# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 4 Conclusions  
This study has provided detailed parametric analysis of velocity and stress profiles for three fractional viscoelastic models, namely Maxwell model, Zener model, and Burgers model. These models were chosen to represent two-, three-, and four-element viscoelastic systems. Two types of parameters were considered: fractional and elastic. We have extended 2D projections of velocity and stress profiles studied in [1] and considered 3D dynamic velocity and stress surfaces in space and frequency domains. It allowed better visual representation of quantities studied. Surface plots simplified comparison of contributions made by varying individual fractional orders for each model considered. Dependent on researchers' needs, proper altering of fractional and elastic parameters can be made. Whether achieving of local/global minimum/maximum is required for practical applications, an optimal operating regime can found and visualized. In case the predictions of fractional models should be compared with classical or Newtonian ones, we have provided corresponding constitutive equations. Fractional Maxwell model was found to be dependent on a pair \((\alpha,\beta)\) of fractional parameters only. The model has demonstrated dynamic behavior with fixed one and varying another fractional parameter, switching from a combination of parabolic and oscillatory behaviors (for lower values of fractional parameters) to purely classic parabolic behavior for the velocity profile (for higher values of fractional parameters). The latter took place only when both tended to unity that corresponded to Newtonian fluid. Fractional Zener model was found to be dependent on three fractional parameters and one elastic parameter. Among these four parameters, the strongest impact is made by the elastic one. As for varying fractional orders, the strongest effect (profile switching rate) was caused by \(\gamma\), while the weakest by \(\alpha\). Moreover, for all combinations of upper limits for fractional orders, only parabolic profiles were observed. Fractional parameters were found to be proportional to the \"rate\" of switching between parabolic+oscillating and parabolic profile as well as determining the position of the dominant resonant peak in frequency domain. Fractional Burgers model was defined by four fractional parameters and one elastic parameter. As changing various parameter combinations, several distinct profile types have been obtained. When elastic parameter was fixed, combinations of three fractional parameters set to lower limit with fourth varying resulted in a combination of parabolic and complex resonant (with plateaus) components of varying amplitudes. No switching to purely parabolic profiles was observed. In contrast to it, when three out of four fractional parameters were set to their upper limits, velocity profiles appeared to be purely parabolic with amplitudes decreasing with fractional orders \(\alpha\) and \(\beta\) increasing. At the same time, peak amplitudes grew monotonically in a frequency domain. When fractional parameters \(\gamma\) and \(\delta\) varied, the shape of the parabolic profile changed. More specifically, in a frequency domain it demonstrated a presence of resonant peak followed by decay and finally reaching a nearly constant value. For changing elastic parameter, new types of velocity profiles emerged. In particular, setting fractional orders to lower limits resulted in a series of changing profiles. It started with a combination of parabolic profile, inclined decaying oscillations that were symmetric with respect to pipe centerline. The latter enclosed a plateau. Next, it transformed to parabolic+oscillatory profile. Further increase in \(\psi\) resulted in parabolic profile with resonance at certain \(\tilde{\omega}\). We ended up with a simple parabolic profile with the amplitude increasing with \(\tilde{\omega}\). For upper limit values of fractional parameters, a dynamic combination of parabolic and M-shape profile was obtained.