## 7 Conclusions  
In this work, we present a numerical study on the development length of a pressure-driven viscoelastic fluid flow (between parallel plates) modelled by the generalised Phan-Thien and Tanner (gPTT) constitutive equation. The governing equations are solved using the finite-difference method, and, a thorough analysis on the effect of the model parameters \(\alpha\) and \(\beta\) is presented. We consider two different definition of the development length: The length from the channel inlet required for the velocity to reach 99% (and 98%) of its fully-developed value. The numerical results showed that the in the creeping flow limit (i.e., \(Re=0\)), the development length for the velocity exhibits a non-monotonic behaviour. The development length increases with \(Wi\), with viscoelastic effects delaying the diffusion and convection of information from the walls to the center of the channel. For low values of \(Wi\), the highest value of the development length is obtained for \(\alpha=\beta=0.5\); for high values of \(Wi\), the highest value of the development length is obtained for \(\alpha=\beta=1.5\). At high \(\alpha\) and \(\beta\) values, the rate of destruction of the junctions is lower than at low \(\alpha\) and \(\beta\) values. This means that when \(Wi\) values are low and the rate of junction destruction is high, information travels slowly from the wall to the center of the channel (compared to when the rate of junction destruction is low). The opposite was expected. Note that in this case, the development lengths are very similar for all tested values of \(\alpha\) and \(\beta\), and therefore the influence of these parameters on the development length is small. These results can be justified by the low value of \(Wi\). As \(Wi\) increases, the highest value of development length is reached with a low rate of destruction of junctions. This result can be justified by the fact that as the rate of destruction of the junctions decreases, the information is transmitted more slowly due to the small number of new contacts between the strands representing the molecules. We also studied the influence of the elasticity number, \(El\), on the development length of the gPTT model. As expected, the results for low values of \(Re\) are consistent with those obtained for creeping flow. For high values of \(\beta\) (1.5), the influence of the elasticity number seems to be neglected by the fluid, since we obtain the same development length for \(El=1\) and \(El=10\). For \(El=0.1\), the results are quite different, which is due to the low values of \(Wi\) compared to the \(El=1\) and \(El=10\) cases. For \(\beta=0.5\), the rate of destruction of junctions increases and the development length decreases by about half. Again, the \(El=1\) and \(El=10\) cases show similar development lengths, which is due to the similar values of \(Wi\) and the almost creeping flow conditions. Figure 13: Influence of the elasticity number, \(El=\frac{Wi}{Re^{2}}\), on the development length of the gPTT model. We consider three different values of \(El\) (\(0.1,\ 1,\ 10\)), \(\alpha=0.5\) and two different values of \(\beta\), \(0.5\) and \(1.5\). **Author Contributions:** Conceptualization, J.B., L.L.F., A.M.A., R.T.L. and A.C.; methodology, J.B., L.L.F., A.M.A., R.T.L. and A.C.; software, J.B., R.T.L. and A.C.; validation, J.B., L.L.F., A.M.A., R.T.L. and A.C.; formal analysis, J.B., L.L.F., A.M.A., R.T.L. and A.C.; investigation, J.B., L.L.F., A.M.A., R.T.L. and A.C.; writing--original draft preparation, J.B., L.L.F., A.M.A., R.T.L. and A.C.; writing--review and editing, J.B., L.L.F., A.M.A., R.T.L. and A.C.; funding acquisition, J.B., L.L.F., A.M.A., R.T.L. and A.C. All authors have read and agreed to the published version of the manuscript.