## 3 Results and Discussion
### Fractional Burgers Model  
Here we discuss the most complex fractional viscoelastic model, Burgers model. For this model \(\xi\) is given by:  
\[\xi^{2}=\frac{\rho\omega^{2}}{E}\Big{[}\frac{1}{(E_{0}/E)\big{(}(i\omega\tau)^{ \gamma}+(i\omega\tau)^{\delta}\big{)}}+\frac{1}{\big{(}(i\omega\tau)^{\alpha}+( i\omega\tau)^{\beta}\big{)}}\Big{]}, \tag{33}\]  
or alternatively in nondimensional form:  
\[\bar{\omega}=\bar{\omega}^{2}\Big{(}\frac{1}{\psi[(i\bar{\omega})^{\gamma}+( i\bar{\omega})^{\delta}]}+\frac{1}{(i\bar{\omega})^{\alpha}+(i\bar{\omega})^{ \beta}}\Big{)} \tag{34}\]  
It is worth noting that for fractional Burgers model the range of fractional orders differs from all other models considered above. In particular, while the range for orders \(\alpha\), \(\gamma\), and \(\delta\) is still from 0 to 1, \(1\leq\beta\leq 2\). This change affects the dynamics of the centerline velocity profile dramatically. Centerline velocity profiles are shown in Figure 11. We first fixed \(\alpha=\gamma=\delta=0.5\), \(\psi=1\) and varied \(\bar{a}\) and \(\beta\). The corresponding plots are shown in Figure 11a,d,g. The profile appears to be independent from the fractional order \(\beta\). The situation changes when we set \(\beta=1.5\) and vary \(\delta\) (Figure 11b,e,h). Velocity profile is very sensitive to the changes in fractional order \(\delta\) when it has relatively low values. Aperiodic oscillations with reducing peak values for increasing values of \(\delta\) are observed. For varying \(\psi\) (Figure 11 c,f,i), trends are overall similar to that for the fractional Poynting-Thomson model with the peaks for the fractional Burgers model being slightly sharper. This similarity can be attributed to the specific values of the fractional orders \(\alpha\), \(\gamma\) and \(\delta\) set in Figure 11c,f,i (\(\alpha=\gamma=\delta=0.5\)). This specific setting has consequences for constitutive equations in both models. That is, the terms containing fractional derivatives of the orders \(\alpha-\gamma\) and \(\alpha-\delta\) become of the order zero. Finally, let us have a look at the shear stresses shown in Figure 12. As can be seen from it, shear stress, too, is independent from the fractional order \(\beta\) (Figure 12a,d,g). Stress dynamics change in Figure 12b,e,h. Quickly decaying aperiodic oscillations are observed. In particular, major peak values decrease monotonically with increasing value of fractional order \(\delta\). Moreover, these peaks become sharper and shift to lower values of \(\tilde{\omega}\) with increasing \(\tilde{a}\). Next, we fixed \(\alpha=\gamma=\delta=0.5\) and \(\beta=1.5\). Here we observed resonant behavior for all values of \(\tilde{a}\) and quickly damped oscillations for low values of \(\psi\) (Figure 12c,f,i). A closer look at expressions for \(\xi\) corresponding to different models reveals their similarity. To illustrate this statement, let us introduce complex compliance, \(J^{*}(\omega)\). For all the models considered, it has the following functional form: \([...]/E\), where an expression in square brackets is model-specific. Both \(\xi\) and \(\tilde{\omega}\) can be expressed in terms of complex compliance:  
\[\xi^{2}=\rho\omega^{2}J^{*}(\omega),\quad\tilde{\omega}=\tilde{\omega}^{2}EJ^ {*}(\omega), \tag{35}\]  
as it was defined in [39]. Thus, generally speaking, \(\xi\) is a complex quantity. Moreover, as we got complex compliance, other physical quantities including creep compliance, complex modulus, and relaxation modulus can be restored. For instance, \(J^{*}(\omega)=1/G^{*}(\omega)\), where \(G^{*}(\omega)\) stands for complex modulus. Then for fractional Maxwell model the loss modulus, \(G^{\"}(\omega)=\text{Im}(G^{*}(\omega))\), reads as [39]:  
\[G^{\prime\prime}(\omega)=E\frac{(\omega\tau)^{\alpha}\sin(\pi\alpha/2)+( \omega\tau)^{2\alpha-\beta}\sin(\pi\beta/2)}{1+(\omega\tau)^{2(\alpha-\beta)}+ 2(\omega\tau)^{\alpha-\beta}\cos(\pi(\alpha-\beta)/2)} \tag{36}\]  
Another insight can be obtained upon examination of Figure 2b-f and corresponding constitutive equations.