Modeling the process of force load generation at the initial periodic change in pressure (a plane problem)

. The article deals with modeling the process of force load generation at an initial periodic change in pressure (a plane problem). The subject of research is a pulsating flow in a flat channel at an initial periodic pressure change. The determination of flow parameters with a periodic change in the inlet pressure; the changes in the structure of the working fluid associated with the release of various particles from the pipe walls, the addition of impurities to prevent leaks, and the high-speed modes, are given in the article considering the law of molecular and molar transfer between layers. Research methods are based on Newton's rheological law, according to which molecular transfer is described by the law of proportionality of stresses to the derivative of the normal velocity; on the method of accounting for molar transfer by proportionality of stresses to the derivative of normal acceleration; on the method of mathematical modeling and the analytical method for their solutions, based on the provisions of operational calculus. An analytical solution to the problem of pulsating fluid motion in a plane-parallel channel is obtained with allowance for single and group transfer of molecules in the flow. The application of the analytical expressions obtained for the velocities is not limited to the critical Reynolds number, i.e. they are applied for any values of this number. Analytical expressions are obtained for the transverse and longitudinal components of the flow velocity. The resulting solution describes two zones of flow: in the first zone, two types of transfer occur, depending on the flow pattern, either molecular or molar transfer of fluid volumes between the layers prevails. In the second zone, only molecular transfer occurs. number, analytical expressions for the velocities describe the fluid flow with a periodic law of pressure change in laminar and turbulent flow modes.


Introduction
Pulsating flows of fluid ensure the existence of biological and social objects and they are an integral part of the technological production processes. The flow in a sudden E3S Web of Conferences 258, 08020 (2021) UESF-2021 https://doi.org/10.1051/e3sconf/202125808020 expansion channel and its flow through an idealized curved coronary artery with a pulsating velocity at the inlet was studied in [1]. A pulsating flow for thermohydraulic analysis of a nuclear reactor in an oceanic environment was investigated in [2]. Modeling a pulsating inlet flow to study the performance of flutterbased energy harvesters [3] and the effect of pseudo-plastic fluid flow in a manifold microchannel heat sink [4] is the evidence of the widespread use of pulsating flow.
Investigations of pulsating fluid flow are conducted by experimental [5,6] and theoretical methods [3,4]. Theoretical studies of this process are conducted using the Navier-Stokes equations. The Navier-Stokes equations are derived with Newton's law, according to which the stress is directly proportional to the derivative of the normal velocity, which describes the molecular transfer of momentum between the layers of the flow. This corresponds to a homogeneous layered fluid flow. Under the pulsating motion, conditions for the accumulation of inhomogeneity are formed in the flow, the alignment of which occurs during the groupped motion of molecules.
In hydraulic drives and other structures, wear products are formed in the working fluid, and to improve performance and prevent leaks in the system, various additives are added to the fluid. Moreover, high speeds and pressures arise in various modes. Taking into account all these factors, hydrodynamic processes cannot be described using classical models.
In this study, to account for the transfer of a substance between layers at the molecular level, it is assumed that the stress is directly proportional to the derivative of the normal velocity, and with all the above circumstances, during molar transfer, the stress is proportional to the derivative of the normal acceleration [7,11]. Taking into account the new factor, the internal structure of the Navier-Stokes equations undergoes substantial changes -the terms in partial derivatives of the third order are formed in the equation [7,11]. There are many methods and algorithms for the numerical solution of these equations [8,9]. Applying these tools to solving problems by involving new complicated equations is laborious work. There are various methods for the analytical solution of problems: the method of separation of variables, the method of linear approximation, the Fourier method [10] or the method of involving the provisions of the operational calculus [7,11]. In the problem considered below, we used the method of involving the provisions of the operational calculus.

Methods
Research methods are based on Newton's rheological law, the equation of continuity, which expresses the law of conservation of mass; the method of mathematical modeling and the analytical methods for their solution, based on the provisions of operational calculus.

Materials
Let us consider a plane-parallel pulsating flow of fluid, taking into account the molecular and molar transfer in the flow. The system of equations of fluid motion, in this case, has the following form [7,11]: E3S Web of Conferences 258, 08020 (2021) UESF-2021 https://doi.org/10.1051/e3sconf/202125808020 Where ɯ,ɭ -are the coordinates; t is time; V1, V2 are the velocity components; Ɋ is the pressure; ȝ is the dynamic viscosity; me is the molar transfer coefficient.
The pressure gradient is given as: The initial and boundary conditions are as follows: x v x y t Ɇ FRQVW of here Ȗ is a parameter that takes into account the instantaneous transition of the velocity at ɯ = 0 from the state of rest in terms of velocity u0=ɫonst. In order to obtain an analytical solution to this problem, we introduce the following function: applying the Laplace transform in the variable t , we obtain which is subject to the following conditions: We introduce function ( , , ) ( , , ) ( ), w x r p u x r p A p and apply the Laplace transform in x to the resulting equation, and a second-order differential equation with respect to the function w is obtained: there is a solution: Here Ɋ, S are the parameters of the Laplace transform in t and ɯ, respectively.
Using the Cauchy theorem [12], we obtain: > @ > @ Performing the inverse transformation sequentially with respect to parameters s and p from [12], we obtain: