The authors have declared that no competing interests exist.

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.

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.

As shown in the "13" way, the practical application of these methods only leads to effective results when the boundary of the area of motion consists of horizontal waterproof and vertical waterproof areas. However, in the real conditions of hydrotechnical construction (channels, reservoirs) directly under cover sediments along with horizontal aquifers of higher permeability (galeks, gravel, coarse-grained sands) are often found and horizontal waterproof areas (impenetrable inclusions, cul-de-pressure rocks), which is fundamentally reflected in the nature of filtration processes. In such situations, the use of Kristoffel-Schwartz integral does not lead to the goal, as in areas of complex flow speed there are already circular polygons that do not boil directly to the straight line.

In contrast to these studies, the following is both a direct continuation and development of the author's previous works (see. The task of the flow of liquid from the pits through the ground array of finite power, underpowered by a well-permeable pressure aquifer containing a waterproof area on its roof, is studied, if there is infiltration on a free surface. The most common case of movement is considered, in which on both waterproof sections of the border the filtration area the flow takes extreme values and the point of zero flow speed goes to the spool (which, apparently, has not been found in the literature so far). 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.

To solve the mixed regional multi-parametric problem, analytical function theory uses the Semibarinova-Kochina method, as well as special-species 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.

On the basis of the built accurate analytical dependencies and through numerical calculations, hydrodynamic analysis of the influence of all physical parameters of the scheme on the picture of the phenomenon is noted and some features of the models being developed are noted. The results of mathematical modeling for all marginal cases are compared to the main filtration scheme. *

Main model. Setting a task. The flat established current from A’A _{0 } (

Groundwater,

We will assume that the movement of groundwater is subject to the law of Darcy with a known filtration factor of

Under such conditions, as it is known, the basic fluid filtration equations can be recorded as

Where

From a mathematical point of view, the challenge is to find the complex potential of the flow of

_{0};

The study is carried out in terms of the values of

Building a solution to the edge problem. Let's turn to the area of complex velocity _{1 } and _{2 }two of which correspond to the extremes of current function on impenetrable areas of

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

_{E} = 0_{G} = 1_{C} = ^{−2 } (0 _{D}^{= } ∞

as well as derivatives

By determining the characteristic performance of the functions of the and

It can be seen that the points ζ = _{A} and _{B}are the common points of function

Recall that along with _{F}_{, }_{N}_{1 } and _{N}_{2 }in the equation (4) the accessory parameters of_{0,}

Replacing variables

^{2}(2

translates the upper half-flatness ζ into a rectangle of the plane:

I'm where

_{E} = 0_{G} = 1_{C} = (1 + _{D} =

and the

_{1}_{,}_{2}(_{0}^{−3}(_{1}(_{2}(_{2}(

Here, sn _{1}((q)_{0}(

Taking into account the ratios (3), (5) and (6), as well as the fact that the functionshas the same appearance.

√

come to the addictions we're in.

in which

(9) Permanent conformal displays of

0

Regulating the position on the boundary of the flow of _{1 } and _{2;};

You can verify that the functions (9) meet the conditions (2) reformulated in terms of

Recording ratios (9) for different parts of the boundary of

and coordinates of

Account control is other expressions for the values

In formulas (11)-(14) subtegrial functions are expressions of the right parts of equality (9) on the corresponding sections of the contour of

Analysis of numerical results for the main filtration model. Views (9) --(14) contain seven unknown permanents: residency prototypes of points

The first means that the speed at the end of the spool turns into infinity, and the second is directly derived from the consideration of boundary conditions (2). After determining the unknown permanents are the exact values

On rice. 1 is a picture of the current, calculated _{0}

Analysis of table and rice data. 3, and allows you to draw the following conclusions.

First of all, the same quality of dependencies of the values _{ ,}

Increase infiltration intensity, impenetrable inclusion and pressure in the underlying layer and reduce the power of the layer, The length of the spool, the

If you make a transformation T’ =1/2+i

0<b’<α’<r’<1/2,

Where b’,α’<r’ there are abscisses of

Calculations show that for any infiltration intensity, the _{*} a single _{H}R: r’ = r_{*}’=1/2plane r’ <r’_{*}

Such a result in the marginal case for the model in question, when the waterproof layer of soil has unlimited power, there is no impenetrable site and infiltration, i.e. at the time of

1 And ε=0(m’ = 0), was first received by N. E. Schukovsky.

и учесть, что при этом

2 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.

3 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.