## 3 Results and Discussion
### Fractional Maxwell Model  
Let us start from the simplest two-parameter fractional Maxwell model. As outlined above, the key quantity that defines the behavior of a model is \(\xi\). For this model \(\xi\) is given by:  
\[\xi^{2}=\frac{\rho\omega^{2}}{E}\Big{[}\frac{1}{(i\omega\tau)^{\alpha}}+\frac{ 1}{(i\omega\tau)^{\beta}}\Big{]}, \tag{24}\]  
or alternatively in nondimensional form:  
\[\bar{\omega}=\hat{\omega}^{2}\Big{[}\frac{1}{(i\bar{\omega})^{\alpha}}+\frac{ 1}{(i\bar{\omega})^{\beta}}\Big{]} \tag{25}\]  
Centerline velocity profiles for fractional Maxwell model are shown in Figure 3. For all the plots \(\alpha=0.5\). The value of \(\bar{a}\) ranges from \(\bar{a}=0.01\) (Figure 3a) to \(\bar{a}=0.1\) (Figure 3c). For this model an additional condition is imposed: \(\alpha\geq\beta\), thus only a half of \(\beta\) range is considered given the value of \(\alpha\) set. As can be seen from this figure, the behavior of centerline velocity changes dramatically from monotonically increasing to resonant or oscillatory. Centerline velocity profile also changes with increasing fractional order (\(\beta\)). For lower values of \(\bar{a}\), an increase of \(\beta\) results in decreasing of centerline velocity up to the constant value (as \(\beta\to 0.5\)). For higher values of \(\bar{a}\) (Figure 3b,c) the system exhibits aperiodic oscillations at low \(\beta\) and demonstrates a switch from resonant behavior to monotonically increasing as \(\beta\) increases. It is also worth mentioning that peak values decrease with increasing \(\beta\). What is the reason for such a dramatic change of a velocity profile dynamics with varying \(\bar{a}^{2}\) In fact, \(\bar{a}^{2}\) turns out to be nothing else but a Reynolds number: \(\bar{a}^{2}=Re\), one of the key flow parameters. Figure 3: Centerline velocity profile for fractional Maxwell model: (**a**) \(\bar{a}=0.01\); (**b**) \(\bar{a}=0.05\); (**c**) \(\bar{a}=0.1\). For all plots \(\alpha=0.5\). Subscript “\(c\)” stands for “centerline”. Now let us have a look at the shear stress dynamics (at the wall) shown in Figure 4. Here again we fix \(\alpha=0.5\) and vary \(\beta\) (\(\alpha\geq\beta\)). Similar to velocity profile, shear stress increases monotonically with \(\tilde{\omega}\) for lower values of \(\tilde{a}\). The trend, however, changes to almost constant with increasing fractional order. As \(\tilde{a}\) increases, the system exhibits more complex behaviors. In particular, there are aperiodic oscillations observed for lower values of fractional order that switch to resonant ones and monotonic decay as the fractional order starts to increase. Moreover, peak values shift to higher values of \(\tilde{\omega}\) with increasing fractional order and the peaks themselves become more dispersive. At the same time, an increase of \(\tilde{a}\) results in resonant peaks becoming sharper and shifting to lower values of \(\tilde{\omega}\). Finally, for relatively high values of \(\tilde{\omega}\) and \(\tilde{a}\), shear stress plots converge and become nearly independent from the fractional order \(\beta\). To better understand physical meaning behind fractional orders \(\alpha\) and \(\beta\), we have examined its influence on centerline velocity and shear stress dynamics in a wider range of \(\tilde{\omega}\) (\(0\leq\tilde{\omega}\leq 1000\)) with the fixed value of \(\tilde{a}\) (\(\tilde{a}=0.1\)) as shown in Figure 5. We have first fixed \(\alpha=0.5\) and varied \(\beta\): \(0.1\leq\beta\leq 0.5\). For both centerline velocity (Figure 5a) and shear stress (Figure 5b) low values of \(\beta\) resulted in oscillatory profiles that in turn reflects the fact that the fractional order \(\beta\) describes elastic properties of the fluid considered. In contrast, if we fix \(\beta=0.5\) and vary \(\alpha\): \(0.6\leq\alpha\leq 1\), the oscillations are damped much quicker for both centerline velocity and shear stress (Figure 5c,d). Finally, let us set \(\alpha=1\) and vary \(\beta\): \(0.25\leq\beta\leq 1\).