# The Role of Elasticity in the Vortex Formation in Polymeric Flow around a Sharp Bend
## 3 Results
### Polymer Flow
#### 3.2.1 Inertial Vortices  
The vortices in Figure 6 correspond to flows at different Reynolds numbers for the same Weissenberg number of 1.5 and a viscosity ratio of 0.1. Larger downstream vortices are predicted as the Reynolds number is increased. With both Wi and \(\beta\) held constant, it should be this variation in Reynolds number that is accountable for the increase in vortex size. It should be noted that increasing the Reynolds number while holding both Wi and \(\beta\) constant is equivalent to reducing the elasticity number. Figure 5: Non-dimensional inertial vortex length for Newtonian flow. The dashed line (logarithmic regression) is added to highlight the trend of the Newtonian vortex length solution. Figure 6: _Cont._It can then be stated that larger Reynolds numbers lead to larger inertial vortices in polymer flow. Conversely, reducing the Reynolds number by increasing the elasticity of the polymer solution should reduce the inertial vortex size. Like Newtonian flow, current predictions indicate that larger Reynolds numbers in polymer flow should lead to larger inertial vortex sizes. This is consistent with laminar Newtonian flow physics, but what role do the polymers play in inertial vortex sizes? A way to evaluate the effect of the polymers on the flow would be to compare a Newtonian inertial vortex to a polymer vortex at the same Reynolds number. Figure 7 compares predicted inertial vortices between Newtonian Flow and Polymer flow at the same Reynolds number but different Weissenberg numbers. The vortex in Figure 7a corresponds to Newtonian flow at a Reynolds number equal to 100. The vortex in Figure 7b corresponds to polymer flow with \(\text{Wi}=0.5\). (El = 0.005). The vortex in Figure 7c corresponds to polymer flow with \(\text{Wi}=1.0\) (El = 0.01). Additionally, the vortex in Figure 7d corresponds to a polymer flow with \(\text{Wi}=1.5\) (El = 0.015). All three polymer inertial vortices are predicted to be comparable in size, within 5%; with the one corresponding to a Weissenberg number of 1.0 being the only one significantly larger than its Newtonian counterpart, but only about 10%, just slightly above the average grid convergence error of 8%. This is an indication that adding polymers to the flow at this high Reynolds number is predicted to have a slight to nonexistent effect on the fluid dynamics of the flow around a bend. The Weissenberg number effect on inertial vortices at high Reynolds numbers is predicted to only be moderate. Figure 6: Downstream inertial vortices, Polymer Flow, \(\text{Wi}=1.5\) and \(\beta=0.1\): (**a**) Re = 50 (El = 0.03); (**b**) Re = 60 (El = 0.025); (**c**) Re = 75 (El = 0.02); (**d**) Re = 100 (El = 0.015). Figure 7: _Cont._  
The contour plots in Figure 8 show two inertial vortices both corresponding to a moderate Reynolds number of 50 also with \(\beta=0.1\). The first vortex, Figure 8a, has a Weissenberg number of 0.5, while the second, Figure 8b, corresponds to \(\text{Wi}=1.0\). Again, these inertial polymer vortices are comparable in size, in this case within 2%, indicating that the influence of the polymer in the size of the inertial vortex is also predicted to be moderate at a moderate Reynolds number. Figure 9 summarizes all inertial vortex length predictions for polymer flow at high Reynolds number. Both solutions for Newtonian flow, RheoTool and Fluent, are added for comparison. Predictions for Wi equal to 0.5 and 1.5 are similar to that of Newtonian Flow. At Wi = 1.5, polymer inertial vortices are predicted to be slightly smaller than the Newtonian counterpart as the Reynolds number is reduced. Figure 8: Downstream inertial vortices, \(\text{Re}=50\). Effect of Weissenberg Number (Polymer flow): (**a**) \(\text{Wi}=0.5\) and \(\text{El}=0.01\); (**b**) \(\text{Wi}=1.0\) and \(\text{El}=0.02\).