the tilt angle (\(\theta\)) increases. This variation in the \(\delta_{a}\) is different from the variation in the velocities of the light fluids that we have collected. Of course, such a simplification introduces a certain amount of error, and we also explain the main reason for this error later during the discussion of the theoretical model.