*3.3. Txy Behaviour*

From the results presented in Section 3.1, we observe an unusual behaviour for the extra-stress tensor component *Txy*, as seen in Figures 4 and 5. It was observed that the maximum value of the extra-stress tensor component *Txy* does not occur at the wall with some parameter combinations. Before the semi-analytical solution was obtained, the research group used a high-order numerical simulation to obtain the laminar solution for the LPTT model in a straight channel. The solutions with both methods were compared and agreed with each other. Therefore, the behaviour of the extra-stress tensor *Txy* was double-checked. The behaviour observed here is really from the viscoelastic model and therefore is necessary to investigate which parameters influence it.

It was observed that the Reynolds number and total viscosity (either with more solvent or polymer viscosity in the mixture) do not affect the behaviour of this extra-stress component. The *Txy* component was affected by the parameters *e*, *ξ*, and the Weissenberg number. An investigation was carried out using different values for these parameters and observing how the *Txy* component is affected.

For the simulations performed, the Reynolds number (*Re* = 1000) and *β* (*β* = 0.5) were kept. Figure 11 shows the variation of the *e* parameter considering *ξ* = 0.1 and *Wi* = 8. The values for the parameter *e* considered were: 0.25, 0.50, 0.75, 1.0, 1.25 and 1.5. Figure 11 shows that as the value of *e* decreases, the maximum value of the tensor *Txy* moves towards the channel centre. This shows that a greater opposition to stretching (higher elongational viscosity) influences the maximum value for the component *Txy* to move away from the wall.

**Figure 11.** *e* influence on the wall-normal variation *y* of the extra-stress tensor component *Txy*. Dimensionless parameters: *Re* = 1000, *β* = 0.5, *ξ* = 0.1 and *Wi* = 8.

Figure 12 presents the variation in the *ξ* parameter values considering *e* = 0.25 and *Wi* = 10. The values for the parameter *ξ* considered were: 0.01, 0.05, 0.1, 0.15, and 0.2. From Figure 12 it is possible to observe that as the value of *ξ* increases, the value of the tensor component *Txy* at the wall decreases. As the value of *ξ* increases, the maximum value of the tensor *Txy* moves towards the channel centre. This shows that the normal stress differences combined with high elongational viscosity exhibit a strong influence on this behaviour.

**Figure 12.** *ξ* influence on the wall-normal variation *y* of the extra-stress tensor component *Txy*. Dimensionless parameters: *Re* = 1000, *β* = 0.5, *e* = 0.25 and *Wi* = 10.

For our last investigation, presented in Figure 13, we performed the variation for the Weissenberg number, considering *e* = 0.25 and *ξ* = 0.2. The values for the parameter *Wi* considered were: 3.0, 4.0, 5.0, 6.0, and 7.0. It is possible to observe, from Figure 13, that as the Weissenberg number increases, the maximum value of the tensor component *Txy* moves towards the channel centre.

**Figure 13.** *Wi* influence on the wall-normal variation *y* of the extra-stress tensor component *Txy*. Dimensionless parameters: *Re* = 1000, *β* = 0.5, *e* = 0.25 and *ξ* = 0.2.

From the analysis carried out, it was possible to verify the influence of these parameters on the behaviour of the extra-stress tensor *Txy* component. Parameter values that emphasize this behaviour were chosen for the simulations. These values comprehend low values for *e*, *ξ* close to 0.2, and Weissenberg numbers higher than 1. This behaviour arises from the combination of high elongational viscosity and a high relationship between normal stress differences and high elasticity. The physical combination of these properties causes the maximum value of the extra-stress tensor *Txy* component to move towards the channel centre. The strong interaction between fluid molecules allied with high elongational viscosity and high elasticity can explain this physical behaviour.

It is worth mentioning that this behaviour happens both for the channel and the pipe, although the simulations showed here were performed only for channels.

#### *3.4. Semi-Analytical Method Limits*

Numerical simulations with different parameter values were performed to observe which ones allow the existence of the solution. It was verified that the Reynolds number does not influence the existence of a solution as long as *Re* > 0. However, the other nondimensional parameters showed an influence. To understand which type of influence and which combinations of values are necessary to obtain a valid solution, different simulations were performed, varying the parameters *e*, *ξ* and *β* (Figure 14), and *e*, *ξ* and the Weissenberg number (*Wi*) (Figure 15).

Adopting fixed values for *Re* and *Wi* and varying the values of the other parameters was obtained the Figure 14. The parameters interval adopted was 0 ≤ *e* ≤ 2, 0 ≤ *ξ* ≤ 0.5 and 0.1 ≤ *β* ≤ 0.9. Figure 14 presents the Valid Solution Region (VSR) where it is possible to obtain the solution of the flow for different values of *β*. The line pointed out as *β* = 0.1 shows the limits of a combination (*e*, *ξ*) values where the solution is valid (VSR). The VSR increases with *β*. For smaller values of *β* (higher polymer viscosity in the mixture), the values of *e* and *ξ* cannot be as large as, for example, the value of *β* being 0.9.

**Figure 14.** Valid solution region for different *β* values with *e* and *ξ* - *Re* = 1000 and *Wi* = 3.0.

To obtain Figure 15, values for *Re* and *β* were kept constant. For the parameter *e*, it was considered the interval (0, 0.75). For the *ξ* variation, it was maintained the same variation performed for the Figure 14, and for the Weissenberg number, the values: 1, 2, 3, 5 and 10 were considered. Figure 15 presents the regions for the limitation of obtaining the solutions. It can be observed that, for *Wi* = 1, it is possible to obtain solutions for small values of the parameter *e*, even for values of *ξ* greater than 0.2. On the other hand, as the value of *Wi* increases, it is possible to observe that the solution becomes more sensitive for smaller values of both *e* and *ξ*. All solutions exist for parameter *e* > 0.75.

**Figure 15.** Valid solution region for different *Wi* values with *e* and *ξ* - *Re* = 2000 and *β* = 0.5.

In general, the solution presented in this paper has limitations when considering low values for *β*. This limitation is due to the impossibility, in these cases, of considering a higher elongational viscosity for the LPTT model (low values of *e*) and also a more significant influence of the differences in normal stresses (higher values for the parameter *ξ*). Therefore, in order to obtain solutions considering high elongational viscosity and also a more significant influence of normal stress differences, it is necessary that the values for the parameter *β* are higher than 0.3, as can be observed in Figure 14.

It is worth mentioning that the valid solution region considering channel flow was also verified for pipe flow, and the results remain the same.

#### **4. Conclusions**

This paper presents a semi-analytical method for the laminar steady channel and pipe flows of the LPTT fluid, with elongational and shear parameter variations. For the verification of the proposed semi-analytical method, its results were compared with the Oldroyd-B model analytical solution, and the solution presented by Alves et al. [16] for the LPTT model without solvent viscosity (*β* → 0). The verification results obtained by the semi-analytical method proposed in this work showed a good agreement compared to both analytical solutions used as references.

The results presented explored the effect of the parameters *e* and *ξ* on the velocity profile and the non-Newtonian extra-stress tensor components. From the analysis, it was possible to verify that the parameter *e* reduces the impact of the tensor components on the velocity profile when it is increased for a high Reynolds number.

The parameter *ξ* has the opposite effect on the maximum value of the streamwise velocity component. As the value of this parameter increases, the velocity profile in the middle of the channel (or pipe) decreases. On the other hand, the extra-stress tensor components *Txx* and *Txy* (or *Txr* for pipe flows) decrease (in absolute value) as parameter *ξ* increases. For the extra-stress tensor component *Tyy* (or *Trr*), its absolute value increases with the parameter *ξ*. The solution for the simplified LPTT model, with *ξ* = 0, for this tensor component is zero.

Another interesting behaviour was observed for the extra-stress tensor component *Txy* (or *Txr* for pipe flows). Its maximum value moves towards the channel centre with a specific combination of the parameters *e*, *ξ*, and the Weissenberg number. It was observed that the combination of high elongational viscosity, the high relationship between normal stress differences, and high elasticity could be responsible for this behaviour.

It was explored for which values of the parameters are present in the flow, it is possible to obtain the solution. In other words, the limitations of the presented solution were explored.

**Author Contributions:** methodology, M.T.d.A., L.F., A.B. and L.S.; investigation, M.T.d.A., L.F., A.B. and L.S.; writing—original draft preparation, M.T.d.A., L.F., A.B. and L.S.; writing—review and editing, M.T.d.A., L.F., A.B. and L.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** Research was carried out using the computational resources of the Center for Mathematical Sciences Applied to Industry (CeMEAI), funded by FAPESP grant 2013/07375-0.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** M.T.d.A., L.F. and L.S. acknowledge the Department of Applied Mathematics and Statistics–University of Sao Paulo, Sao Carlos. A.B. acknowledges the Department of Mathematics and Computer Science–Sao Paulo State University, Presidente Prudente.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

