# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 3 Results and Discussion
### Stress Profiles for Fractional Burgers Model  
Finally, we will have a look at the dynamics of stresses for fractional Burgers model. In picking the values for parameters affecting system behavior, we pursue the logic developed for the velocity profiles. In particular, we first fix the value of the elastic parameter at \(\psi=10\), set limiting values for three out of four fractional parameters and vary the fourth one. Then vary \(\psi\) for various limiting values of fractional orders. Once again, let us start from varying fractional order \(\alpha\) (\(\alpha=0.2\),..., 1). Simultaneously, we set \(\beta=1.2\), \(\gamma=0.2\), \(\delta=0.2\) for lower limit (animation bsf1 [[https://youtu.be/Pa3](https://youtu.be/Pa3) jHar4 Ld8]([https://youtu.be/Pa3](https://youtu.be/Pa3) jHar4 Ld8)) and \(\beta=1.2\), \(\gamma=1\), \(\delta=1\) for the upper limit (animation bsf2 [[https://youtu.be/mTf29](https://youtu.be/mTf29) Vn1 TSQ]([https://youtu.be/mTf29](https://youtu.be/mTf29) Vn1 TSQ)). When compared to 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 fractional Zener model, both similarities and differences can be outlined. For upper limit values, stresses are independent from \(\alpha\) and do not change in frequency domain. For lower limit, plots have similar structure and are independent from \(\alpha\). At the same time, for Zener model major peak is sharper and higher, while for Burgers it has lower peak amplitude and is way more dispersive. Next, we fix \(\alpha=1\) and consider the impact of fractional order \(\beta\). Velocity profile for lower limit of the remaining fractional orders (\(\gamma=0.2\), \(\delta=0.2\)) is given in animation bsf3 [[https://youtu.be/UdAFrW3](https://youtu.be/UdAFrW3) Ut-8]([https://youtu.be/UdAFrW3](https://youtu.be/UdAFrW3) Ut-8). It differs dramatically from a similar plot for fractional Zener model (animation zsf3 [[https://youtu.be/jSkii6](https://youtu.be/jSkii6) WKFlg]([https://youtu.be/jSkii6](https://youtu.be/jSkii6) WKFlg)). In particular, while aperiodic oscillations are also observed in both frequency and space domains, no switching to V-shape profile occurs. Instead, amplitude of major peak slowly increases with increasing \(\beta\). As for the upper limit (\(\gamma=1\), \(\delta=1\)), static V-shape profile is observed. Now let us have a look on the influence made by varying fractional order \(\gamma\). More specifically, we fix \(\alpha=1\), consider lower (\(\beta=1\), \(\delta=0.2\)) and upper (\(\beta=2\), \(\delta=1\)) limits of fractional orders \(\beta\) and \(\delta\). Finally, we vary \(\gamma\) in a range: \(\gamma=0.2\),..., 1. For lower limit, plot is shown in animation bsf5 [[https://youtu.be/eRX9](https://youtu.be/eRX9) wVxeW_Y]([https://youtu.be/eRX9](https://youtu.be/eRX9) wVxeW_Y). It exhibits aperiodic oscillations in both space- and frequency domains. As \(\gamma\) increases, the value of the major peak increases too. Moreover, the peak itself shifts to higher values of \(\bar{\omega}\). When compared to appropriate Zener model (animation zsf5 [[https://youtu.be/M72](https://youtu.be/M72) Hy5s1D_U]([https://youtu.be/M72](https://youtu.be/M72) Hy5s1D_U)), a striking difference can been seen. More specifically, for Zener model, increasing \(\gamma\)results in switching profile from oscillating to V-shape. For the upper limit, plot is given in animation bsf6 [[https://youtu.be/owV0e5](https://youtu.be/owV0e5) Nvc9Q]([https://youtu.be/owV0e5](https://youtu.be/owV0e5) Nvc9Q). As \(\gamma\) increases, it demonstrates switching between oscillatory and V-shape profiles. When compared to appropriate Zener model (animation zsf6 [[https://youtu.be/A-yh4](https://youtu.be/A-yh4) HRlXz4]([https://youtu.be/A-yh4](https://youtu.be/A-yh4) HRlXz4)), the major difference is that for the latter only V-shape profile is present.