# X-ray Particle Tracking Velocimetry in an Overflowing Foam
## 3 Results
### Horizontal Velocity Component at the Weir  
The tracer particles were carried by the foam flow and followed its streamlines sufficiently well. This essential conclusion can be drawn from X-ray images with the entire field of view, as exemplarily shown in Figure 7 for three tracer particles in one measurement run: the motion paths from the starting positions towards the weir are running very smooth. Therefore, we assume in the following that the velocity of each tracer particle along its motion path corresponds to the local flow velocity with negligible uncertainty. Figure 8 plots the short segments of the motion paths \(y^{i}\) (\(x^{i}\)) at the horizontal weir (\(-5\) mm \(<x<+5\) mm) as well as the corresponding horizontal velocity component \(v_{x}\left(x\right)\) for a total of 21 tracer particles, recorded in repeated measurement runs at the surfactant concentration \(c_{1}=1.0\) g L\({}^{-1}\). All of these tracers started at slightly different distances in the horizontal (\(x_{0}\)) and vertical (\(y_{0}\)) directions from the weir but in the same central measuring plane (\(z_{0}=0\) mm). As the expected flow profile was similar to a plug-flow, we assume that the tracer particles did not move significantly along the X-ray beam direction (\(v_{z}\approx 0\) mm s\({}^{-1}\)) and stayed very close to this measuring plane (\(z\approx 0\) mm) when approaching and passing the weir. The tracer particles passed the weir at different heights between \(y\approx 1\) mm and a maximum of 10 mm. On the one hand, the motion paths of tracer particles moving close to the weir crest (\(y\lesssim 2\) mm) run almost horizontally. On the other hand, with increasing height \(y\), the motion paths are characterised by a parabola-like shape at a higher velocity level on average. The horizontal velocity component \(v_{x}\) of the different tracer particles is between approximately 30 mm s\({}^{-1}\) and a maximum of 100 mm s\({}^{-1}\). Most tracer particles, particularly those with parabola-like motion paths, are slightly accelerated as they pass the weir. The strong fluctuations in \(v_{x}\) of the tracer particle having the lowest height at the weir (\(y\approx 1\) mm, indicated by the olive-green star symbol in Figure 8) are most likely due to the fact that this tracer particle slightly touched the weir crest several times. Figure 7: Tracking of tracer particles carried by the overflowing foam in different measuring planes \(z\) of the three-dimensional configuration: using small tubes, three tracer particles were added into the foam flow at different but fixed starting positions \(\left(x_{0},y_{0},z_{0}\right)\) in each measurement run in order to distinguish them from each other. The X-ray image, here a minimum intensity projection over time, visualises the motions paths of the three tracer particles during one measurement run. The focus was on the horizontal weir of the foam flow channel. As indicated in the three-dimensional sketch, the side walls at \(z=\pm 50\) mm were transparent for both visible light and X-rays to a large extent. To emphasise the correlation between the height or vertical position \(y\) above the weir and the horizontal velocity component \(v_{x}\) along the motion paths from Figure 8, they are plotted against each other in Figure 9, and additionally the region of interest is split into three segments in the horizontal direction. The correlations suggested by the two plots in Figure 8 become very clear from Figure 9: in general, the horizontal velocity component \(v_{x}\) increases in the horizontal direction \(x\) while passing the weir as well as in the vertical direction \(y\) above the weir crest. However, the velocity gradient \(dv_{x}/dy\) in the vertical direction is much more pronounced than \(dv_{x}/dx\) in the horizontal direction. Measurement data at \(y\gtrsim 0\) mm, i.e., near the weir crest, are lacking, as the plotted data points represent the centre of mass of each tracer particle.