Longitudinal Flow Dynamics of a Rheological Maxwell Model of a Frac Type Material in a Pipe of Variable Cross Section

1. Introduction

The author learned his first scientific knowledge about the longitudinal and transverse dynamics of deformable ideally elastic media from Professor Dr. and mathematician Danilo P. Raškpvić at exceptional lectures and from his exellant university monographs (see References [1,23]). In particular, the author emphasizes the analogies he spoke about and emphasized the analogous dynamics, which are described by the same mathematical descriptions, and among them the analogies of the dynamics of torsional and longitudinal oscillations of rods (see Reference [4], as well as the basic original and previously published References [5,6,7], of Mihail Petrović, the founder of the Serbian school of mathematics, who completed his studies as the most successful student of generation of mathematics and physics at the University of the Sorbonne and was one of the three doctoral students of Julius Henri Poincaré, also see Reference [8]).

The author later, as a professor of elastodynamics for mechanical engineering students, published References [9,10,11], and one of them contains a chapter on classical models of rheological materials, see Reference [9], as well as on the analytical mechanics of discrete hereditary systems, see Reference [121.

In the 1980s, the Electronic Industry of Niš had in its production program the production of ultrasonic piezoelectric transducers for various purposes. Two magisters of science in the field of electronics, Miodrag Prokić and Dragan Šarković, worked on the construction and testing of piezoelectric transducers, at whose suggestion the author of this paper was engaged as an external consultant at EI Niš, and in the field of mechanical engineering and the theory of oscillations and the theory of elasticity. From this collaboration, a number of References were published [13, 14, 15,1, and among them is the first published Reference [13] on the construction and testing of ultrasonic piezoelectric transducers. The structure of ultrasonic piezoelectric transducers contains a circular piezoelectric plate that is connected to an electrical source and polarized and represents an exciter of ultrasonic longitudinal oscillations in a rod of variable cross-section, which we call a concentrator or amplifier of the amplitude of ultrasonic longitudinal oscillations at the tip of the smallest cross-section of that rod of variable cross-section. On the other hand, there is a so-called metal counterweight of the structure.

Many years later, we published Reference [16], on the properties of fluid sprayers based on excitation of longitudinal oscillations and the operation of ultrasonic piezoelectric transducers, and Dragan Šarković, M.Sc., completed his doctoral dissertation under the mentorship of Vukota Babović and successfully defended it at the Physics Group at the Faculty of Natural Sciences and Mathematics in Kragujevac. From this collaboration, Perić Ljubiša, under the mentorship of the author of this paper, completed and wrote both his magister's science thesis and his doctorate on coupled mechanical and piezoelectric state tensors and defended it at the Faculty of Mechanical Engineering in Niš. Dragan Mančić, under the mentorship of Milan Radmanović, completed and defended his doctoral dissertation at the Faculty of Electronics in Niš.

The very construction of ultrasonic piezoelectric transducers was the inspiration for the study of longitudinal oscillations of rods, by the author of this paper and Aleksandar Filipovski, who later published the following References on these scientific results [17,18,19]. Under the mentorship of the author of this paper, in 1995, Aleksandar Filipovski completed and defended his magister's science thesis, entitled ''Energy analysis of longitudinal oscillations of rods of variable cross-section'', Reference [20], at the Faculty of Mechanical Engineering in Niš, and one of the members of the Masgiter's science thesis defense committee was academician Božidar Vujanović. Much later, in 2002, at the initiative of the author of this paper, they published a co-authored paper Reference [21].

At the invitation of one of the editors, Bilen Emek Abali, the author of this paper, wrote and published in 2020, Reference [22], on Forced Longitudinal Fractional Type Vibrations of a Rod with Variable Cross Section, in the Springer Nature book Developments and Novel Approaches in Nonlinear Solid Body Mechanics.

Four years later, in 2024, the author of this paper, driven by strong scientific inspiration, returned to research in the field of applications of fractional calculus in mechanics, with a sudden and spontaneous scientific inspiration, based on previously acquired knowledge of classical rheological models and carrying in his mind the ideas of applying fractional calculus, and in a very short time achieved new scientific results, which he formulated in References [23,24,25], published in a specialized journal called ''Fracral and Fractional'', and with the exceptional support of the Journal's Editorial Board and especially one of the associate editors. This series of new papers also includes the published Reference [26], as well as Rerenne [27,28].

It is also useful to see the overview References [29,30] for insight into other results related to the content of this manuscript.

In the published References [22,23], new scientific results of theoretical-analytical research of the dynamics of rheological models of fractional type of elasto-viscous material in a rod of variable cross-section are presented. This is a material of fractional type, properties of subsequent elasticity (post-elasticity). The material is a rheological Kelvin-Voigt model of a material of fractional type with properties of elasto-viscosity and subsequent elasticity (post-elasticity), which are described in detail in Reference [26]. These properties lead to the appearance of rheological longitudinal oscillations of fractional type in the rod. We described this motion only by a partial differential equation of fractional order and determined its approximate analytical solutions and corresponding approximate expressions for describing the eigen and forced rheological modes of the elastoviscous rheological fractional type. We considered different boundary conditions and for different cross-sections of a rod of variable cross-section. We derived an ordinary differential equation of fractional order in terms of time functions in each amplitude form of oscillation and gave its approximate analytical solutions, with the accompanying approximate analytical expressions of the eigen and forced modes. We determined the amplitude functions. We graphically displayed the surfaces of the eigen and forced modes as a function of the exponent of the differentiation of the fractional order and time. References [20,22,23,26] contain differential equations of the amplitude functions of longitudinal rheological oscillations of a rod of ideally elastic material, as well as tables of eigenvalues for different cross-sections and different boundary conditions. Detailed descriptions of the amplitude functions of longitudinal oscillations of a rod of ideally elastic material for different cross-sections and different boundary conditions are also given.

In this paper, unlike the previous one, we will present new scientific results of studying the flow (creep) dynamics of the rheological Maxwell model of materials, fractional type, viscoelastic fluids with the property of relaxation of the normal stress of the material, in a pipe of variable cross-section. These properties lead to the appearance of longitudinal flows (creeping) of the rheological Maxwell model of materials of fractional type. We described this flow (creeping) motion only by a partial differential equation of fractional order in terms of normal stresses and determined its approximate analytical solutions and the corresponding approximate analytical expressions for the eigen and forced modes of the normal stress flow (creeping) in terms of the eigen time functions for the corresponding eigen amplitude modes. We considered different possible boundary conditions for the normal yield (creeping) stresses of the rheological Maxwell model of fractional-type materials and for different cross-sections of pipes of variable cross-section.

2. Constitutive Differential Relation of the Fractional Order of the Rheological Maxwell Model of Materials, Fractional Type

In the Maxwell rheological model of materials, fractional type, the following rheological elements are sequentially related:

* The ideally elastic solid Hooke element of the constitutive relation between the normal stress ${\sigma }_z\left(t\right)$ and axial dilatation $\epsilon {}_z\left(t\right)$, in a state of axial strain, has the form:

$${\sigma }_z=E\epsilon {}_z$$

(1)

* an ideally viscous fluid generalized Newtonian element, of fractional type, constitutive differential relations of fractional order of normal stress ${\sigma }_z\left(t\right)$ and ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]$ fractional type of axial dilatation $\epsilon {}_z\left(t\right)$, in the state of axial flow, (creeping) is of the form:

$${\sigma }_{z,\alpha }=E_{\alpha }{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right], 0<\alpha \le 1$$

(2)

where ${\mathrm{D}}_t^{\alpha }\left[•\right]$ is the fractional order differential operator, the exponent of the fractional order of differentiation$\alpha$, on the interval $0<\alpha \le 1$.

In this paper, we will use the following differential operator of non-integer (fractional) order ${\mathrm{D}}_t^{\alpha }\left[•\right]$, which we define with the following derivative and integral:

$${\mathrm{D}}_t^{\alpha }\left[{\epsilon }_z\left(t\right)\right]={\displaystyle \frac{d^{\alpha }{\epsilon }_z\left(t\right)}{d{t}^{\alpha }}}={{\epsilon }_z}^{\left(\alpha \right)}\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(1-\alpha \right)}}{\displaystyle \frac{d}{dt}}{\displaystyle {\displaystyle \underset{0}{\overset{t}{\int }}}\frac{{\epsilon }_z\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha }}}d\tau ,\hspace{1em}for\hspace{1em}0<\alpha \le 1$$

(3)

where $\mathsf{\Gamma }\left(1-\alpha \right)$ is the special Gamma function, which is defined in integral form (see Reference [31]):

$$\mathsf{\Gamma }\left(1-\alpha \right)={\displaystyle {\displaystyle \underset{0}{\overset{+\infty }{\int }}}{e}^{-t}}t{}_{-\alpha }dt,\hspace{1em}1-\alpha >0$$

(4)

or in general form as a function of a variable $x$, in the form:

(5)

The constitutive differential relation of the rheological Maxwell model of the fractional type of viscoelastic material is composed using the sum ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]$ of the fractional type dilation velocities, ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_{z,1}\right]$ and ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_{z,2}\right]$, in the longitudinal direction ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]$ and ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_{z,2}\right]$ is equal to: is:

$${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]={\mathrm{D}}_t^{\alpha }\left[\epsilon {}_{z,1}\right]+{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_{z,2}\right]$$

(6)

That is, how: $\epsilon {}_{z,1}={\displaystyle \frac{{\sigma }_z}{E}}$ follows ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_{z.1}\right]={\displaystyle \frac{1}{E}}{\mathrm{D}}_t^{\alpha }\left[{\sigma }_z\right]$ and ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_{z.2}\right]={\displaystyle \frac{{\sigma }_z}{E_{\alpha }}}$ it follows that heaven is a sum (see Figure 1.a) axial dilatation analysis): ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]={\displaystyle \frac{1}{E}}{\mathrm{D}}_t^{\alpha }\left[{\sigma }_z\right]+{\displaystyle \frac{{\sigma }_z}{E_{\alpha }}}$.

Finally, the constitutive differential relation of the rheological Maxwell model of fractional-type material is in the form:

$${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]={\displaystyle \frac{1}{E}}{\mathrm{D}}_t^{\alpha }\left[{\sigma }_z\right]+{\displaystyle \frac{{\sigma }_z}{E_{\alpha }}}$$

(7)

when the fractional-type rate ${\mathrm{D}}_t^{\alpha }\left[\sigma {}_z\right]$ of change of normal stress approaches to zero ${\mathrm{D}}_t^{\alpha }\left[\sigma {}_z\right]\to 0$, the material described by the modified fractional Maxwell model behaves like a viscoelastic fluid. This is because the deformation, i.e. axial dilation $\epsilon {}_z\left(t\right)$ of the body: ${\sigma }_z\to E_{\alpha }{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]$, increases indefinitely without any additional load. Upon unloading, the deformation in the ideally elastic (Hookean) element fully recovers, while the deformation resulting from the flow (creeping) in the fractional-type viscous (modified Newtonian fluid fractional type) element ${\mathbf{N}}_{\mathsf{\alpha }}$, connected in series, remains unrecovered.

If this material, a the fractional-type modified Maxwell model ${\mathit{M}}_{\alpha }$,, fractional type, is suddenly loaded to some value of normal stress, ${\sigma }_{z,0}$, the corresponding elastic deformation occurs instantaneously in Hooke's ideal elastic element ${\epsilon }_{z,0}={\displaystyle \frac{{\sigma }_{z,0}}{\mathit{E}}}$. This occurs due to the sudden application of load at the very beginning of the observation period, the flow (creeping) behavior of the serially connected fractional-type viscous element (modified Newtonian fluid, fractional type) in the rheological fractional type Maxwell model does not immediately manifest. If the development of deformation (dilatation) is constrained—i.e., if the fractional-type rate of dilatation tends to zero ${\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]\to 0$—then the normal stress becomes a time-dependent function that must be determined.

When the rate of change of the normal stress ${\mathrm{D}}_t^{\alpha }\left[\sigma {}_z\right]$ of the fractional-type modified rheological Maxwell complex assembly ${\mathit{M}}_{\alpha }$ tends to zero ${\mathrm{D}}_t^{\alpha }\left[\sigma {}_z\right]\to 0$, then the normal mechanical stress tends to a value proportional to the rate of dilation of the fractional type ${\sigma }_z\to E_{\alpha }{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]$:

$${\mathrm{D}}_t^{\alpha }\left[\sigma {}_z\right]\to 0$$

(8)

In order to determine the dependence of normal stress on time, when we keep the material model of the modified rheological Maxwell complex model, fractional type ${\mathit{M}}_{\alpha }$, at some constant rate of dilatation, fractional type $\left\{{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]{}_{z,,0}\right\}=const$, we write that:

$${\displaystyle \frac{1}{E}}{\mathrm{D}}_t^{\alpha }\left[{\sigma }_z\right]+{\displaystyle \frac{{\sigma }_z}{E_{\alpha }}}=\left\{{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]{}_{z,,0}\right\}=const$$

(9)

By applying the previous condition (9), we obtain a differential equation of fractional order, which can be solved using the Laplace transform. We then apply the Laplace transform to the functional relationship defined by equation (26)—the fractional-order differential equation. As a result of this transformation, we obtain the following expression: ${\displaystyle \frac{1}{E}}\mathrm{L}\left\{{\mathrm{D}}_t^{\alpha }\left[{\sigma }_z\right]\right\}+{\displaystyle \frac{1}{E_{\alpha }}}\mathrm{L}\left\{{\sigma }_z\right\}=\mathrm{L}\left\{{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]{}_{z,,0}\right\}$.

The approximate analytical solution of the constitutive differential equation (9) of fractional order is in the form (for details of solving the fractional type differential equation (9), see References [23,26]):

$${\sigma }_z\left(t\right)={\mathrm{L}}^{-1}\mathrm{L}\left\{{\sigma }_z\right\}\approx E_{\alpha }\left\{{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\right]{}_{z,,0}\right\}\cdot \left\{1+{\displaystyle \sum _{k=0}^{\propto }{\left(-1\right)}^k{\left(\frac{E_{\alpha }}{E}\right)}^k}{\displaystyle \frac{t^{\left(2-\alpha \right)k+1}}{\mathsf{\Gamma }\left(2k+2-\alpha k\right)}}\right\}$$

(10)

Fig. 1 presents generalized complex rheological Maxwell models of fractional type for ideal materials, incorporating a generalized Newton viscous element of fractional type. In particular, Fig. 1.a illustrates the generalized fractional Maxwell model of a viscoelastic fluid, along with the decomposition and analysis of axial dilatation and normal stress states.

The normal stress relaxation surface is shown, based on the approximate analytical expression (10), representing the time-dependent normal stress response of the rheological Maxwell model, fractional type, in a three-dimensional coordinate system: normal stress ${\sigma }_z$ time t and exponent α of fractional derivative in interval from zero to one, 0<α≤1.

From the previous solution (10), as well as from the surface plot shown in Fig. 1.b, it is evident that the normal stress ${\sigma }_z\left(t\right)$decreases asymptotically over time and tends toward zero, as illustrated in Fig. 1.b. This gradual reduction in normal stress under constant dilatation is known as normal stress relaxation of fractional type. The constitutive differential relation of the fractional order, viscoelastic rheological material - Maxwell's fractional type model is a fractional order differential equation fractional type in the form:

$${\mathrm{D}}_t^{\alpha }\left[{\sigma }_z\left(t\right)\right]+{\displaystyle \frac{E}{E_{\alpha }}}{\sigma }_z\left(t\right)=E{\mathrm{D}}_t^{\alpha }\left[\epsilon {}_z\left(t\right)\right]$$

(11)

by the normal stress along the entire longitudinal direction of the rheological Maxwell model of the fractional type of material. The solution is easily obtained using the Laplace transform and the convolution theorem (for details of solving equation (11) see References [23,26]).

3. Partial Differential Equation of Longitudinal Flow Dynamics of the Rheological Maxwell Model of a Material, Fractional Type, in a Pipe of Variable Cross-Section

Normal force to the cross section of the rheological Maxwell model of material, fractional type, with variable cross section at coordinate $z\in \left(0.\mathcal{l}\right)$ is $N\left(z,t\right)=A\left(z\right)\sigma \left(z,t\right)$, whail it’s value in cross section on coordinate $z+dz\in \left(0.\mathcal{l}\right)$ is (ssee Fig. 2): Normal force to the cross section $A\left(z\right)$ of the rheological Maxwell model of material, fractional type, with variable cross section $A\left(z\right)$ at coordinate $z\in \left(0.\mathcal{l}\right)$ is $N\left(z,t\right)=A\left(z\right)\sigma \left(z,t\right)$, while it's value in cross section on coordinate $z+dz\in \left(0.\mathcal{l}\right)$ is (see Figure 2):

$$N\left(z+dz,t\right)=A\left(z\right)\sigma \left(z,t\right)+{\displaystyle \frac{\partial }{\partial z}}\left[A\left(z\right)\sigma \left(z,t\right)\right]dz$$

(12)

Let’s denote with $w\left(z,t\right)$ longitudinal displacement of the cross section$A\left(z\right)$ of the rheologic Maxwell material fractional tyoe at longitudinal coordinate $z\in \left(0.\mathcal{l}\right)$. According to the D’Alambert’s Principles of dynamical equilibrium of all acting forces to the system, following differential equation could be written for dynamical equilibrium of the forces of one elementary part of rheologic Maxwell material fractional type material is in form (for details see References [17,21,22]):

$$\rho A\left(z\right)dz{\displaystyle \frac{{\partial }^{2}w\left(z,t\right)}{\partial t^2}}=-N\left(z,t\right)+N\left(z+dz,t\right)+q\left(z,t\right)A\left(z\right)dz={\displaystyle \frac{\partial N\left(z,t\right)}{\partial z}}dz+q\left(z,t\right)A\left(z\right)dz$$

(13)

where $\rho$ is visco-elastic rheologic Maxwell model fractional type material’s density, $A\left(z\right)$ is area of cross section of material flowand$q\left(z,t\right)$ is distributed volume force. Substituting equation (12) into equation (13) leads to:

$${\displaystyle \frac{{\partial }^{2}w\left(z,t\right)}{\partial t^2}}-{\displaystyle \frac{1}{\rho A\left(z\right)}}{\displaystyle \frac{\partial }{\partial z}}\left[A\left(z\right)\sigma \left(z,t\right)\right]={\displaystyle \frac{1}{\rho }}q\left(z,t\right)$$

(14)

We can, now, write two basic relations that describe the dynamics flow (creeping) of the rheological Maxwell model of a fractional-type material flowing (creeping) in a pipe of constant or variable cross-section: the first is the constitutive differential relation of the fractional order, and the second is the dynamic equilibrium equation of an element of the Maxwell model of a fractional-type material (14):

$${\displaystyle \frac{1}{E}}{\mathrm{D}}_t^{\alpha }\left[{\sigma }_z\left(z,t\right)\right]+{\displaystyle \frac{{\sigma }_z\left(z,t\right)}{E_{\alpha }}}=\left\{{\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{\partial w\left(z,t\right)}{\partial z}}\right]\right\}\ne const, {\epsilon }_z\left(z,t\right)={\displaystyle \frac{\partial w\left(z,t\right)}{\partial t}}, 0<\alpha \le 1$$

(15)

$${\displaystyle \frac{{\partial }^{2}w\left(z,t\right)}{\partial t^2}}-{\displaystyle \frac{1}{\rho A\left(z\right)}}{\displaystyle \frac{\partial }{\partial z}}\left[A\left(z\right)\sigma \left(z,t\right)\right]={\displaystyle \frac{1}{\rho }}q\left(z,t\right)$$

(16)

From these equations (15) and (16) we eliminate the longitudinal displacement $w\left(z,t\right)$, to obtain a partial differential equation of fractional order in terms of the nodal stress of rheological flow (creeping) of dynamics flow of the rheological Maxwell model of material, fractional type.

We differentiate the first fractional order differential equation (15) twice with respect to time, and we differentiate the second partial differential equation (16) with respect to fractional order time and with respect to the coordinate $z\in \left(0.\mathcal{l}\right)$, so that we obtain:

$${\displaystyle \frac{1}{E}}{\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{{\partial }^2{\sigma }_z\left(z,t\right)}{\partial t^2}}\right]+{\displaystyle \frac{\frac{{\partial }^2{\sigma }_z\left(z,t\right)}{\partial t^2}}{E_{\alpha }}}=\left\{{\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{{\partial }^{3}w\left(z,t\right)}{\partial z\partial t^2}}\right]\right\}\ne const, 0<\alpha \le 1$$

(17)

$${\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{{\partial }^{3}w\left(z,t\right)}{\partial z\partial t^2}}\right]-{\displaystyle \frac{\partial }{\partial z}}〈{\displaystyle \frac{1}{\rho A\left(z\right)}}{\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{\partial }{\partial z}}\left[A\left(z\right)\sigma \left(z,t\right)\right]\right]〉={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial }{\partial z}}〈{\mathrm{D}}_t^{\alpha }\left[q\left(z,t\right)\right]〉$$

(18)

Then, we substitute the first obtained equation (17) into the second (18) and we obtain a partial differential equation of fractional order in terms of the normal stress $\sigma \left(z,t\right)$ in the form:

$${\displaystyle \frac{1}{E}}{\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{{\partial }^2{\sigma }_z\left(z,t\right)}{\partial t^2}}\right]+{\displaystyle \frac{\frac{{\partial }^2{\sigma }_z\left(z,t\right)}{\partial t^2}}{E_{\alpha }}}-{\displaystyle \frac{\partial }{\partial z}}〈{\displaystyle \frac{1}{\rho A\left(z\right)}}{\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{\partial }{\partial z}}\left[A\left(z\right)\sigma \left(z,t\right)\right]\right]〉={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial }{\partial z}}〈{\mathrm{D}}_t^{\alpha }\left[q\left(z,t\right)\right]〉$$

(19)

This fractional order partial differential equation (19) in terms of normal stress describes the longitudinal dynamics flow (creeping) of the rheological Maxwell model of a fractional type material, in a pipe of constant or variable cross-section as a function of the longitudinal coordinate .

4. Decomposition of the Partial Differential Equation of Longitudinal Flow Dynamics of the Rheological Maxwell Model of a Material, Fractional Type, in a Pipe of Variable Cross-Section

Let us assume the solution of the partial differential equation (19) of fractional order of the longitudinal dynamics-flow of the rheological Maxwell model of the material, of fractional type, in a pipe of variable cross-section under normal stress in the form of the product of two functions, one $T\left(t\right)$ that depends on time $t$, and the other $Z\left(z\right)$ that depends on the longitudinal coordinate $z\in \left(0.\mathcal{l}\right)$ in the form:

$${\sigma }_z\left(z,t\right)=Z\left(z\right)T\left(t\right)$$

(20)

Let's also assume that we can also assume the load in the form $q\left(z,t\right)=Z\left(z\right)Q\left(t\right)$.

Then, the resulting equation is in the following form:

$${\displaystyle \frac{Z\left(z\right)}{E}}{\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{{\partial }^2\left(T\left(t\right)\right)}{\partial t^2}}\right]+Z\left(z\right){\displaystyle \frac{\frac{{\partial }^2\left(T\left(t\right)\right)}{\partial t^2}}{E_{\alpha }}}-{\displaystyle \frac{\partial }{\partial z}}〈{\displaystyle \frac{1}{\rho A\left(z\right)}}{\displaystyle \frac{\partial }{\partial z}}\left[A\left(z\right)\left(Z\left(z\right)\right)\right]{\mathrm{D}}_t^{\alpha }\left[T\left(t\right)\right]〉={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial Z\left(z\right)}{\partial z}}〈{\mathrm{D}}_t^{\alpha }\left[Q\left(t\right)\right]〉$$

(21)

Then, we separate the resulting equation (21) into two separate ordinary differential equations, one in terms of a function in terms $Z\left(z\right)$ of coordinates $z\in \left(0.\mathcal{l}\right)$, and the other of fractional order in terms of a function in terms $z\in \left(0.\mathcal{l}\right)$ of time $t$. For that reason we, dividing the differential equation (21) by $Z\left(z\right)$ and separate the left and right sides into terms $T\left(t\right)$ that depend only on time, i.e., only on the coordinate $Z\left(z\right)$, and it follows:

$${\mathrm{D}}_t^{\alpha }\left[{\displaystyle \frac{{\partial }^2\left(T\left(t\right)\right)}{\partial t^2}}\right]+{\displaystyle \frac{E}{E_{\alpha }}}{\displaystyle \frac{{\partial }^2\left(T\left(t\right)\right)}{\partial t^2}}\mp k^{2}E{\mathrm{D}}_t^{\alpha }\left[T\left(t\right)\right]={\displaystyle \frac{E}{\rho }}{\displaystyle \frac{1}{Z\left(z\right)}}{\displaystyle \frac{\partial Z\left(z\right)}{\partial z}}〈{\mathrm{D}}_t^{\alpha }\left[Q\left(t\right)\right]〉$$

(22)

$$-{\displaystyle \frac{\partial }{Z\left(z\right)\partial z}}〈{\displaystyle \frac{1}{\rho A\left(z\right)}}{\displaystyle \frac{\partial }{\partial z}}\left[A\left(z\right)\left(Z\left(z\right)\right)\right]〉==\pm k^2$$

(23)

We have separated the partial differential equation (21) of fractional order, in the form of two separate ones, (22) in (23), one ordinary differential equation in terms of the function in terms $Z\left(z\right)$ of the coordinate $z\in \left(0.\mathcal{l}\right)$, and the other of fractional order in terms of the function $T\left(t\right)$ in terms of time $t$, where we have introduced one unknown constant $t$, which we determine from the condition that the function $Z\left(z\right)$ in terms of the coordinate $z\in \left(0.\mathcal{l}\right)$ as well as the solution for the flows normal stress $\sigma \left(z,t\right)$ of floe (creeping) of rheological Maxwell's material satisfy the boundary conditions. Now we can write the previous system of equations (22)-(23) in the following convenient form:

$$T^{″}\left(t\right)+{\displaystyle \frac{E_{\alpha }}{E}}{\mathrm{D}}_t^{\alpha }\left[T^{″}\left(t\right)\right]\mp k^2{E}_{\alpha }{\mathrm{D}}_t^{\alpha }\left[T\left(t\right)\right]={\displaystyle \frac{E_{\alpha }}{\rho }}{\displaystyle \frac{Z^{\prime }\left(z\right)}{Z\left(z\right)}}〈{\mathrm{D}}_t^{\alpha }\left[Q\left(t\right)\right]〉$$

(24)

$$-{\displaystyle \frac{\partial }{Z\left(z\right)\partial z}}〈{\displaystyle \frac{1}{\rho A\left(z\right)}}{\displaystyle \frac{\partial }{\partial z}}\left[A\left(z\right)\left(Z\left(z\right)\right)\right]〉==\pm k^2$$

(25)

In the previous system of ordinary differential equations (24) –(25), the first of which is of fractional order, we have introduced the eigen characteristic number $+k^2$, which we determine from the boundary conditions, which will be explained in the following sections of the article.

The previous first ordinary differential equation (24) in terms of the eigen time function is an ordinary differential equation of fractional order and from it we determine the eigentime functions $T\left(t\right)$. The second equation (25) is an ordinary differential equation in terms of functions $T\left(t\right)$ that depend on the longitudinal coordinate$Z\left(z\right)$ and from it we determine the eigen amplitude functions, which must satisfy the boundary conditions. The sign in front of the constant is determined from the condition that the solution for the eigen amplitude function that depends on the longitudinal coordinate , satisfies the conditions of physical reality of the task of mathematical solution of the ordinary differential equation (25).

In the paper, we assumed that it is possible to differentiate over whole derivatives and derivatives of fractional order by changing the order of differentiation.

8. Concluding Considerations on the Flow Dynamics of the Rheological Maxwell Model of a Material, Fractional Type, in a Pipe of Variable Cross-Section

We have presented new scientific analytical results of the theoretical study of the flow (creeping) dynamics of the rheological Maxwell model of a viscoelastic material, fractional type, in a pipe of variable cross-section. Here we emphasize that in the study we neglected the influence of the formation of a boundary layer during flow in the region at the boundary layer of the pipe of the rheological Maxwell model of a viscoelastic material, fractional type. In particular, we neglected the occurrence of friction forces at the boundary of the boundary layer in contact with the pipe surface, assuming that this boundary layer is of negligible thickness in relation to the cross-section of the flow stream of the rheological Maxwell model of a viscoelastic material, fractional type, i.e. the cross-section of the pipe, and that these friction forces are negligible in relation to the flow force - the product of the normal stress and the cross-sectional area of the flow stream in each of its cross-sectio

We have presented comparative plots of the surfaces of eigen and forced forces of fractional type for the dynamics of oscillation of the rheological Kelvin-Voigt model of an elastophisto-viscous material, fractional type, and the dynamics of flow of the rheological Maxwell model of a viscoelastic material, fractional type. In the structural formula of these two rheological models of materials of Kelvin-Voigt and Maxwell fractional type, they contain the same basic rheological elements, Hooke's ideally elastic and Newton's ideally viscous of fractional type, the first in their parallel connection, and the second in the order connection. It is connected with the property of subsequent elasticity (post-elasticity), and the second with the property of normal stress relaxation.