# A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows
## 4 Results
### Planar Poiseuille Flow  
A basic validation of the proposed method is performed for a viscoelastic planar Poiseuille flow. The numerical results for the transient as well as the steady flow are compared to the corresponding analytic solution. The computational domain has height and length \(h\) and it is shown in Figure 4. The upper boundary is treated as a wall with the no-slip condition \(\mathbf{u}=0\) and the lower boundary has a symmetry condition. At the left and right boundaries, the Dirichlet conditions \(p=\Delta p\) and \(p=0\) are imposed, respectively, while the cyclic conditions are used for the velocity and viscoelastic stress. Thus, the numerical model represents an infinitely long channel of height \(2 lt\), which is subjected to a constant pressure drop \(\Delta p\). The simulation starts from rest, with velocity and viscoelastic stress equal to zero. A Weissenberg number for the flow is defined as \(\text{Wi}=\lambda U/h\), where \(U\) is the mean steady flow velocity, and a Reynolds number as \(\text{Re}=2\rho\text{I}h/(\mu+\eta)\). The flow is simulated for \(\beta=1/9,1/18,1/27\) and \(\text{Wi}=0.1,1\). Here, \(U=0.1\,\text{m}/\text{s}\), \(\rho=1\,\text{kg}/\text{m}^{3}\) and \(\eta=1\,\text{Pa}\) are constant, while \(\text{Wi}\) and \(\beta\) are varied, respectively, by changing \(\lambda\) and \(\mu\) between simulations. In order to ensure sufficient temporal resolution, the flow is simulated using different time step lengths. These simulations are performed for the lowest viscosity ratio \(\beta=1/27\), for which the transient variations are the largest. A uniform Eulerian grid with the cell size \(\Delta x=H/20\) is used. In Figure 5, the simulated centerline velocity, i.e., at \(x=h/2\), \(y=0\), obtained with the time step lengths \(10^{-4}\,\text{s},10^{-5}\,\text{s}\) and \(10^{-6}\,\text{s}\), are shown for \(\text{Wi}=0.1\) and \(\text{Wi}=1\). The velocity that is obtained with the longest step length is clearly different from those obtained with the two shorter step lengths. However, the two shorter lengths produce results that practically overlap on the scale of comparison. Thus, the step length \(\Delta t=10^{-5}\,\text{s}\) is considered to be sufficiently small and it is used to obtain the remaining results reported in this section. [PERSON] et al. presented an analytic solution of the transient solution for the flow considered [46]. In Figure 6, the obtained centerline velocities for the \(\beta=1/9,1/18,1/27\) are shown for \(\text{Wi}=0.1\) and \(\text{Wi}=1\) and compared to the analytic solution. The results demonstrate the strong influence of the relationship between the solvent viscosity and polymeric viscosity on the transient flow. The numerical results overlap the analytic solution, showing  
Figure 4: Schematic of the planar Poiseuille flow. Figure 5: Centerline velocity in channel flow simulated using different time steps. that the proposed method treats the transient flow dynamics correctly. Furthermore, the results confirm that the spatial resolution \(\Delta x/h=20\) is sufficient for this case. In order to assess grid dependency in more detail, the flows for \(\beta=1/27\) and \(\text{Wi}=0.1,1\) are simulated until a steady flow is reached while using varying spatial resolution. The flow being steady is ensured through the condition  
\[\frac{||\phi_{n-1}-\phi_{n}||_{L_{2}}}{||\phi_{n}||_{L_{2}}}<10^{-10}, \tag{24}\]  
where \(||\bullet||_{L_{2}}\) denotes the \(L_{2}\)-norm over the cells of the Eulerian grid, \(\phi\) is velocity or viscoelastic stress, and the subscript \(n\) denotes the quantity at the \(n\)th time step. The error measurements with respect to the analytic steady state solution are calculated as  
\[E_{\phi}=\frac{||\phi_{s}-\phi_{a}||_{L_{2}}}{||\phi_{a}||_{L_{2}}}, \tag{25}\]  
where \(\phi_{s}\) and \(\phi_{a}\) denote the simulated and analytic solution, respectively.