# X-ray Particle Tracking Velocimetry in an Overflowing Foam
## 3 Results
### Horizontal Velocity Component at the Weir
Nevertheless, considering the millimetre-sized tracer particles, we conclude that foam bubbles being in direct contact with the weir also have a horizontal velocity component larger than zero. This was confirmed by supporting optical observation with the naked eye for foam bubbles at the transparent side wall of the foam flow channel. Figure 10 extends Figure 9 by plotting the horizontal velocity component \(v_{x}\) over the vertical position \(y\) in four further measuring planes, which are close (\(z_{0}=\pm 35\) mm) and very close (\(z_{0}=\pm 45\) mm) to the left and right side walls of the foam flow channel, for both surfactant concentrations \(c_{1}=1.0\) g L\({}^{-1}\) (plots on the left) and \(c_{2}=0.5\) g L\({}^{-1}\) (on the right). At the higher surfactant concentration, the findings obtained for the central measuring plane \(z_{0}=0\) mm can be partly transferred to the other planes closer to the side walls. We identified a positive velocity gradient \(dv_{x}/dy\) above the weir for tracer particles starting from \(z_{0}=-35\) mm. Very close to the left side wall (\(z_{0}=-45\) mm), the gradient is less pronounced. For \(z_{0}=+35\) mm and \(z_{0}=+45\) mm, \(v_{x}\) and \(y\) do not correlate clearly, although a symmetrical flow configuration would be expected. In addition to the velocity gradient \(dv_{x}/dy\) within the respective measuring plane \(z_{0}\), Figure 10 reveals a gradient \(dv_{x}/dz\) between the different planes, which is caused by the side walls. While the highest velocities of up to 100 mm s\({}^{-1}\) were measured in the central plane (\(z_{0}=0\) mm), the tracer particles closer to the side walls moved much slower, with less than 50 mm s\({}^{-1}\). The velocity gradient between the two measuring planes close to the wall on each side, i.e., between \(z_{0}=\pm 35\) mm and \(z_{0}=\pm 45\) mm, is smaller and difficult to estimate, particularly as measurement data at \(y>6\) mm in \(z_{0}=\pm 45\) mm are lacking. The plots on the right in Figure 10 show \(v_{x}\) against \(y\) in the five different measuring planes at the lower surfactant concentration \(c_{2}=0.5\) g L\({}^{-1}\). In direct comparison to the higher concentration, the velocity level is generally lower. Particularly in the central plane  
Figure 8: Tracer particles carried by the overflowing foam (\(c_{1}=1.0\) g L\({}^{-1}\)) in the central measuring plane (\(z_{0}=0\) mm) at the weir: along their motion paths, the horizontal velocity component \(v_{x}\) is plotted over the horizontal position \(x\). The same tracer particle is indicated by the same symbol and colour in both plots. (\(z_{0}=0\) mm), \(v_{x}\) is less than 50 mm s\({}^{-1}\). Further, neither a velocity gradient \(dv_{x}/dy\) within the respective measuring plane nor a gradient \(dv_{x}/dz\) between the planes is clearly visible. Very close to the side walls (\(z_{0}=\pm 45\) mm), \(v_{x}\) is in a similar range for both concentrations. All in all, the X-PTV measurements indicate the symmetry of the flow field at the weir with respect to the central measuring plane. However, it should be noted that the velocity of each tracer particle along its motion path represents a snapshot of the overflowing foam, and the accumulated data were acquired in repeated measurement runs. Figure 9: Tracer particles in the central measuring plane (\(z_{0}=0\) mm) at the weir: the horizontal velocity component \(v_{x}\) is plotted against the vertical position \(y\) above the weir. The overall velocity level is slightly increasing, while the liquid foam (\(c_{1}=1.0\) g L\({}^{-1}\)) with the tracer particles approaches the weir crest (\(-5\) mm \(<x<-2\) mm, plot on the top), flows over it (\(-2\) mm \(<x<+2\) mm, plot in the middle), and moves on (\(+2\) mm \(<x<+5\) mm, plot on the bottom). More importantly, \(v_{x}\) increases in vertical direction \(y\) by tendency, emphasised by the grey dashed trend line in each plot. The same tracer particle is indicated by the same symbol and colour in all three plots.