Technology for creating a device for laminar flow of water in pipes

Аннотация

The article discusses the mathematical modeling of the movement of viscous incompressible fluids through a bundle of tubes located inside the outer pipe. The laminar and turbulent modes of this movement are considered, and the physical meaning of their occurrence is also analyzed. The fluid flow through n tubes of length L and radius r located inside the outer tube is considered. Calculation formulas are derived for calculating the maximum velocity of this flow, the volume of fluid passing through the cross section of the tube, the coefficient of resistance to friction in the tube along the length of the flow, and also the maximum value of the tangential stress. The results of the study of the relationship of the coefficient of resistance to friction in the tube with the Reynolds number are presented. A description is given of a device created according to the results of a study that brings the disordered flow of liquids into a laminar state. Keywords: Reynolds number, laminar flow, turbulent flow, parabolic flow, friction force, integral, coordinate, pipe, viscosity, density, main flow velocity, average speed, maximum speed, radius, Hooke, Gegin, Poiseuille, Darcy-Weisbach, fluid volume, drag coefficient. The motion of real fluids is often very different from that of laminar flow. They have a special property called turbulence. As the Reynolds number increases in real fluid flows in pipes, channels, and boundary layers, the transformation of a laminar-shaped flow into a turbulent flow is clearly observed. This transition of laminar flow to turbulent flow is sometimes called turbulence, which is fundamental in the whole field of hydrodynamics. Initially, such a transition was observed in the flow of straight pipes and channels. Information on the forces acting on a fluid for flow in a cylindrical tube is given in the article [2; pp. 36-47]. Consider the motion of a tube of constant diameter along its entire length and the flow of fluid through n tubes of length L and radius r placed 182 Scientific research results in pandemic conditions (COVID-19) inside the tube. In real liquids, the liquid sticks to the walls of the tubes and exerts a repulsive stress on the flowing surface. This creates a force called internal friction, which is the viscosity of liquids. Viscosity is the property of gases and liquids to resist the action of external forces that cause the fluid to move. The presence of experimental stresses and the adhesion of the fluid to solid walls cause the real fluids in motion to be qualitatively different from ideal fluids. We now calculate the forces acting on the fluid in the pipe, taking into account the n tubes. Due to the adhesion, the velocity in the tube wall is zero, and the velocity between the tubes reaches its maximum value. Some concentric layers move so that the velocity is axial everywhere and the flow is laminar. At a sufficiently long distance from the starting point of the tubes, the velocity distribution of the flow in the tube does not depend on the longitudinal coordinates along the radius. The movement of the fluid in the tube is due to the fact that the pressure decreases along the axis of the tube, but the pressure does not change in the cross section perpendicular to the axis of each tube. The motion of each element of the fluid accelerates due to the pressure drop and slows down due to the shear stress caused by friction [3; pp. 36-38]. The pressure p is assumed to be constant, i.e., across the entire tube section [4; pp. 59-60]. There are compressive forces and effects on the cylinder along the main axis, which correspond to the inlet and outlet bases of the cylinder, respectively, and there is an actuating force acting on the side surface of the cylinder. It is necessary to determine the maximum flow rate in this cylinder, the volume of liquid flowing through the cross section of the pipe, the coefficient of frictional resistance of the pipe along the length of the pipe, and the maximum value of the test voltage. By balancing the forces acting on the fluid in the tube (Figure 1), we obtain the following equation as a condition of equilibrium in the direction of motion: n n n . (1) py2 py2 2yL 0 l i1 i1 i1 Figure 1. N tubes are placed inside the tube 183 Scientific research results in pandemic conditions (COVID-19) The projection of the internal friction force is obtained with a positive sign because the velocity gradient is negative (the flow velocity of the layer decreases with increasing coordinate). From formula (1) we can determine the test voltage p  p y  0 l  (2) L 2 In this case, the flow rate decreases with increasing coordinate and becomes zero due to the viscosity. Therefore, we can say that according to du Newton's law of  friction. dy Substituting this expression into (3), we get: du p  p y   0 l  dy L 2 Henceforth du p  p y  0 l  (4) dy L 2 Now, assuming that when y r, it is u(y)0, we integrate Equation (4) with this initial co