# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 3 Results and Discussion
### Velocity Profiles for Fractional Burgers Model  
Finally, let us examine the most complex model presented in this study, fractional Burgers model. Here five parameters should be considered, namely four fractional orders \(\alpha,\beta,\gamma,\delta\) and elastic parameter \(\psi\). Here we follow the pattern used above to maintain consistency. In particular, we first fix elastic parameter value and study the influence of fractional parameters on model dynamics and then include varying elastic parameter in consideration. To demonstrate the model to be universal, we picked a different value of elastic parameter, \(\psi=10\), when varying fractional orders \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). It is worth remembering that for fractional Burgers model the range of fractional order \(\beta\) is different (\(1\leq\beta\leq 2\)) from those for fractional Maxwell and Zener model (\(0\leq\beta\leq 1\)). As previously, let us start from a varying fractional order \(\alpha\) (\(\alpha=0.2\),..., 1). We fix \(\beta=1.2\) and consider lower (\(\gamma=0.2\), \(\delta=0.2\)) and upper (\(\gamma=1\), \(\delta=1\)) limits of the fractional orders \(\gamma\) and \(\delta\). Corresponding plots are shown in animations buz1 [[https://youtu.be/tLqceVl3](https://youtu.be/tLqceVl3) ywk]([https://youtu.be/tLqceVl3](https://youtu.be/tLqceVl3) ywk) and buz2 [[https://youtu.be/JlldYSZ-hbk](https://youtu.be/JlldYSZ-hbk)]([https://youtu.be/JlldYSZ-hbk](https://youtu.be/JlldYSZ-hbk)). For lower limit values of fractional orders \(\gamma\) and \(\delta\), velocity profile appears to be independent from the fractional order \(\alpha\). Moreover, profile itself looks different from those fractional Maxwell and Zener models (for varying \(\alpha\)). In particular, as nondimensional frequency increases, the velocity profile changes from parabolic to a resonant with plateaus. This difference, however, can be attributed to the fact that we picked a different value of elastic parameter \(\psi\). When upper limit values of \(\gamma\) and \(\delta\) are considered, velocity profile changes to parabolic with gradually increasing amplitude for the entire frequency domain. As \(\alpha\) increases, no changes in profile shape are observed until \(\alpha\sim 0.6\). Starting from it, peak profile value starts decreasing and reduces by approximately 40%. 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 buz3 [[https://youtu.be/TUZE3](https://youtu.be/TUZE3) HmaUg8]([https://youtu.be/TUZE3](https://youtu.be/TUZE3) HmaUg8). While the initial profile looks similar to the previous case, its dynamics with \(\beta\) differs. More specifically, profile consists of parabolic and resonant profiles. As \(\beta\) increases, peak value of parabolic profile slowly increases. Upon reaching \(\beta\sim 1.8\) it remains almost unchanged. In contrast to it, as shown in animation buz4 [[https://youtu.be/Cp6h-9S-7](https://youtu.be/Cp6h-9S-7) hA]([https://youtu.be/Cp6h-9S-7](https://youtu.be/Cp6h-9S-7) hA), for \(\gamma=\delta=1\), there is only parabolic profile present for the entire frequency domain. Its peak amplitudes decrease gradually with increasing \(\beta\) dropping by approximately 70% from their maximum values. Moreover, as \(\beta\) surpasses a certain value (\(\beta\sim 1.7\)), profile remains almost unchanged up to upper limit (\(\beta=2\)). Now let us have a look at 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 for fractional orders \(\beta\) and \(\delta\). Finally, we vary \(\gamma\) in a range: \(\gamma=0.2\),..., 1. For the lower limit (animation buz5) [[https://youtu.be/X2](https://youtu.be/X2) DWL3 tJDV8]([https://youtu.be/X2](https://youtu.be/X2) DWL3 tJDV8) we start with a combination of parabolic and resonant profiles.