# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 3 Results and Discussion
### Stress Profiles for Fractional Burgers Model
In fact, divided by \(E\), these are complex compliances, \(J^{*}(\omega)\) (\(\xi^{2}=\rho\omega^{2}J^{*}(\omega)\)), as they were defined in literature (see, for example, [22] for fractional Maxwell and fractional Zener models). As \(\xi\) enters both velocity and stress profiles, it delivers deeper physical meaning and establishes its role in fluid flow characterization. Moreover, with complex compliance introduced, other physical quantities can be readily obtained, including creep compliance, complex, and relaxation moduli. All these quantities are routinely measured in experiments for material properties characterization. Fractional orders, their mutual influence, and underlying physical meaning should also be considered. Each fractional order (except for \(\beta\) in fractional Burgers model) varies in a range from 0 to 1. As outlined earlier, it allows to balance material properties between purely elastic and purely viscous. The same is true for all the differences of fractional orders. However, when both elastic and viscous behavior are present, damped oscillations of flow velocity are observed (in nondimensional frequency domain). Velocity profiles for models considered in this study are not symmetrical with respect to fractional orders. To better understand this phenomenon, let us look again at velocity profiles for fractional Maxwell model. Why does profile behavior in frequency domain differ in animationsmuz1 [[https://youtu.be/wNkkfdEOS4Q](https://youtu.be/wNkkfdEOS4Q)]([https://youtu.be/wNkkfdEOS4Q](https://youtu.be/wNkkfdEOS4Q)) and muz2 [[https://youtu.be/2](https://youtu.be/2) dWZ6 KNwL5I? ]([https://youtu.be/2](https://youtu.be/2) dWZ6 KNwL5I?) In animation muz1 an initial situation (\(\alpha=\beta=0.2\)) results in \(\alpha-\beta=0\) in the left-hand side of the constitutive equation. At the same time, in the right-hand side \(\alpha=0.2\). Consequently, damped (\(\alpha\
eq 0\)) oscillations are observed. When \(\alpha\) increases, \(\alpha-\beta\
eq 0\). It starts contributing in oscillatory behavior of the flow. That is why damped oscillations are present for \(\alpha=1\). Now examine animation muz2. At the very beginning we have \(\alpha=1\) and \(\alpha-\beta=0.8\). When \(\beta\) starts increasing, oscillations in frequency domain gradually disappear. Why? Because we are on the track to pure dashpot that is reached for \(\alpha=\beta=1\) (\(\alpha=1\) in a right-hand side of (7) and \(\alpha-\beta=0\) in a left-hand side of (7)). Finally, if we examine Figure 2, simple-to-complex approach in constructing fractional viscoelastic models can be restored. All the models are constructed via connecting fractional elements in series/ parallel. For instance, fractional Zener model is readily obtained from Maxwell one via adding fractional element in parallel. More complex models can be built in a similar fashion. Fractional Burgers model, for example, can be obtained from fractional Kelvin-Voigt one, as outlined in [1]. Fractional viscoelastic models are principally different from their integer-order counterparts. An increase in the number of fractional elements is made to properly describe the specific class of materials. For integer-order models, adding more elements often serves to increase model accuracy. Thus, fractional viscoelastic models not only help to describe behaviors missed by integer-order ones but also simplify problem solving and material characterization.