# The Role of Elasticity in the Vortex Formation in Polymeric Flow around a Sharp Bend
## 3 Results
### Polymer Flow
#### 3.2.2 Elastic Vortices  
As the Reynolds number is reduced, the downstream or inertial vortices are predicted to disappear. However, as the Reynolds number is reduced further, vortices are predicted to form upstream of the corner bend. We refer to these vortices forming upstream of the bend corner as elastic vortices. At low Reynolds numbers, it is the Weissenberg number that is predicted to determine the size of the elastic vortex. To evaluate how changing the Weissenberg number affects vortices that form upstream of the bend corner, Figure 11 compares an elastic vortex for Wi = 1 and El = 1:10\({}^{6}\) to a vortex for Wi = 1.5 and El = 1:5-10\({}^{6}\). Both simulations correspond to a viscosity ratio of 0.1 and a Reynolds number of 1:10\({}^{-6}\). It is clear that the two vortices are  
Figure 10: Non-dimensional inertial vortex length for polymer flow vs. elasticity. Figure 9: Non-dimensional inertial vortex length for polymer flow vs. Reynolds number. predicted to be of different sizes, with the vortex for Wi = 1.5 being larger. Since the elasticity number and the Weissenberg number are linked, the vortices are predicted to be different in size, most likely due to the difference in Weissenberg numbers. Lastly, we evaluate the effect of the viscosity ratio. As explained previously, this ratio can take any value between 0 and 1. Since a \(\beta\) value of 1 defines a purely Newtonian fluid, we expect that fluids with viscosity ratios less than 1 should exhibit polymer behavior similar to increasing the Weissenberg number in polymer flow. The elastic vortex should therefore be larger for smaller \(\beta\) values because of its reliance on elasticity to form. The elastic vortices in Figure 12 show the effect of decreasing \(\beta\); from \(\beta\) = 0.4 to \(\beta\) = 0.1. The second elastic vortex, lower viscosity ratio, is larger than expected, indeed, significantly larger, roughly three times. Reducing \(\beta\) is predicted to have a similar effect to that of increasing the Weissenberg number, increasing the size of the elastic vortex. However, the presence of elasticity is not sufficient to produce elastic vortices; it was found that elastic vortices are only predicted to form for viscosity ratios not exceeding a certain value. In other words, a dilute polymer solution, mostly solvent with a viscosity ratio near 1, is always predicted to have Newtonian-like flow behavior without a vortex forming upstream of the bend corner. Conversely, a concentrated polymer solution, mostly polymer with a viscosity ratio near 0, is predicted to have secondary flows, vortices upstream of the bend corner, for a large range of Reynolds numbers. These predictions are in line with the experiments of [PERSON] [5] who found that stable elastic vortices form within the flow of semidilute DNA solutions in a 90\({}^{\circ}\) micro bend channel above a certain threshold for the solution elasticity. Figure 12: Upstream elastic vortices, effect of viscosity ratio (polymer flow) for Wi = 1 and El = 1\(\cdot\)10\({}^{6}\); (**a**) elastic vortex for \(\beta\) = 0.4; (**b**) elastic vortex for \(\beta\) = 0.1. Based on these findings we created a map to evaluate the effect of the three parameters, Wi, \(\beta\), and El on the formation of elastic and inertial vortices. Figure 13 shows a phase diagram mapped for viscosity ratio versus elasticity number for Wi = 1.0. This value was chosen so the inverse of the elasticity number, El, is exactly the Reynolds number, Re. Three regions form in this phase diagram: the 'devoid region', no vortex forms; the 'inertia region', downstream or inertial vortices form; and the 'elastic region', upstream or elastic vortices form. As can be seen in the diagram, the boundary between the elastic region and the devoid region is a horizontal line at about \(\beta\) = 0.6. This is interesting because it suggests that there is a critical viscosity ratio needed for the elastic vortex to form.