# A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows
## 4 Results
### Die swell
In the light of this discussion, the results that are obtained on grid M2 and M3 are in relatively good agreement. This is particularly true far downstream of the channel exit, where the free surface produced by M2 and M3 are very close, while M1 produces a different result. The results indicate that M2 is of sufficient spatial resolution to predict the swell ratio \(S_{r}\). Therefore, the results presented in the remainder of this section have been  
Figure 10: Simulated profiles for \(S_{R}=2.5\) across the channel at \(x=5h\), obtained with grid M1, compared to the analytic solution. Figure 9: Grid M1 used in the grid dependence study, as defined in Table 1.  
obtained while using grid M2. In addition, the simulation with grid M2 has been repeated with half the time step size, such that \(U\Delta t/\Delta x_{\text{min}}=0.005\), which produced equivalent results. The simulations are repeated with grid M2 for \(S_{R}\in\{1.0,1.5,2.0,2.5\}\). In Figure 12, the contour \(\alpha=0.5\) is shown for \(S_{R}=1\) at different times. Figure 13 shows the corresponding results for \(S_{R}=2.5\). Similar behavior is observed in both cases. The emerging viscoelastic fluid increases in diameter directly upon exiting the channel. The magnitude of the increase is strongly influenced by the level of elasticity, as qualitatively predicted by Tanners theory. At a certain distance from the channel exit, approximately for \(x/h>3\), the emerging viscoelastic fluid reaches a terminal diameter. The swell ratio is calculated for the simulations as \(S_{r}=h_{\text{max}}/h\), where \(h_{max}\) is taken as the position of the free surface at \(x/10\). The calculated swell ratios that were obtained with the proposed method are compared to available numerical data from the literature. The data used are the original FEM simulations reported by [PERSON] (2011), the marker-and-cell simulations by [PERSON] et al. (2014), the pseudo-VOF simulation by [PERSON] et al. (2022), and the three different VOF methods used by [PERSON] et al. (2014). Table 2 briefly summarizes the data. For a full description of the cases and numerical methods, the reader is referred to the respective studies. However, a brief discussion concerning the differences between the results is given below. Figure 11: Free surface position in the expansion zone of the die swell geometry, simulated for grids M1, M2, and M3 for \(S_{R}=2.5\). Figure 12: Snapshots of die swell simulation with \(S_{R}=1.0\), interface between viscoelastic phase (green) and Newtonian phase (white) visualized by \(\alpha=0.5\). The calculated swell ratios as a function of the recoverable shear for the flows simulated with the proposed method are compared to the results from the literature in Figure 14. It is remarked that [PERSON] et al. employed a different definition of the Weissenberg number and swell ratio. However, all of the data that are presented in Figure 14 have been adopted to the definition used in the current work. Indeed, a certain spread among the data is observed. The differences are not unexpected, though, as there are slight differences between how the different data have been obtained. This is in terms of the numerical method used as well as the simulation setup. Furthermore, [PERSON] et al. even obtained different results while using three slightly different numerical methods for exactly the same flow. [PERSON] and [PERSON] and [PERSON] et al. did not consider a fluid surrounding the viscoelastic phase, as the per construction of their numerical methods. [PERSON] et al. used a similar definition of the surrounding fluid as in the current work, but with larger density. [PERSON] et al. stated that the surrounding fluid is treated as air with \(\mu_{\text{air}}\to 0\) and \(\rho_{\text{air}}\to 0\), but the magnitudes were not reported. Furthermore, the value of viscosity ratio of the viscoelastic fluid \(\beta\) varied between their simulations. This is in contrast to a constant \(\beta=1/9\), which has been used in the other simulations included in this work.