## 3 Results and Discussion
### Two-Phase Flow Results
#### 3.2.2 Vorticity Fields  
Figure 15 shows the vorticity field near the wall, while Figure 16 shows the vorticity field at the center. In each figure, the bubbles are positioned the same as in the velocity fields. The magnetic particle flow receives a very small effect from the wall due to the external magnetic force.  
Figure 16: Vorticity fields of bubbly flow at the center. The N\({}_{2}\) is located (**a**) above, (**b**) above and below, (**c**) below.  
Figure 14: A comparison of velocity profiles of the slug flow (**a**) near the wall and (**b**) at the center.  
Figure 15: Vorticity fields of bubbly flow near the wall. The N\({}_{2}\) is located (**a**) above, (**b**) middle, (**c**) below.  
Figure 17a shows the vorticity field near the wall below the slug; Figure 17b shows the vorticity field between the slug and the wall; Figure 18a shows the vorticity field of the area at the center with a narrow gap between the slugs; and Figure 18b shows the vorticity field of the area at the center with a wide gap between slugs. Near the wall, the slug vorticity is shown to be lower than the bubbly vorticity. The vorticity at the center is higher than that near the wall. These results are due to the high difference in velocity between the N\({}_{2}\) and water.  
Figure 19a shows the average vorticity near the wall, while Figure 19b shows the average vorticity at the center, taking in the calculated area compared to the bubbly area, the slug, the theoretical result, and the single-phase nonmagnetic particle and magnetic particle vorticity. Near the wall, the bubbly flow, slug flow, and single-phase magnetic particle values were found to be 44.2%, 10% and 2.3%, respectively, compared to the single-phase nonmagnetic particle value. The experimental results of the two-phase and single-phase magnetic particles near the wall had lower values than the theoretical and the single-phase nonmagnetic results. Since the cross section of the flow field width was uniform, the slug flow had a lower vorticity than the bubbly flow, even when the N\({}_{2}\) velocity was high.  
At the center, the bubbly flow, slug flow, and single-phase magnetic particle had values of 23.8%, 70.4% and 3.3%, respectively, compared to the single-phase nonmagnetic particle value. The experimental results of the bubbly flow and the single-phase magnetic particle showed lower values than the theoretical and the nonmagnetic results. The vorticity of the slug flow was higher than that of the  
Figure 17: Vorticity fields of slug flow near the wall. The slug is located (**a**) above, (**b**) beside.  
Figure 18: Vorticity fields of slug flow at the center. The slug is located with a (**a**) narrow (**b**) wide gap.  
bubbly flow. However, all of the values at the center were lower than those near the wall. According to these results, the vorticity was reduced by the external magnetic field near the wall and at the center.  
Figures 20-22 are velocity uncertainty fields. The uncertainty of the measured velocity was analyzed using the method developed by [PERSON] [36]. The uncertainty of the velocity was estimated to be up to 2.6%.  
Figure 19: A comparison of the bubbly flow, slug flow, theoretical, and single-phase nonmagnetic and magnetic vorticity values (**a**) near the wall and (**b**) at the center.  
Figure 20: Uncertainty fields of the bubbly flow near the wall. The N\({}_{2}\) is located (**a**) above, (**b**) middle, (**c**) below.