# Exchange Flows in Inclined Pipes with Different Viscosity Ratios
## 4 Conclusions  
In this work, we consider the effect of viscosity ratio (\(\beta\)) and inclination angle (\(\theta\)) on the exchange flow of two fluids occurring inside a pipe. We found that when these two factors are changed, the flow regimes, as well as the velocities of the two fluids inside the pipe, are changed. The main findings are summarized below. Under different \(\beta\), after tilting the pipeline to a certain angle, we find that the light fluid will appear in a total of four flow regimes: CAF, ECAF, TSBS, and SBS. With the  
Figure 8: Comparison of experimentally measured and theoretical velocities with \(\theta\) for different \(\beta\). The solid line is the change in experimentally measured velocity, and the dashed line is the change in model-predicted velocity; the \(\beta\) are (**a**) \(\beta\) = 5.46, (**b**) \(\beta\) = 9.3, (**c**) \(\beta\) = 82.67, (**d**) \(\beta\) = 238.66, (**e**) \(\beta\) = 467.77.  
increase in \(\beta\), the critical angle for the flow state to change into a side-by-side flow will become larger. When the tilt angle is greater than \(16^{\circ}\), no matter how large the \(\beta\) is, the flow state of the light fluid is a side-by-side flow at this time. We also find that the width of the rising light fluid to the width of the pipe (\(\delta_{a}\)) increases and then decreases as the \(\theta\) becomes larger. When the \(\beta\) is large, the \(\delta_{a}\) at a high \(\beta\) is larger than at a low \(\beta\). For a range of three \(\beta\). As the pipe is tilted, at a low \(\beta\), the portion flowing away from the wall becomes smaller as the angle of \(\theta\) increases. In the medium \(\beta\) range, as the \(\beta\) increases, the maltose aqueous solution does not produce any more detached maltose aqueous solution at the beginning of the flow. At a high \(\beta\), the flow pattern is essentially the same for all \(\theta\). Finally, we have also studied the change in velocity of the light fluid. It was found that the velocity of the watery fluid first becomes larger and then smaller as the \(\theta\) increases. However, the extent of the velocity change is not significant when \(\beta\) is small, and the velocity change is more obvious when \(\beta\) is large. For a final comparison, regarding the theoretical model we built previously, we found that the comparison is good only when \(\theta=0^{\circ}\), and the difference in the comparison is larger after the pipe is tilted. Our experimental study fills the gap in the literature regarding the effect of the viscosity ratio on the exchange flow of two fluids in an inclined pipe. We can clearly understand the flow state of the two fluids in the pipe. This is very important for further research--for example, in a study of two complex fluids inside a pipe. However, our theoretical model does not compare well with the experimental velocity when the pipe is inclined. However, the comparison results are very reliable when the pipe is vertical. Our theoretical model can predict the fluid flow velocity in a pipe for any parameter. This velocity is not limited to the velocity of an ascending fluid, but also indicates the velocity of a downward-flowing fluid, which requires our model to be extended. We can use this to predict the velocity of fluids in vertical or near-vertical pipes, both in nature and in industrial production. Examples include the velocity of lava eruption in a crater channel and the velocity of oil flow in a submarine oil flow pipeline. Our experiments also provide insight into the design of industrial pipelines. For example, they show how the piping can be set up to help us obtain relatively high efficiency (such as the rate of oil recovery in subsea oil pipelines) or a lower rate of hazards (such as in the exchange of gases in ventilation ducts in the event of a fire). Conceptualization, Z.Q. ; methodology, W.H. ; formal analysis, W.H. and K.H. ; investigation, Z.N. ; resources, Z.Q. ; data curation, Z.Q. and W.H. ; writing--original draft preparation, K.H. ; writing--review and editing, Z.Q., W.H.