# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 2 Problem Formulation
### Constitutive Equations  
First, we consider fractional Maxwell model (Figure 2a). Its constitutive equation reads as:  
\[\sigma(t)+\tau^{\alpha-\beta}{}_{a}D_{t}^{\alpha-\beta}\sigma(t)=E\tau^{\alpha}{} _{a}D_{t}^{\alpha}\varepsilon(t),\quad 0\leq\alpha\leq 1,\quad 0\leq\beta\leq 1, \quad\alpha\geq\beta \tag{2}\]  
Here \(\alpha\) and \(\beta\) are the fractional orders, \(\tau,E\) are given by:  
\[\tau=\Big{(}\frac{E_{1}\tau_{1}^{\alpha}}{E_{2}\tau_{2}^{\beta}}\Big{)}^{\frac {1}{\alpha-\beta}},\quad E=E_{1}\Big{(}\frac{\tau_{1}}{\tau}\Big{)}^{\alpha}, \quad E=E_{2}\Big{(}\frac{\tau_{2}}{\tau}\Big{)}^{\beta}, \tag{3}\]  
where \(E_{1},\tau_{1},E_{2},\tau_{2}\) stand for the Young's moduli, relaxation times of elements \"1\" and \"2\", respectively. The ranges imposed for \(\alpha\) and \(\beta\) have both physical and mathematical meaning. Physically, changing \(\alpha\) and \(\beta\) from 0 to 1 allows balancing between viscous and elastic properties of fluid. Mathematically, all fractional orders should be non-negative, in order to ensure that we deal with fractional derivatives, not fractional integrals. An additional relation bounds \(\alpha\) and \(\beta\), so that the entire range for \(\beta\) can only be considered provided \(\alpha=1\). The limiting integer values of fractional orders deserve special attention, as they allow restoring various classical cases. In particular, \(\alpha=1\) and \(\beta=0\) correspond to the classical Maxwell fluid:  
\[\sigma+\tau\dot{\sigma}=E\tau\dot{\varepsilon}, \tag{4}\]  
while \(\alpha=1\) and \(\beta=1\) results in a constitutive equation for the Newtonian fluid:  
\[\sigma=\frac{E\tau}{2}\dot{\varepsilon} \tag{5}\]  
Here we omitted time dependence of stresses and strains for brevity and introduced dot as a time derivative operator. A closer look at the definition of \(\tau\) reveals that the case \(\alpha=\beta\) is somewhat special. Indeed, it corresponds to a critical gel. It means that \(\tau\) blows up when \(\alpha=\beta\). However, it happens when the base (expression in brackets) is greater than unity. If the opposite is true, then \(\tau\) is bounded. However, the detailed analysis of all the specifics it brings is beyond the scope of this study. Next, we consider fractional Zener model shown in Figure 2b. In this case constitutive equation is:  
\[\sigma(t)+\tau^{\alpha-\beta}{}_{a}D_{t}^{\alpha-\beta}\sigma(t) =E\tau^{\alpha}{}_{a}D_{t}^{\alpha}\varepsilon(t)+E_{0}\tau^{ \gamma}{}_{a}D_{t}^{\gamma}\varepsilon(t)+E_{0}\tau^{\gamma+\alpha-\beta}{}_ {a}D_{t}^{\gamma+\alpha-\beta}\varepsilon(t),\\ 0\leq\alpha\leq 1,\quad 0\leq\beta\leq 1,\quad 0\leq \gamma\leq 1,\quad\alpha-\beta\geq 0,\quad\gamma+\alpha-\beta\geq 0 \tag{6}\]  
Here \(\gamma\) is the fractional order of element \"3\" and \(E_{0}\) is defined as:  
\[E_{0}=E_{3}\Big{(}\frac{\tau_{3}}{\tau}\Big{)}^{\gamma}, \tag{7}\]  
where \(E_{3},\tau_{3}\) stand for [PERSON]'s modulus and relaxation time of element \"3\" and all other notations are the same as above. At a glance, here we end up with two additional relations for the fractional orders \(\alpha\), \(\beta\), and \(\gamma\): \(\alpha-\beta\geq 0\) and \(\gamma+\alpha-\beta\geq 0\). A closer look to this inequalities reveals that if \(\alpha-\beta\geq 0\) is held, the second one is satisfied automatically. That is, of course, only true for non-negative values of the fractional orders. To get classical Zener fluid, one should assume \(\alpha=1\), \(\beta=0\), and \(\gamma=0\).