Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries

Abstract

In this work we discuss the connection between classical and fractional viscoelastic Maxwell models, presenting the basic theory supporting these constitutive equations, and establishing some background on the admissibility of the fractional Maxwell model. We then develop a numerical method for the solution of two coupled fractional differential equations (one for the velocity and the other for the stress), that appear in the pure tangential annular flow of fractional viscoelastic fluids. The numerical method is based on finite differences, with the approximation of fractional derivatives of the velocity and stress being inspired by the method proposed by Sun and Wu for the fractional diffusion-wave equation [ Z.Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics 56 (2006) 193-209]. We prove solvability, study numerical convergence of the method, and also discuss the applicability of this method for simulating the rheological response of complex fluids in a real concentric cylinder rheometer. By imposing a torsional step-strain, we observe the different rates of stress relaxation obtained with different values of α and β (the fractional order exponents that regulate the viscoelastic response of the complex fluids).

Development Length of Fluids Modelled by the gPTT Constitutive Differential Equation

“…In order to achieve a closed system of equations, a constitutive equation for the extrastress tensor, σ, is required. Recently, Ferrás et al [27] proposed a new differential model based on the Phan-Thien-Tanner constitutive equation [26] (see also [28]). This new model considers a more general function for the rate of destruction of junctions, the Mittag-Leffler function, where one or two fitting parameters are included, in order to achieve additional fitting flexibility.…”

Identification of the derivative order in fractional differential equations

“…19 To be more precise, consider the spatial cylindrical domain governing the -component of the fluid velocity u (r, t). In Ferras et al, 20 the authors proposed a fractional order model governing the fluid velocity u (r, t) :…”

Recent Advances in Complex Fluids Modeling

“…In Refs. [11,12,17], the beneficial fitting qualities of this constitutive model framework are discussed in detail. Here, we are interested in determining to what extent the properties of the Mittag-Leffler function can be used to improve the fitting quality of differential models, and this will be discussed in the next subsection.…”

“…The model was derived from a Lodge-Yamamoto type of network theory for polymeric fluids, in which the network junctions are not assumed to move strictly as points of the continuum but instead they are allowed a certain effective slip as well as a rate of destruction that depends on the state of stress in the network. Phan-Thien proposed that an exponential function form would be quite adequate to represent the rate of destruction of junctions and in [17] it was shown that the Mittag-Leffler function could improve the quality of model fits to real data by allowing different forms for the rates of destruction.…”