# Exchange Flows in Inclined Pipes with Different Viscosity Ratios
## 3 Results and Discussion
### Velocity Changes
#### 3.2.2 Comparison of Experimental and Theoretical Model Velocities for Light Fluids  
In our previous work, we established that the steady state exchange flow of two incompressible Newtonian fluids in a pipe inclination, \(\theta\), from the vertical direction can be described by the Stokes equation for radial co-ordinates [12]:  
\[\mu_{d}\frac{1}{r}\frac{\mathrm{d}}{dr}\Big{(}r\frac{\mathrm{d}u_{d}}{dr}\Big{)} =\frac{dp}{dz}+\rho_{d}g\cos\theta,\quad r\in[\delta,R] \tag{1}\]  
\[\mu_{a}\frac{1}{r}\frac{\mathrm{d}}{dr}\Big{(}r\frac{\mathrm{d}u_{a}}{dr} \Big{)}=\frac{dp}{dz}+\rho_{a}g\cos\theta,r\in[0,\delta] \tag{2}\]  
where \(R\) denotes the radius of the pipe; \(\mu_{a}\) and \(\mu_{d}\) denote the viscosity of the light and heavy fluids; \(u_{a}\) and \(u_{d}\) denote the velocities of the light and heavy fluids; \(\rho_{a}\) and \(\rho_{d}\) denote the densities of the light and heavy fluids; \(g\) denotes the acceleration of gravity; \(\theta\) denotes the angle of inclination of the pipe; and \(\delta\) denotes the radius of the rising fluid. The pressure drop is an unknown constant to be determined. The boundary conditions include no slippage of the pipe wall and the disappearance of radial stresses on the symmetric line at the centre of the pipe, as well as the continuity of velocities and shear stresses at the fluid-fluid interface. We define the dimensionless quantities here:  
\[\hat{r}=\frac{r}{R},\quad\hat{\delta}=\frac{\delta}{R},\quad\hat{u}=\frac{u}{U} \tag{3}\]  
Here, we define \(U=\Delta\rho gR^{2}/\mu_{d}\) and \(\Delta\rho=\rho_{d}-\rho_{a}\) as the density difference. By combining Equations (1) and (2) and removing the superscripts, we obtain the dimensionless velocity equations for rising and heavy fluids:  
\[u_{d}(r)=\frac{p}{4}\Big{(}r^{2}-1\Big{)}-\frac{\delta^{2}}{2}\cos\theta\log r,\quad r\in[\delta,1] \tag{4}\]  
\[u_{a}(r)=M\frac{P-\cos\theta}{4}\Big{(}r^{2}-\delta^{2}\Big{)}+\frac{P}{4}( \delta^{2}-1)-\frac{\delta^{2}}{2}\cos\theta\log\delta,\quad r\in[0,\delta] \tag{5}\]  
\[P=(\frac{dp}{dz}+\rho_{d}g\cos\theta)/(g\Delta\rho),M=\frac{\mu_{d}}{\mu_{a}} \tag{6}\]  
Figure 7: Variation in light fluid velocity (V) with tilt angle (\(\theta\)) for \(\beta\) = [5.46, 9.3, 82.67, 238.66, 467.77]. The rising flux in a closed pipe containing an incompressible fluid must be in perfect balance with the falling flux, and we can use this to obtain an expression for P. Finally, we obtain an expression for the rising velocity by integrating the velocity profile after bringing in the magnitude:  
\[V=(\frac{P}{4}(\delta^{2}-1)-\frac{\delta^{2}}{2}\cos\theta\log\delta-\frac{ \delta^{2}}{8}M(P-\cos\theta))\frac{\Delta\rho gR^{2}}{\mu_{d}} \tag{7}\]  
\[P=\delta^{2}\frac{2(\delta^{2}-1)-M\delta^{2}}{\delta^{4}-1-M\delta^{4}}\cos\theta \tag{8}\]  
The parameters obtained from our experiments are shown in Table 3, and the parameters are brought into Equation (7) to derive the velocities for a comparison with the velocities measured by our experiments. Figure 8a-e show the results of the comparison between the experimentally measured velocities and the model-calculated velocities for the five viscosity ratios (\(\beta\)) at different times. We can find that the experimental velocities for the five \(\beta\) compare well with the theoretical velocities when the pipe is tilted at \(\theta=0^{\circ}\) because, at this time, the fluid in the pipe flows upward in the centre of the pipe, which agrees with the conditions of the model. Once the pipe begins to tilt, the model differs more from the experimental measurements. We observe two main areas where the differences are large. 1. First, of all, we found that in the velocity results of the model, as the pipe is tilted, the velocity becomes smaller and then larger, whereas this did not happen in the velocities measured in our experiments.