# A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows
## 4 Results
### Die swell  
The die swell effect is a phenomenon that occurs for the free surface flows of viscoelastic fluids flowing in a pipe or channel and exiting through circular or slit dies. [PERSON] et al. [47] described the origin of the die swell effect by viewing the viscoelastic fluid as a bundle of elastic threads. In the channel flow, the threads are stretched by the normal stress component \(\tau_{xx}\), where \(x\) is the flow direction. When the fluid exits the channel, the threads are allowed to relax and shorten in length. Consequently, the diameter of the emerging fluid increases. Flows that exhibit the die swell effect are commonly used to benchmark and validate numerical methods [11; 13; 14; 16; 17; 18; 19; 20; 22; 24]. In addition to the free surface, the flow has extensional characteristics around the channel exit as well as a stress singularity at the channel exit corner. These flow features make the flow suitable for testing robustness and accuracy of new numerical methods. Therefore, in this work, a planar die swell flow is simulated. The case parameters are selected to allow for a meaningful comparison with numerical data from the literature. A schematic description of the computational domain can be found in Figure 8. A channel of height \(2h\) and length \(10h\) is considered. A symmetry boundary condition is used at \(y=0\), such that effectively half of the domain is simulated. The expansion zone is \(5h\) high and \(12h\) long, giving the domain a total length of \(22h\). The length of the channel and size of the expansion zone are chosen to be sufficiently large for boundary effects not to affect the flow near the channel exit. The exterior boundaries of the expansion zones are treated as outlets with the pressure condition \(p=0\), and an immersed boundary is used to impose the no-slip condition for the channel walls. Gravity is neglected. The viscoelastic fluid in the channel is an Oldroyd-B fluid with the viscosity ratio \(\beta=1/9\). The surrounding fluid is a Newtonian fluid with much lower viscosity and density than the viscoelastic fluid. In this work, these parameters are set to \(\mu_{\text{air}}=10^{-6}\mu\) and \(\rho_{\text{air}}=10^{-3}\rho\), respectively. [PERSON] et al. used a similar definition [24], but with \(\rho_{\text{air}}=10^{-2}\rho\). The simulation of the flow starts from rest, with the channel filled with viscoelastic fluid. Transient flow is simulated for a sufficiently long physical time for the flow in the channel and the expansion region to fully develop, as well as for the free surface flow to exit through the outlet with a uniform velocity profile. At the inlet, fully developed flows are imposed for the velocity and viscoelastic stress. For fully developed channel flow of an Oldroyd-B fluid, it can be shown that  
\[u(y)=\frac{3}{2}U\bigg{(}1-\frac{y^{2}}{h^{2}}\bigg{)}, \tag{26}\]  
\[\frac{\partial u}{\partial y}=-\frac{3 Uy}{h^{2}}, \tag{27}\]  
Figure 8: Die swell domain. \[\tau_{xy}(y)=\eta\frac{\partial u}{\partial y}, \tag{28}\]  
\[\tau_{xx}(y)=2\lambda\tau_{xy}(y)\frac{\partial u}{\partial y}=2\lambda\eta\left( \frac{\partial u}{\partial y}\right)^{2}, \tag{29}\]  
where \(U\) is the mean velocity. A derivation of the expressions can be found in Appendix A. The channel half height \(h\) and velocity \(U\) are taken as characteristic length and velocity scales for the flow. Thus, a Weissenberg number for the flow may be defined as  
\[\text{Wi}=\frac{\lambda U}{h}, \tag{30}\]  
and a Reynolds number as  
\[\text{Re}=\frac{2\rho Uh}{\mu+\eta}. \tag{31}\]  
A dimensionless time is also defined as \(t^{*}=tU/h\). The swell ratio \(S_{r}=h_{\text{max}}/h\) is the main quantity of interest, in terms of validation by comparison. The swell ratio is a function of the so-called recoverable shear \(S_{R}\), which is defined as [11]  
\[S_{R}=\left|\frac{N_{1}}{2\tau_{xy}}\right|_{y=h}=\left|\frac{\tau_{xx}-\tau_ {yy}}{\tau_{xy}}\right|_{y=h}=\frac{3\lambda U}{h}=3\text{Wi}, \tag{32}\]  
where \(N_{1}=\tau_{xx}-\tau_{yy}\) is the first normal stress difference.