Simulation of Groundwater Movement from Pits Fenced by Zchukovsky's Sprunts

The hydrodynamic production considers the filtration of liquid from the pits, fenced with spools of Zhchukovskiy, through a layer of soil, underrunable by a well-permeable pressure aquifer, on the roof of which contains an impenetrable area. To study infiltration on the free surface of groundwater, a mixed regional multi-parametric task of the theory of analytical functions is formulated, which is solved by the Semibarinova-Kochina method and methods of conformal display of areas of special species characteristic of the tasks of underground hydromechanics. Refs. 24. Il. 4. Table. 1. DOI: Coming soon Corresponding author: N. Bereslavsky, Petersburg State University of Civil Aviation, Russian Federation, 196210, St. Petersburg, St. Pilots, 38, Email: eduber@mail.ru


Introduction
kovsky. In a number of works, free filtration was studied, i.e. the current without support, and in some cases -pressure, i.e. the presence of a free surface was neglected. In all of these studies, infiltration was not recorded. In addition, different techniques were used to solve the problems: the function of Zchukovsky and the way Vedernikov-Pavlovsky, bringing the case to the conformal display of straight polygons, followed by the use of the Kristoffel-Schwartz formula. [1] As shown in the " 13 polygons that do not boil directly to the straight line. [2] In contrast to these studies, the following is There are extreme cases of current associated with the absence of both support, impenetrable inclusion or infiltration, and the case of degeneration of pits in the semi-endless left flooding strip, previously studied by V.
V. Vedernikov. Results are provided for a diagram that occurs in the absence of critical points when the flow rate at the end of the spool is finite; the resulting solution is a certain analogue of the classic task of Zhchukovskiy. [3] To solve the mixed regional multi-parametric problem, analytical function theory uses the Semibarinova-Kochina method, as well as specialspecies methods developed for areas of special species, which are very typical of underground hydromechanics.
Taking into account the specifics and characteristics of the movement allows you to present solutions through special, and in some cases elementary functions, which makes their application simple and convenient. [4] On the basis of the built accurate analytical On the roof of this layer is a waterproof area, D'D simulated horizontal segment length 2L. Due to thesymmetry of the motion picture, we will limit ourselves to the study of the right half of the ABCDEGR filtrationarea. [6] Groundwater, d flowing ARG spool under the influence of pressure difference in the pit and the underlying well-permeable aquifer, rise behind it to a certain height of the RG and, overcoming the point of M zero speed on the spool, form a free surface of GE ,which receives infiltration waters with the intensity of the e (0<e< 1),attributed to the ground filtration factor of the k= const. . [7] We will assume that the movement of groundwater is subject to the law of Darcy with a known filtration factor of z and occurs in a homogeneous and isotropic soil, which is considered incompressible. [8] Under such conditions, as it is known, the basic fluid filtration equations can be recorded as

….(1)
Where φ is the potential for filtration speed; u and v -projections on the axis of filtration rate coordinates; h -pressure; p -pressure in the stream; γ is the specific weight of the liquid [9] From a mathematical point of view, the challenge is to find the complex potential of the flow of z (φ the potential of speed, ψ is the function of the current) as an analytical within the filtration area of the function of the complex coordinates z under the following boundary conditions: [10] AB : y = 0, φ = −H; BC : x = 0, ψ = 0; CD : y = −T, ψ = 0; AG : x = l, ψ = Q; (2) The study is carried out in terms of the values of z and ω which are related to the actual values of the same through the means of equality. [11] . Building a solution to the edge problem. Let's turn to the area of complex velocity w ( Figure 2) corresponding to the edge conditions (2). This area, which is a circular polygon with three incisions, tops N 1 and N 2 two of which correspond to the extremes of current function on impenetrable areas of AB and DE, belongs to the class of polygons in the polar grids.
However, unlike the "15" limit, an additional corner point appears on the boundary of the traffic area, point B (see Figure 1); the total number of special points becomes nine, making the task much more difficult. [12] To solve the regional problem, the Semibarin-Kochina method, which is based on the application of the analytical theory of linear differential equations of the Fuchs class. Entered Auxiliary parametric variable ζ and z functions : conformally displaying the upper half-flatness on the z area when dots are matched [13] as well as derivatives .
By determining the characteristic performance of the functions of the and the "regular special points" near the regular special points, we will find that they are linear combinations of the two branches of the next function of Riemann. [14] …. (3) It can be seen that the points ζ = ζ A and ζ= ζ B are the common points of function Y, representing the last symbol of Riemann, which corresponds to the linear differential equation of the Fuchs class with seven regular special points, very typical for the tasks of underground hydromechanics. [15] ….. (4) Recall that along with ζ the ζ F, ζ N1 and ζ N2 in the equation (4) the accessory parameters of 0, λ 1 and 2 remain unknown in setting the task and must be determined in the course of its solution.
Replacing variables ζ = sn 2 (2Kτ,k) (5) translates the upper half-flatness ζ into a rectangle of the plane: I'm where , ρ K(k) is a full elliptical integral of the first kind with module ( k, 23 Here, sn τ = sn (u.k) is Jacobi's elliptical function (sinus) at module k, θ 1 ((q),θ 0 ((q) -theta function with the parameter q q exp, which is uniquely associated with the module k , q, γ -some suitable permanent. [16] Taking into account the ratios (3), (5) and (6), as well as the fact that the functions has the same appearance.
…. (9) in which N is a large-scale constant simulation, sn-unknown residents of points A and B of the area of q.
(9) Permanent conformal displays of q, z and γ, which are bound by a ratio (8), are subject to the conditions   Table 1.
The results of the calculations of D and Z values However, the greatest impact on the depth of D is an impenetrable area: table data show that when L width increases by only 28%, the depth d increases by almost Note that for q 0. 7,T 6 and 6.. 5, S q 3, L q 15, H q 5 and l q 10, in which the value of D becomes negative, the H is 1.2191, 1.2020, 1.0140, 1.0443, 1.0638 and 1.0442 respectively.     Re D C and D points merge in the area of complex speed, C w, its left semi-flatness is cut off, The circular incision of the EG goes into the right half-flatness, and the original area is transformed into a ∞ circular triangle.
. The solution for this extreme case is derived from formulas (9)- (14), if they put k q 0 and take into account that in this case elliptical functions degenerate into trigonometry, and theta functions break off on their first members or constants: , ….. (15) where There are prototypes of M,R,C points on the absciss of the plane. τ. case of = 0 L, we will focus on the absenceof infiltration.
Thus, it turns out to solve the problem, first considered by V. V. Vedernikov, only in another way.
A case of the final speed of the flow at the end of the spool. As part of the edge task (2), consider the case where the flow rate at the end of the spool v R , < ε 0 qlt; Then in the area of complex speed w both vertical incisions disappear, the left half-flat is cut off, as before at L q 0,however, unlike the latter, the MR section is transferred to the first quadrant.
The parametric solution of the problem formally has the same form (9) with the replacement of integrals Y 1,2 ((q) and permanent conformal display of q and q for the following: Y 1,2 (τ) = θ− 0 1 (τ)θ 1 (τ ± iγ)exp ( ±iπτ), (16) α = β = (1 + iρ)/2. (17) The solution to a similar problem in the absence of backup arises from submissions (9), (16), (17) at γ of γ*. The functions degenerate into hyperbolic, and theta functions, which this time are characterized by theq'=0 parameter, break off on their first members or constants. Thus, in the extreme case of the scheme studied, it turns out to solve the problem of Shchukovsky, but only in a different way. 4 The author thanks the reviewers for helpful tips and comments that contributed to the improvement of the work.