Study on the process of droplet formation when liquid flows out of a capillary

In this article, the law of interaction of air and fibrous materials emanating from two opposite pipelines is described in the theory of singular points by S.A. Chapligin, N.E. Zhukovsky function, theoretically studied using K. Schwarz's integral formula, Lopital's rule, complex potential field and canonical field.


Introduction
It is known that the air flow is widely used in the technological process of cotton processing. The air flow is widely used [1], especially when transporting cotton, fiber, and also when transporting waste from technological processes. In this area, B. Levkovich, O. Ishmurotov, S. Kadyrkhodzhaev, A. Suslin, R. Burnashev, M. T. Khodzhiev, R. Murodov, B. Mardonov and Kh. Akhmedkhodzhaev and others conducted extensive scientific and practical research on the transportation of cotton products in pneumatic vehicles, but studies of pneumatic vehicles used for dust removal and studies of dust concentrations emitted from dust collectors have not been sufficiently conducted [2][3][4][5][6][7][8][9][10][11][12][13]. With this in mind, today, in the in-depth study of the composition of dust, great attention should be paid to the issue of separating their constituents during the cleaning process [1]. In particular, the analysis of the existing technology for dust cleaning of air shows that no scientific and practical studies of the cleaning process, taking into account the fractional composition of dusty air, have not been carried out. The problem of air flow and separation of fibrous materials from all of the above technological processes is a very urgent problem, and its solution is extremely important. In solving this problem, we tried to create a technology for their separation, based on a sharp decrease in the speed of heavy particles in the air, based on the force of interaction of oppositely directed air flows [2]. The creation of a theoretical study of this process is one of the important tasks. To do this, we will take into account the regularity of the interaction of air and a fibrous mixture coming from two opposite tubes in this process. In it, we consider a theoretical study of this issue, mainly based on the technological scheme presented in Fig. 1 [3].

Materials and Methods
A mixture of two pipes in a horizontal position collides with each other during movement and propagates along the upper and lower vertical pipes, as shown in the diagram in Figure  1. As a result of the collision of opposite currents around the point of their collision Е, a steady state current is formed ( Fig. 2 and 3). As a result, the state of suspension formation in Fig. 2   0 0 ED С , the symmetric state of the two mean currents of the G z was investigated on the basis of the circuit shown in Fig. 3. As a result of the study, the following indicators and their parameters will be determined: -intermediate stagnation of waste and fibrous materials moving in the stream; -hopper for collecting waste and fibrous materials Lc width of the lower channel; -determine the distance L ox (between points Е о and E) from the point of inflow of the air flow to the point of the beginning of stagnation; -Determination of the radii of curvature of the arc radii determine the amount of waste and fibrous materials entering the collection bin; -velocities 1 V and 2 V of upper and lower vertical currents after capture of waste.
To simplify the derivation of theoretical research results, we assume that the problem is two-dimensional and does not depend on time. When solving such problems, we rely on the methods of the theory of functions of a complex variable and ideal fluids [4,5,6]. The problem is solved in a parametric form. As an auxiliary field, we take the upper half-plane with a parametric variable and denote it as  (Fig. 5). In the reflection process, we assume that parameters will have first-order poles and zeros at points dt dW -we will plot this result by plotting a function at the poles and zeros: (1) in this: n q -(АА) flow rate of the mixture in the pipe ( We introduce a function similar to N.E.Zhukovsky's.  From (1) and (2) we obtain the following.
In this case, the limit value of the N.E. Zhukovsky function has the following form. According to Fig. 5, Using the K. Schwartz integral formula, we obtain the following [3].
or, to put it more fully, from the boundary conditions of Zhukovsky's function: In this case, taking into account the formulas (5) - (7), we calculate I 1 (t) and I 2 (t): t=ξ+iη, η=0 In that case from (5): we check the correctness of formula (9). 1.
according to the Lopital rule,   Consequently, the law of interaction between air and fibrous mixtures from two opposite pipes: (AA) is directly related to the density of fibrous air in the pipe head and the velocities of air and fibrous mixtures in the channels, which is proved in the above calculations.
It can be seen that expressions (1), (9), (10) and (11) (AA) allow one to accurately express the law of interaction between air and a fibrous mixture leaving the pipe. Now let's calculate the parameters of movement in the flow of air-fiber mixtures in two opposite pipes. Situation 1. In this case, we perform the following calculations: g 1 =0.33; Let be g 2 =1; L A =0.4, from the pipe to the point when two media are separated.
The above, mathematically and theoretically substantiated existing parameters of the flow of air and mixtures of fibers in opposite pipes, can be taken as a scientific and theoretical basis for solving the problem of flowing around air and separating from it mixtures of fibers isolated from technological processes cotton ginning.