*Article* **Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: General Solutions**

**Dmitry Gritsenko 1,\* and Roberto Paoli 1,2**


Received: 29 October 2020; Accepted: 2 December 2020; Published: 18 December 2020

**Abstract:** Fractional calculus is a relatively old yet emerging field of mathematics with the widest range of engineering and biomedical applications. Despite being an incredibly powerful tool, it, however, requires promotion in the engineering community. Rheology is undoubtedly one of the fields where fractional calculus has become an integral part of cutting-edge research. There exists extensive literature on the theoretical, experimental, and numerical treatment of various fractional viscoelastic flows in constraint geometries. However, the general theoretical approach that unites several most commonly used models is missing. Here we present exact analytical solutions for fractional viscoelastic flow in a circular pipe. We find velocity profiles and shear stresses for fractional Maxwell, Kelvin–Voigt, Zener, Poynting–Thomson, and Burgers models. The dynamics of these quantities are studied with respect to normalized pipe radius, fractional orders, and elastic moduli ratio. Three different types of behavior are identified: monotonic increase, resonant, and aperiodic oscillations. The models developed are applicable in the widest material range and allow for the alteration of the balance between viscous and elastic properties of the materials.

**Keywords:** Riemann–Liouville fractional derivative; viscoelasticity; pipe flow; fractional Maxwell model; fractional Kelvin–Voigt model; fractional Zener model; fractional Poynting–Thomson model; fractional Burgers model
