Modeling of air velocity in a fluid transported through a pipeline

The article considers the movement of liquid with an air bubbles along a finite pipeline. Taking into account the Stokes equation, the Newtonian frictional stress and the Reynolds equation, a differential equation for the change in the velocity of an air bubble in a pipeline that transports a liquid at a given velocity speed is derived. It is assumed that the pipeline has no connection with the atmosphere. The analytical solution of the differential equation of the air bubble velocity in the moving fluid of the pipeline using the exponential integral is given. A fluid with dynamic viscosity of  = 1.79 ∙ 10-3, Pa∙s and density of ρp = 1027.3, kg/m3 was used for modeling. The angle of inclination of the pipeline was 45 degrees, the velocity of the liquid flow in the pipeline was taken as 0.1, 1.0, 2.0 and 5.0 m/s. The total length of the pipeline was H = 2 m. The air bubble was taken in the form of a sphere with an initial radius of 1, 5, and 10 mm. Air was blown into the fluid flow. When an air bubble enters the pipeline, its initial velocity corresponds to the velocity of the fluid flow. The diameter of the air bubble has almost no effect on its velocity speed. Аналіз Analysis of the velocity simulation results shows that as the velocity of the fluid increases, the velocity of the air bubble increases. The nature of the change of the air bubble velocity depending on the fluid velocity shows that there is a condition when the fluid velocity and the air bubble velocity will be equal. As the fluid velocity increases, the bubble velocity difference between the air entry point and the air exit point decreases. The decrease in the velocity difference is caused by the pressure force on the surface of the air bubble, which is caused by the velocity of the fluid and the air bubble itself. As the dynamic viscosity of the fluid increases, the air bubble velocity decreases. The developed dependence describes the physical process of the flow of Newtonian fluids and gases in the pipeline.