# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 3 Results and Discussion
### Velocity Profiles for Fractional Burgers Model
As \(\gamma\) increases, peak value amplitudes for both components gradually decrease. Simultaneously, the profile degenerates to a parabolic for the entire frequency domain. These trends are observed up to \(\gamma\sim 0.6\). Upon passing this value, profile remains parabolic. Its peak amplitude starts slowly increasing up to the limit value (\(\gamma=1\)). When it comes to upper limit ((animation buz6) [[https://youtu.be/hs7](https://youtu.be/hs7) pUv2 nybo]([https://youtu.be/hs7](https://youtu.be/hs7) pUv2 nybo)), dynamic parabolic profile is observed for the entire frequency domain. It, however, has some specifics that distinguishes it from previous cases. That is, for lower values of \(\gamma\), peak values of parabolic profiles approach resonance at a certain \(\tilde{\omega}\), then starts decaying, and finally reaches nearly constant value. The situation, however, changes when \(\gamma\) starts increasing. Peak values of parabolic profile start reducing until peak itself disappears. Then for the entire frequency domain parabolic profile is observed with amplitudes gradually increasing with \(\tilde{\omega}\). The last, but the least fractional order to vary is \(\delta\). Other fractional orders are as follows: \(\alpha=1\); \(\beta=1\), \(\gamma=0.2\) (lower limit) and \(\beta=2\), \(\gamma=1\) (upper limit). Finally, \(\delta=0.2\),..., 1. Lower limit case is given in animation buz7 [[https://youtu.be/wmkBITIXXIOI](https://youtu.be/wmkBITIXXIOI)]([https://youtu.be/wmkBITIXXIOI](https://youtu.be/wmkBITIXXIOI)). Similar to animation buz5 [[https://youtu.be/X2](https://youtu.be/X2) DWL3 tJDV8]([https://youtu.be/X2](https://youtu.be/X2) DWL3 tJDV8), where we varied \(\gamma\), a combination of parabolic and resonant profiles is observed. With \(\delta\) increasing it degenerates to a parabolic profile for the entire frequency range with a resonance at a certain \(\tilde{\omega}\). For the upper limit (animation buz8 [[https://youtu.be/UUQ6](https://youtu.be/UUQ6) tFeV3 yc]([https://youtu.be/UUQ6](https://youtu.be/UUQ6) tFeV3 yc)) theprofile is overall similar to animation buz6 [[https://youtu.be/hs7](https://youtu.be/hs7) pUv2 nybo]([https://youtu.be/hs7](https://youtu.be/hs7) pUv2 nybo), where varying \(\gamma\) was considered. Finally, we consider the impact made by a varying elastic parameter \(\psi\). Here \(\psi\) changes in a range \(\psi=0.1\),..., 100. We start from \(\alpha=1\), \(\beta=1\), \(\gamma=0.2\) and \(\delta=0.2\) (animation buz9 [[https://youtu.be/t_BV2](https://youtu.be/t_BV2) foEg5w]([https://youtu.be/t_BV2](https://youtu.be/t_BV2) foEg5w)). The profile dynamics appears to be the most complex considered so far. For the lowest values of \(\psi\) we have a combination of parabolic profile, inclined decaying oscillations that are symmetric with respect to pipe centerline. The latter enclose a plateau. As \(\psi\) increases, this complex combination transforms to parabolic+oscillatory profile observed for fractional Maxwell and Zener models. Next, it turns to a combination of parabolic and resonant profiles. With further increase in \(\psi\), parabolic profile with resonance at certain \(\bar{\omega}\) emerges. Finally, it becomes a simple parabolic profile with the amplitude increasing with \(\bar{\omega}\). The next set of fractional parameters considered is: \(\alpha=1\), \(\beta=2\), \(\gamma=0.2\), and \(\delta=0.2\) (animation buz10 [[https://youtu.be/Fyel0](https://youtu.be/Fyel0) IDrbU]([https://youtu.be/Fyel0](https://youtu.be/Fyel0) IDrbU)). The variety of the profiles remains the same. However, in this case switching between them occurs slower compared to the previous one. Moreover, relative peak amplitudes turn out to be higher.