# Numerical Simulation of KBKZ Integral Constitutive Equations in Hierarchical Grids
## 4 Results
### Mesh Independence in Channel-Flow  
The numerical method described in Section 3 was applied to simulate the flow of a KBKZ fluid in a 2D planar channel (see Figure 4) of length \(10L\) and height \(L\), where \(L=0.006\,\mathrm{m}\). At the channel entrance, a dimensionless parabolic velocity profile given by \(u(y)=4y(1-y)\) was used. The scaling parameters were the centerline velocity, \(U=0.1\) ms\({}^{-1}\), and the fluid simulated was FLUID S1, whose parameters are described in Table 2. In this flow, we had \(Re=0.34\), \(De=1\), \(\epsilon=0.1\), \(s_{c}=0.1s\) and the number of deformation fields was \(N=100\). In order to verify the mesh convergence of the results, the flow was simulated using several meshes (see Tables 3 and 4 and Figure 5). \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Refined Meshes—Two Levels** \\ \hline
**Meshes** & **Larger \(d_{x}\)** & **Smaller \(d_{x}\)** \\ \hline MRI & 0.125 & 0.0625 \\ MRII & 0.0625 & 0.03125 \\ MRIII & 0.03125 & 0.015625 \\ \hline \multicolumn{3}{c}{**Refined Meshes—Three Levels**} \\ \hline
**Meshes** & **Larger \(d_{x}\)** & **Middle \(d_{x}\)** & **Smaller \(d_{x}\)** \\ \hline MRVI & 0.125 & 0.0625 & 0.03125 \\ MRV & 0.0625 & 0.03125 & 0.015625 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Non-uniform meshes used in the 2D planar channel. Figure 4: Dimensionless representation of the channel domain. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **FLUID S1** \\ \hline  & \(\rho_{0}=801.5\) kg/m\({}^{3}\), \(\alpha=10\), \(\beta=0.7\), \\  & \(\lambda_{ref}=0.06\) s \(\eta_{0}=1.424\) Pa\(\cdot\)s \\ \hline \(k\) & \(\lambda_{k}\) (s) & \(a_{k}\) (Pa) & \(\eta_{k}\) (Pa\(\cdot\)s) \\ \hline
1 & 0.6855 & 0.058352 & 0.0400 \\
2 & 0.1396 & 1.664756 & 0.2324 \\
3 & 0.0389 & 14.560411 & 0.5664 \\
4 & 0.0059 & 99.152542 & 0.5850 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fluid parameters used in this work. Adapted from [15]. Figure 5 shows the non-uniform meshes, where we can see the structure of the mesh. In Figure 5a, the mesh \(MRI\) (with two levels of refinement) is depicted, and in Figure 5b, we can see the mesh \(MRI\) (with three levels of refinement). The \(u\)-profiles are illustrated in Figure 6, where the mesh convergence can be seen. We adopted mesh \(MV\) as a reference mesh (black line) and the solutions of the refined meshes (full symbols) and uniform meshes (empty symbols) are shown. In this figure, we also show the OpenFoam system profile using the mesh MV. We saw good agreement between the solutions obtained in both systems. Our results for the tensor components \(\tau_{xx}\) and \(\tau_{yy}\) are presented in Figure 7, where we can also see good agreement between the numerical solutions of the HiGFlow and the OpenFOAM systems. To verify the convergence, we show the errors (\(L_{1}\), \(L_{2}\), \(L_{\infty}\)) and the orders in Table 5. The errors are calculated using the following equations:  
\[L_{1}=\frac{\sum_{0}^{n}|u(i)^{MV}-u(i)^{*}|}{\sum_{0}^{n}|u(i)^{*}|},\ L_{2}= \sqrt{\frac{\sum_{0}^{n}\left(u(i)^{MV}-u(i)^{*}\right)^{2}}{\sum_{0}^{n}{u(i) ^{*}}^{2}}}\text{ and }L_{\infty}=\frac{max|u(i)^{MV}-u(i)^{*}|}{max|u(i)^{*}|}\]  
where \(u(i)^{MV}\) is the solution in mesh \(MV\) and \(u(i)^{*}\) is the solution in meshes \(MI-MIV\) and \(MRI-MRV\), \(u(i)\) is the \(u_{x}\) profile in points \(i=(x(i),y(i))\) in which \(x(i)=5\) and \(y(i)=i*0.125\), \(i=0,1,\cdots,8\). The orders \(Q\) for uniform meshes \(Q=\frac{log\left(E_{M2}\right)E_{M1}}{log\left(h_{2}\backslash h_{1}\right)}\) show values close to 2 (\(Q\approx 2\)), which is the correct value that we expected to observe, since the velocity is calculated using an implicit Euler method. The values \(E_{M2}\) and \(E_{M1}\) are the errors (in the norms \(L_{1}\), \(L_{2}\) or \(L_{\infty}\)) for two consecutive meshes (the \(d_{x}\) value in \(M_{2}\) is lower than in \(M_{1}\)) and \(h_{2}\) and \(h_{1}\) are the \(d_{x}\) values in their respective meshes.