# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 3 Results and Discussion
### Stress Profiles for Fractional Zener Model  
We further consider the dynamics of stress profiles for fractional Zener model. We first fix \(\psi=1\) and examine the influence of fractional parameters only. We consider the contribution made by the different parameters in the same order as for velocity profiles, i.e., varying \(\alpha,\beta,\gamma\) and \(\psi\). Let us start from the fractional order \(\alpha\). Corresponding plots are shown in animations zsf1 [[https://youtu.be/SOV4_XRiL1s](https://youtu.be/SOV4_XRiL1s)]([https://youtu.be/SOV4_XRiL1s](https://youtu.be/SOV4_XRiL1s)) and zsf2 [[https://youtu.be/ajgZpM0](https://youtu.be/ajgZpM0) JQS8]([https://youtu.be/ajgZpM0](https://youtu.be/ajgZpM0) JQS8). For the first one we set \(\beta=0.2\), \(\gamma=0.2\) and vary \(\alpha\): \(\alpha=0.2\),..., 1. As previously, profile is symmetric with respect to the pipe centerline. Moreover, it exhibits abruptly decaying aperiodic oscillations in a frequency domain. As \(\alpha\) increases, peak amplitudes of the stress initially reduce, followed by increase for \(\alpha>0.5\). Different from fractional Maxwell model (see animation msf1 [[https://youtu.be/3](https://youtu.be/3) YnnhJUy1 Wg]([https://youtu.be/3](https://youtu.be/3) YnnhJUy1 Wg)), the rate of peak amplitude change appears to be almost the same for \(\alpha<0.5\) and \(\alpha>0.5\). The stress profile for upper limit values of \(\beta\) and \(\gamma\) is shown in animation zsf2 [[https://youtu.be/ajgZpM0](https://youtu.be/ajgZpM0) JQS8]([https://youtu.be/ajgZpM0](https://youtu.be/ajgZpM0) JQS8). It reveals that the stress is independent from the fractional order \(\alpha\) and remains almost the same for the entire frequency domain. Next, we examine the influence of fractional order \(\beta\). More specifically, \(\alpha\) is fixed (\(\alpha=1\)) with \(\beta\) varying (\(\beta=0.2\),..., 1) and limiting values of \(\gamma\). Stress dynamics for \(\gamma=0.2\) is given in animation zsf3 [[https://youtu.be/jSkii6](https://youtu.be/jSkii6) WKFlg]([https://youtu.be/jSkii6](https://youtu.be/jSkii6) WKFlg). Different from the case with varying \(\alpha\), here we observe switching from oscillatory to constant behavior in a frequency domain as \(\beta\) increases. Stress profile for upper limit of \(\gamma\) (\(\gamma=1\)) is shown in animation zsf4 [[https://youtu.be/zpyOc_J-Xnc](https://youtu.be/zpyOc_J-Xnc)]([https://youtu.be/zpyOc_J-Xnc](https://youtu.be/zpyOc_J-Xnc)). It turns out to be very slightly increasing at the pipe walls with increasing \(\beta\) in a frequency domain, while linearly increasing from the pipe center up to its walls. Let us now vary fractional order \(\gamma\). We set \(\alpha=1\); \(\beta=0.2\) (animation zsf5) [[https://youtu.be/M72](https://youtu.be/M72) Hy5s1D_U]([https://youtu.be/M72](https://youtu.be/M72) Hy5s1D_U) or \(\beta=1\) (animation zsf6 [[https://youtu.be/A-yh4](https://youtu.be/A-yh4) HRlXz4]([https://youtu.be/A-yh4](https://youtu.be/A-yh4) HRlXz4)) and vary \(\gamma\): \(\gamma=0.2\),..., 1. Here again we observe a switch from oscillatory to V-shape profile with increasing \(\gamma\) (\(\beta=0.2\)). When \(\beta=1\), V-shape profile is present for the entire range of \(\gamma\). It slightly decays at the pipe walls. However, this decay vanishes with increasing \(\gamma\) to to being completely independent from \(\hat{\omega}\). Finally, we looked at the difference made by varying elastic parameter \(\psi\). We fixed \(\alpha=0.5\), \(\beta=0.2\), vary \(\psi\) (\(\psi=0.01...100\)). The lower limit case (\(\gamma=0.2\)) is given in animation zsf7 [[https://youtu.be/qlLd7](https://youtu.be/qlLd7) LQ6 mtw]([https://youtu.be/qlLd7](https://youtu.be/qlLd7) LQ6 mtw). Switching between oscillatory and V-shape profiles is observed. However, its dynamics is different from the cases of varying fractional orders. More specifically, the peak stress values first increase with increasing \(\psi\). This trend is observed until switching to V-shape profile, when stress amplitudes start gradually decrease.