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

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.

Sampling Points-Independent Identification of the Fractional Maxwell Model of Viscoelastic Materials Based on Stress Relaxation Experiment Data

“…is derived based on the spectral representation given by Equation (21). Differentiation of (21) on both sides with respect to τ r yields…”

“…To prove the boundness of the partial derivatives with respect to the relaxation time parameter τ r and the orders α and β of the stress and strain derivatives, the spectral representation (21) of the FMM will be applied together with the property concerning the absolute boundness, uniform on the set T × G 1 , of some definite integrals, which results from the following known [62] property concerning the absolute integrability of the product of absolutely integrable and bounded functions.…”

“…Describing the rheological properties of polymers by the FMM [18,19] is well known. However, the FMM was also applied, for example, for computational modeling and analysis on the damping and vibrational behaviors of viscoelastic composite structures [20], viscoelastic flow in a circular pipe [21], effect of temperature on the dynamic properties of mixed surfactant adsorbed layers at the water/hexane interface [22,23] and constitutive equations of the Mn-Cu damping alloy [24]. Fractional viscoelasticity described by the Maxwell model turned out to model both exponential and non-exponential relaxation phenomena in real materials.…”

Fluid flow during phase transition: From viscous fluid to viscoelastic solid via variable-order calculus

“…63 Since their advent, fractional-and variable-order models of viscoelastic response have also been used to analyze flows of viscoelastic/ polymer fluids under various circumstances. Among those who utilized analytical approaches to solve problems of isothermal nonreacting flows were Wenchang and Mingyu and others in their works, 64,65 Shah and Qi, 66 Ming et al, 67 Sun et al, 68 Gritsenko and Paoli in their works, 69,70 and Letelier and Stockle. 71 Those who utilized numerical approaches were to solve problems of the same kind were Qiao et al, 72 Mosavi et al, 73 Chen et al, 74 while Ferras et al utilized both theoretical and numerical methods in their work.…”