## 6 Development Length of a gPTT Fluid
### Creeping Flow
#### 6.2.2 Case of \(\mathcal{L}_{98\%}\)  
In addition to the error arising at the entrance corner due to a singularity, we also have the problem of the development length, which takes into account \(99\%\) of the fully developed maximum velocity and may not work so well in an intermediate mesh as \(M_{2}\), for higher \(Wi\). This leads to an increased difficulty for the numerical method to capture \(\mathcal{L}_{99\%}\) for the mesh \(M_{2}\). Therefore, to capture the essence of the development length for higher \(Wi\) values, we considered another development length, \(\mathcal{L}_{98\%}\) (length from channel entry required for the velocity to reach \(98\%\) of its fully-developed value), which is less restrictive. The results obtained for \(\mathcal{L}_{98\%}\) are shown in Figure 9 for mesh \(M_{2}\) and Table 2 for the three meshes (along with the extrapolated development length value). Figure 8: Development length as a function of \(Wi\) considering \(99\%\)\(U_{max}\) (\(\mathcal{L}_{99\%}\)) and mesh \(M_{2}\). The results obtained follow the same trend observed in the case \(\mathcal{L}_{99\%}\), with some minor differences. The development length is initially higher for the case \(\alpha=\beta=0.5\) than for the case \(\alpha=\beta=1\), until \(Wi=0.2\), where the growth rate of \(\mathcal{L}_{99\%}\) becomes larger with \(Wi\) for \(\alpha=\beta=1\). For \(\mathcal{L}_{99\%}\), the two development lengths are quite similar. For \(Wi=1\), we obtain a development length of \(2.679\) for \(\alpha=\beta=1.5\) and a development length of \(1.313\) for \(\alpha=1.5,\ \beta=0.5\) (and \(1.368\) for \(\alpha=\beta=0.5\)). Again, these results are consistent with the idea that the higher the rate of destruction of junctions, the smaller the development length (information travels faster across the channel). Figure 10-12 show the different velocity profiles obtained at 10 different sections of the channel. The first numerical velocity profile is taken at \(x/H=0.1\) (\(x/L=0.01\)) and the last profile is taken at the middle of the channel (\(x/H=5\) or \(x/L=0.5\)). Figure 10: Velocity profiles obtained at 10 different sections of the channel for \(Wi=1.0\), \(\alpha=1\), \(\beta=1\). Figure 9: Development length as a function of \(Wi\) considering \(98\%\)\(U_{max}\) (\(\mathcal{L}_{98\%}\)) and mesh \(M_{2}\). Figure 10 shows the reference velocity profiles obtained for the classical case of an exponential PTT model (\(\alpha=\beta=1\)). The velocity profile evolves from a plug profile in the center of the channel (for profiles near the inlet) to the typical parabolic profile (in the fully developed region). Note the overshoots near the walls that occur when the fluid is still developing. This is due to the different characteristic times of the fluid and the diffusion of information moving from the walls (\(y/H=0\) and \(y/H=1\)) to the center of the channel (\(y/H=0.5\)). Figure 11 shows the velocity profiles obtained for \(\alpha=0.5\), \(\beta=0.5\) and \(\alpha=0.5\), \(\beta=1.5\). It can be seen that the overshoots are stronger near the inlet (compared to the exponential PTT model) and that a lower maximum velocity is obtained when the fluid is fully developed. The influence of the parameter \(\beta\) is residual. Figure 12 shows the velocity profiles obtained for \(\alpha=1.5,\beta=0.5\) and \(\alpha=1.5\), \(\beta=1.5\). In Figure 12a it can be seen that the overshoots near the inlet resemble the case of the exponential PTT model, and that the maximum velocity increases again. The main  
Figure 11: Velocity profiles obtained at 10 different sections of the channel for \(Wi=1.0\). (**a**) \(\alpha=0.5\), \(\beta=0.5\); (**b**) \(\alpha=0.5\), \(\beta=1.5\). Figure 12: Velocity profiles obtained at 10 different sections of the channel for \(Wi=1.0\). (**a**) \(\alpha=1.5\), \(\beta=0.5\); (**b**) \(\alpha=1.5\), \(\beta=1.5\). difference is that the plug profile is now less pronounced. When \(\beta\) increases (Figure 12b), one can observe a dramatic change in the evolution of the velocity profiles.