# Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study
## 1 Introduction  
Fractional calculus in general and fractional derivative in particular has been gathering an increasing researchers interest recently. This mathematical tool provides the means for significant improvement of predictive power for numerous practical applications as summarized in a recent literature [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. Heat conduction [5; 18], anomalous diffusion [4; 7], and viscoelastic properties of fluids and solids [3; 9; 10; 13] are just a few examples of fundamental phenomena where fractional calculus finds immediate practical applications and provides promising results. While fractional models in general demonstrate better fit of experimental data compared to their classical counterparts, those involving variable-order fractional operators (VO-FC) are the handy tools for the cutting-edge research. VO-FC not only provides an effective tool to alter the system properties described by the fractional orders but also allows accounting for fractional orders variations with time, coordinate, and internal system-specific parameters.  
One of the fields where fractional calculus has already proven its efficiency is the prediction of dynamics of viscoelastic flows in various constrained geometries, with circular pipe being one of the most popular ones. With theoretical foundations laid in [19], researchers considered numerous specific problems. The models studied here were addressed by [PERSON] in [20] ([PERSON]), by [PERSON] in [21] ([PERSON]) to name a few. Fractional viscoelastic models, including Maxwell and Zener ones, were considered in close detail in series of works by [PERSON], [PERSON], and others [22; 23]. Those authors provided in-depth physical analysis of mathematical concepts introduced via fractional calculus formalism, suggested, developed, and explained various ladder models. This brief literature overview is intended to mention the most recent achievements in a field of FC and to point out studies thatspecifically targeted fractional models studied here. A more detailed consideration that includes historic retrospective can be found in our previous study [1].  
In this paper, we capitalize on the approaches developed in [1] to examine the behavior of fractional viscoelastic Maxwell, Zener, and Burgers models in application to pipe flow. We present an in-depth parametric analysis of these models and outline various operating regimes for each of them. Different from [1], here we consider 3D profiles of flow velocity and stress. It allows for the exploration of flow dynamics in spatial and frequency domains simultaneously and catches the specifics that cannot be easily visible in 2D projections. We consider various combinations of fractional orders, investigate their mutual influence, and establish an approach for viscoelastic flow optimization. The paper is organized as follows. We first provide domain definition. The governing equations for fluid flow and constitutive equations for the model considered are given in Sections 2.1 and 2.2, respectively. The main results of this study appear in Section 3. In particular, Section 3.1 describes the general approach for seeking the solution. Sections 3.2-3.7 present and discuss space-frequency dynamics of velocity and stress profiles. In particular, Section 3.2 presents velocity for fractional Maxwell model. Stress profiles for fractional Maxwell model are considered in Section 3.3. Zener model behavior is studied in Sections 3.4 and 3.5 (velocity and stress, respectively). Sections 3.6 and 3.7 are devoted to fractional Burgers model. The results of this study are summed up in Section 4. Hereafter we use the following notions. Under \"classic\" we understand either model or profile corresponding to integer values of fractional orders. \"Fractional order\" and \"fractional parameter\" is essentially the same. We also use \"frequency domain\" to refer to nondimensional frequency.