Pulsing Flows of a Viscous Incompressible Liquid in a Pipe with Elastic Walls

As you know, the recent intensive introduction into practice of flexible pipelines made of polymer synthetic materials, pulsating fluid flow in elastic pipes is of great importance. As you know, the recent intensive introduction into practice of flexible pipelines made of polymer synthetic materials, pulsating fluid flow in elastic pipes is of great importance. By solving the problem, the necessary hydrodynamic parameters will be determined, such as pressure distributions, velocities, flow rates, the speed of propagation of the pulse wave pressure and their decay. For the first time in this article, a decrease in hydraulic resistance in a pulsating flow through pipes due to the elasticity of the wall will be determined. The dependence of the dimensionless value of the pressure pulse wave on the vibrational number was investigated  .The speed of the pulse wave was compared with the speed of Moens-Korteweg c , and significant differences were revealed between them occurring at lower values of the Womersley oscillatory parameter, at large values of which significant differences are not observed. The dependence of the reciprocal damping per wavelength on the vibrational number , was also investigated; it was shown that the damping is free at smaller values of the Womersley vibrational parameter, practically equal to zero, and at large values of which it asymptotically approaches unity.


Introduction
In recent years, the intensive introduction into practice of flexible pipelines made of polymer synthetic materials is of great importance [1][2][3], pulsating fluid flows in elastic pipes. Also in this area, pulsating fluid flows in pipes are of no small importance, taking into account the various mechanical properties of the wall [7][8][9][10][11][12][13][14][15][16][17]. Based on this consideration, this article will investigate the pulsating flow of a viscous fluid in an elastic tube. By solving the problem, the necessary hydrodynamic parameters will be determined, such as pressure distributions, velocities, flow rates, the speed of propagation of the pulse wave pressure and their attenuation. For the first time in this article, a decrease in hydraulic resistance in a pulsating flow through pipes due to the elasticity of the wall will be determined.

Problem statement and solution methods
Let us formulate a simplified problem, which is of no small importance in studies of the pulsating flow of a viscous fluid in pipes with elastic walls [4][5][6][7][8]. For this, we assume that the relative amplitude of the wall deformation to the radius is too small compared to unity, i.e.
To deform the pipeline wall on the basis of the accepted assumption at small wall deformations, it is sufficient to use the Lightfoot equation [3] 2 2 2 2 1 Note that the adhesion of the liquid and the permeability of the pipe wall are determined by the boundary conditions for the velocity components: If the wall deformation is small, then we can assume that where x V -average flow rate.
Then the relationship between pressure and average velocity is described by the equation Thus, the simplified system of equations of motion of a viscous fluid in pipes with elastic walls will take the final form: To solve a simplified problem under the conditions that in the initial and final sections of the pipe, the fluid pressure is set in a complex form, as is done in the previous paragraph, which correspond to the case under consideration, i.e.
Here: 0 n p and nL pvibration amplitudes;circular vibration frequency; nharmonic number.
The solution to the system of equations (9) is sought in the form in t x p x t p x e   Then the system of equations takes the following form: 1 , The solution of the system of equations (11), (12) and (13) taking into account the boundary conditions (4) is written in the form Multiply both sides of formula (14) Differentiating (17) with respect to x and substituting its value from equation (13)  xL  The solution to equation (18), taking into account the boundary conditions (19), has the following form:

Analysis of the results obtained and their discussion
From the obtained formulas (20)     In fig. 1.shows the dependence of the dimensionless value of the pulse pressure wave on the vibrational number  . It was found that the speed of propagation of the pressure pulse wave increases with an increase in the elastic modulus of the surrounding tissue and an increase in wavelength. Here, the pulse wave velocity is also compared with the Moens-Kortewegvelocity c  , and revealed significant differences between them occur at lower values of the Womersley vibrational parameter, at large values of which significant differences are not observed.
In fig. 2.shows the dependence of the reciprocal of the damping, referred to the wavelength, on the vibrational number . it is shown that the damping is free at lower values of the Womersley vibrational parameter, practically equal to zero, and at large values of it, asymptotically approaches unity. The presented simplified model is suitable for determining the speed of propagation of a pulse wave and attenuation of pulses. However, it is not acceptable for determining the hydraulicresistance in an elastic pipe, since in this case the impedance / p Q x       does notdepend on the wall elasticity coefficient. In order to determine the hydraulic resistance in an elastic pipe, it is necessary to solve the problem in a twodimensional formulation, that is, taking into account the orthotropy of the wall deformation, using the linearized Navier-Stokes equations for the flow of a viscous fluid.