# A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows
## 4 Results
### Die swell
In the simulations that are reported in this section, the quantities \(U\), \(\eta\), \(\mu\), and \(\rho\) are constant, while \(S_{R}\) is varied by changing \(\lambda\) between simulations. The Reynolds number is \(\text{Re}=0.5\) for all simulations. [PERSON] [48; 49] developed a theoretical solution for the amplitude of the die swell as a function of the recoverable shear. For an Oldroyd-B channel emerging from a slit the solution reads  
\[S_{r}=0.19+\left(1+\frac{S_{R}{}^{2}}{3}\right)^{\frac{1}{2}}. \tag{33}\]  
The theory does not take the stress singularity at the channel exit corner into account and is, therefore, only expected to yield a good approximation of the swell ratio when the influence of elasticity is small, i.e., for small Wi. Nevertheless, it is often included for comparison in numerical studies of the die swell effect and it is included in this study. The influence of the spatial discretization on the numerical results is assessed through a grid dependence study and the flow is simulated for \(S_{R}=2.5\) with three different grids, denoted as M1, M2, and M3. A grid is defined by a base cell size \(\Delta x_{\text{base}}\) and a set of refinements, as described in Section 3. The grids M1, M2, and M3, respectively, have the base cell sizes \(\Delta x_{\text{base}}=h/5,h/10,h/20\). The largest cells are located in the expansion zone, far from the channel exit. One level of refinement is used in the channel and around the channel opening and two levels of refinement near the exit corner. As an example, grid M1 is shown in \[u(y)=\frac{3}{2}U\bigg{(}1-\frac{y^{2}}{h^{2}}\bigg{)}, \tag{26}\]0 The area around the channel exit corner has been zoomed for clarity. Table 1 summarizes the grids used. A constant time step length is used, such that \(U\Delta t/\Delta x_{\text{min}}=0.01\), where \(\Delta x_{\text{min}}\) is the smallest cell size of the grid. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Grid** & \(h/\Delta x_{\text{base}}\) & \(h/\Delta x_{\text{channel}}\) & \(h/\Delta x_{\text{corner}}\) & **Num. Cells Total.** \\ \hline M1 & 5 & 10 & 20 & 4829 \\ M2 & 10 & 20 & 40 & 19,685 \\ M3 & 20 & 40 & 80 & 78,740 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of grids that are used to study the grid dependence of the die swell simulations. First, the velocity as well as the viscoelastic normal and shear stress are compared to the analytic steady solution inside the channel. In Figure 10, the corresponding profiles obtained using grid M1 across the channel at \(x/h=5\), halfway from the inlet to the channel exit. Already at the lowest grid resolution, the simulated profiles overlap the analytic solution. The results that are obtained with M2 and M3 are visually the same and the figures are, therefore, omitted. The results validate that the constitutive equations are solved correctly and that the immersed boundary method used to impose the boundary conditions at the channel wall works as intended for the solution algorithm for the viscoelastic stresses. In Figure 11, the position of the free surface that is visualized by the contour \(\alpha=0.5\) is shown for \(S_{R}=2.5\) for the grids that are defined in Table 1. Some small surface oscillations are observed in the results from the two finer grids M2 and M3 in the region \(x/h<2\). As reference, [PERSON] et al. [24] reported similar behavior for their corresponding simulations. In their simulations, small self-sustained surface oscillations appeared at a certain level of recoverable shear, in their case \(S_{R}>1.5\) for the Oldroyd-B fluid, and for sufficiently high resolution. Similar observations were also made for simulations with the Giesekus model in their study. The oscillations were damped out further downstream and they did not influence the calculated swell ratio \(h_{\text{max}}/h\). Similar characteristics are observed in the current work, as in Figure 11. [PERSON] et al. attributed the oscillations to numerical difficulties around the free surface due to the nature of the non-sharp interface in the VOF method.