# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 3 Results and Discussion
### Velocity Profiles for Fractional Zener Model  
Fractional Zener model is described by four parameters, namely three fractional orders \(\alpha,\beta,\gamma\) and elastic parameter \(\psi\) (\(\psi=E_{0}/E\)). Let us first set \(\psi=1\) and examine contributions made by fractional orders \(\alpha,\beta,\gamma\). In the first pair of animations (zuz1 [[https://youtu.be/T8](https://youtu.be/T8) LLEwgiC4k]([https://youtu.be/T8](https://youtu.be/T8) LLEwgiC4k) and zuz2 [[https://youtu.be/dXQCrXTNLQ](https://youtu.be/dXQCrXTNLQ)]([https://youtu.be/dXQCrXTNLQ](https://youtu.be/dXQCrXTNLQ))) we investigate the influence of fractional order \(\alpha\) (\(\alpha=0.2\),..., 1) for two limiting cases of \(\beta\) and \(\gamma\) (\(\beta=\gamma=0.2\) and \(\beta=\gamma=1\), respectively). In both cases we observe parabolic profiles along the pipe diameter with the amplitude increasing when so does nondimensional frequency. However, in case of lower limits for fractional orders \(\beta\) and \(\gamma\), the parabolic profile is followed by complex oscillatory pattern. Moreover, peak values of initial parabolic profile also oscillate. The situation is strikingly different if we consider the upper limit for fractional orders \(\beta\) and \(\gamma\) (\(\beta=\gamma=1\)). Here parabolic profile occupies the entire range of nondimensional frequency considered. Moreover, it is independent from fractional order \(\alpha\). Let us now have a look at the impact of the fractional order \(\beta\). More specifically, we fix fractional orders \(\alpha=1\), \(\gamma=0.2\) (animation zuz3 [[https://youtu.be/bjlLWJxFWy8I](https://youtu.be/bjlLWJxFWy8I)]([https://youtu.be/bjlLWJxFWy8I](https://youtu.be/bjlLWJxFWy8I))) and \(\alpha=1\), \(\gamma=1\) (animation zuz4 [[https://youtu.be/9](https://youtu.be/9) VaIO3 HKW8Y]([https://youtu.be/9](https://youtu.be/9) VaIO3 HKW8Y)), while varying \(\beta\): \(\beta=0.2\),..., 1. Initial velocity profiles look somewhat similar to the previous case. That is, for lower limit value of \(\gamma\) profile is initially parabolic and switches to oscillatory in both space- and frequency domains. For the upper limit of \(\gamma\), increasing parabolic profile along pipe diameter occupies entire frequency domain. However, it all changes when \(\beta\) start increasing. In particular, velocity profile gradually becomes parabolic along pipe diameter for the entire frequency range thus delivering a case of Newtonian fluid (animation zuz3 [[https://youtu.be/bjlLWJxFWy8I](https://youtu.be/bjlLWJxFWy8I)]([https://youtu.be/bjlLWJxFWy8I](https://youtu.be/bjlLWJxFWy8I))). For the upper limit value of \(\gamma\) (animation zuz4 [[https://youtu.be/9](https://youtu.be/9) VaIO3 HKW8Y]([https://youtu.be/9](https://youtu.be/9) VaIO3 HKW8Y)), the amplitudes of parabolic profile gradually decrease (up to roughly 60% from the maximum value) with increasing \(\beta\). Next, we investigate the effect of varying fractional order \(\gamma\). More specifically, we fix \(\alpha=1\) for both lower limit of \(\beta\) (\(\beta=0.2\), animation zuz5 [[https://youtu.be/j8](https://youtu.be/j8) AGHtMGXV8]([https://youtu.be/j8](https://youtu.be/j8) AGHtMGXV8)) and upper limit of \(\beta\) (\(\beta=1\), animation zuz6 [[https://youtu.be/-wJNeA8](https://youtu.be/-wJNeA8) hTXY]([https://youtu.be/-wJNeA8](https://youtu.be/-wJNeA8) hTXY)), while varying \(\gamma\): \(\gamma=0.2\),..., 1. While the plots exhibit some similarities with the case of varying fractional order \(\beta\), their dynamics is different. In particular, the rate of amplitudes decay for both parabolic and oscillating components is much higher. The same can be said when the profile turns to a parabolic for the entire frequency domain. As for the upper limit value of \(\beta\), the amplitudes of parabolic profile also decay relatively faster (up to roughly 35% from the maximum value). Finally, we consider the impact of elastic parameter \(\psi\).