The mathematical description of the movement of water in the systems of irrigation pipelines, based on the equation of continuity of the medium, the system of Navy-Stokes equations. The resulting mathematical package describes a system with distributed parameters and is performed based on the condition of dynamic balance at the point of flow, taking into account the dependence on the nature of the flow and the physical properties of the environment. Calculation is executed with use of functions Besseliya. Methodology for calculating the hydrodynamic component of water movement in irrigation water supply systems. Pipelines are universal in nature and can be used in the calculation, construction and assessment of the stability of water supply hydraulic systems; the technique can be used to describe the object of operation in the construction of control systems for the hydraulic parameters of the water supply system.
To solve specific problems in irrigation water supply systems, quasi-one-dimensional models of unsteady flows can be used. In such models, the state of the flow of the working medium at each moment of time is characterized by the values ??of pressure, velocity and density averaged over the cross section, while the enumerated hydrodynamic quantities obtained by averaging over the flow cross section with the coefficients of momentum, kinetic energy and hydraulic resistance are introduced into the equations. Theoretical and experimental studies show that the instantaneous averaging coefficients of hydrodynamic quantities differ from quasi-stationary values ??.Linear mathematical model of unsteady fluid motion in a pipe.
When a fluid flows, the medium continuity condition obtained by
when selecting an elementary volume with an inhomogeneous medium
The resulting system of equations is called Navier-Stokes .
This system of equations describes the conditions of dynamic equilibrium at the point of flow under the condition that the real fluid of the medium is replaced by a continuous medium in which stresses are not normal to the areas on which they arise. The values ??of the derivatives characterizing the presence of additional stresses, apart from pressure, depend on the nature of the flow and the physical properties of the medium. When describing the unsteady motion of the working medium in a cylindrical round pipe, the flow is assumed to be axisymmetric with sufficiently small changes in temperature and pressure so that the viscosity of the medium can be assumed to be constant.
The bulk viscosity of the medium in the processes under study may not be taken into account. Under the assumptions made, equation (2) is described in cylindrical coordinates, the x axis of which is directed along the pipe axis, and the coordinate r is measured along the radius of the pipe cross section, is reduced to two Navier–Stokes equations:
Pre-irrigation soil moisture is determined by the requirements of plants, with its decrease, short-term inhibition of plants is possible; after irrigation, the moisture content depends on the water-holding capacity of the soil, it is usually not adjusted to the FPV (limiting field moisture capacity) in order to avoid water losses due to seepage. The smaller the humidity control range, the lower the norm, which is favorable for plants and soil, but watering should be more frequent. Irrigation rates are also affected by the thickness of the soil layer and the lithological structure of the underlying soils. On thin soils underlain by well-permeable soils, the norms are reduced to reduce the deep filtration of irrigation water. The terrain also has an effect: at large slopes of the earth's surface, irrigation rates are less because of the danger of soil erosion. On saline soils, irrigation rates are higher than on non-saline ones. From the experience of land irrigation, the limits of irrigation norms have been established for different irrigation methods: with surface irrigation - 60 ... 100 mm, with sprinkling - 20 ... 70, with drip - 5 ... 10, with subsoil - 40 ... 80 mm .
The timing and norms of irrigation are set by various methods. On operating irrigation systems, irrigation dates are set by measuring moisture reserves at key points, or by plant condition, which is less accurate; irrigation norms - based on experience.
When designing irrigation systems, a design irrigation regime is developed for the year of estimated availability. The rates and timing of irrigation are determined by the balance method of A. N. Kostyukov, in which the balances of moisture reserves in the settlement layer are analyzed sequentially for short settlement periods - phases of plant development, decades, weeks. The water reserves in the calculation layer are determined at the end of each calculation period, and if the reserves become less than the allowable, watering is prescribed, it is convenient to do this graphically. An example of determining by a graphical method the norms and terms of irrigation according to the graphs of the dependence of soil moisture and irrigation over time is shown in Fig.2.
1 - Line of optimal humidity Wmax; 2 - line of the minimum allowable humidity for a given crop Wmin; 3 - watering; 4 - precipitation; q - irrigation intensity; β - soil moisture; M - irrigation rate
At the beginning of the growing season, soil moisture is usually higher than optimal, but gradually it decreases (Fig. 2). At the moment of approaching the permissible minimum, the first watering should be given, which will increase the humidity for a while, until a second watering is required after a period of time t if precipitation does not fall. The process continues until the end of the growing season. The duration of one irrigation, i.e., the time from the beginning to the end of irrigation, is called the irrigation period, which depends on the technique and technology of irrigation, organization and productivity of labor, the area of ????irrigated plots, and the type of crops. The duration of the irrigation period can be from 3...5 to 12...15 days.
With the addition of equation (26), the system of equations (18), (19) will be closed and, under given boundary conditions, completely describes the unsteady laminar motion of a viscous compressible fluid in a plastic pipe. When using equation (26), the number of terms in its left and right parts should be limited in accordance with the required accuracy of the calculation. In a number of cases it is sufficient to take the first two or three terms on the left and on the right. Most real periodic processes in a hydraulic system can be represented by the sum of a finite number of harmonic components (harmonic oscillations). When the working medium fluctuates in a pipeline or in any other pressure channel, the distribution of flow velocities over the flow cross section differs from the law that describes this distribution in the case of steady-state movement of the medium. Thus, when a laminar fluid flow oscillates in a round cylindrical pipe, the parabolic distribution of velocities is violated, which, as is known from hydraulics, is characteristic of a laminar steady motion of a fluid in a pipe .
The system of equations (19), (21), (24), (25), (26), (27), (28), (29), (30) is used to describe hydrodynamic systems with distributed parameters.
 Popov, D. N. Dynamics and regulation of hydro and pneumatic systems: a textbook for universities in Specialties \"Hydropneumoautomatics and hydraulic drive\" and \"Hydraulic machines and means of automation\" / / D.N. Popov. - 2nd ed., revised. and additional – M.: Mashinostroenie, 1987. – 464 p. : ill.
 Altshul, A. D. Hydraulics and aerodynamics: a textbook for universities / /A. D. Altshul, L. D. Zhivotovsky, L. P. Ivanov. – M.: Stroyizdat, 1987. – 414 p. : ill.
 Bronshtein, I. N. Handbook of mathematics for engineers and university students// I. N. Bronstein, K. A. Semendyaev. - 13th ed., Spanish. – M.: Nauka, 1986. – 544 p.
 Popov, D.N. Non-stationary hydromechanical processes// D.N. Popov. – M: Engineering, 1982. - 240 p.