# Exchange Flows in Inclined Pipes with Different Viscosity Ratios
## 3 Results and Discussion  
### Exchange Flow Experiments in Pipes  
Before investigating the flow of two fluids with large differences in viscosity ratios in an inclined pipe, we first examined the flow of two fluids in a vertical pipe. One is to compare the experimental results for the subsequent experiments after tilting the pipe, and the other is to benchmark the experimental results of [PERSON] et al. [1]. The experimental results obtained are shown in Figure 2, where we observe the existence of three different flow regimes for the falling heavy fluid at different ranges of viscosity ratios. The colour plots in Figure 2a show the experiments with viscosity ratios of 5.46 and 9.3 (\(\beta\) = 5.46 and 9.3), respectively, where we observe that the heavy fluid rapidly detaches from the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **\(\rho\) (kg/m\({}^{3}\))** & **\(\mu\) (mpa.s)** \\ \hline Syrup & 1323 & 818.6 \\
80 wt\% glycerol-syrup & 1270 & 150.1 \\ Dil.syrup & 1285 & 88 \\ Dil.syrup & 1224.3 & 9.9 \\ Dil.syrup & 1174.1 & 3.43 \\ v.dil.syrup & 1100 & 1.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental fluid parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Hidden Meaning** & **Unit** \\ \hline \(\mu_{a}\) & Viscosity of rising fluid (light fluid). & mpa.s \\ \(\mu_{d}\) & Viscosity of downward flowing fluids (heavy fluids). & mpa.s \\ \(\rho_{a}\) & Density of light fluids. & kg/m\({}^{3}\) \\ \(\rho_{d}\) & Density of heavy fluids. & kg/m\({}^{3}\) \\ \(\beta\) & Viscosity ratio (ratio of high viscosity to low viscosity). & kg/m\({}^{3}\) \\ \(\Delta\rho\) & Density difference (high density minus low density). & kg/m\({}^{3}\) \\ \(\delta_{a}\) & The ratio of the width of the light fluid to the diameter of the pipe. & kg/m\({}^{3}\) \\ \(\delta_{d}\) & Ratio of minimum width of heavy fluid to pipe diameter. & kg/m\({}^{3}\) \\ \(F_{a}\) & Inertial force on a light fluid. & \\ \(F_{d}\) & Inertial force on heavy fluid. & \\ \(F_{af}\) & Axial inertia forces on light fluids. & \\ \(F_{ar}\) & Radial force on light fluid. & \\ \(F_{df}\) & Axial inertia force on heavy fluid. & \\ \(F_{dr}\) & Radial force on heavy fluid. & \\ \(f_{a}\) & Viscous resistance to light fluids. & \\ \(f_{d}\) & Viscous resistance to heavy fluids. & \\ u & Experimentally measured rate of ascent of a light fluid. & m/s \\ V & Rising velocity of light fluid calculated by theoretical model. & m/s \\ Q & Fluid flow. & m\({}^{3}\)/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The meaning of the symbols that appear in the text and the corresponding units. pipe wall and descends along the centre of the pipe at these two viscosity ratios, forming an approximate mirror image of the lighter fluid, and in Figure 2b, the colour plots show the experiments with viscosity ratios of 82.67 and 238.66 (\(\beta\) = 82.67 and 238.66); at this time, we observe that the heavy fluid eventually separates from the wall of the tube and then splits into small pieces falling along the centre of the tube wall; since the colour contrast of the separated pieces is not obvious, we have depicted the shape of the pieces in the red areas. The colour plot in Figure 2c shows the experiment with a viscosity ratio of 467.77 (\(\beta\) = 467.77), where we find that the heavy fluid still sticks to the wall. At all viscosity ratios (\(\beta\)), the light fluid always flows upward along the centre of the pipe. This is consistent with the experimental results observed by [PERSON] et al. [1] (black-and-white plot in Figure 2). Figure 3 shows the main forces on the light and heavy fluids during flow in an inclined pipe. This facilitated us to subsequently analyse the reasons for the change in the flow pattern. Figure 3a illustrates the forces on the light fluid. The forces affecting the flow of the light fluid are the viscous force \(f_{a}\) and the inertial force \(F_{a}\) (which is the equivalent of the difference between the buoyancy force on the light fluid and its gravity).