International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

In the paper a mathematical model of the non-stationary motion of a viscous liquid mixture through the vertical straight pipe of the circular cross section is proposed. During the model construction weak compressibility of the mixture is considered. The Navier-Stokes equations system is taken as a basis. Such model can be used in the description of oil motion in a vertical well.

In the paper we solve the problem of transporting viscous weakly compressible liquid through the pipeline of the circular cross-section under non-stationary conditions. This paper is based on previous author’s results, presented on WS of XIII MTM Congress.

The Navier-Stokes equations are the basis for mathematical model. The liquid kinematic viscosity and its density are considered to be weakly changing with time. The non-stationarity is caused by specific boundary conditions, depending on time. The obtained results allow to optimize the control of a viscous weakly compressible liquid flows in the pipeline systems.

In this paper the analytical solution of the differential equation, describing viscous incompressible liquid motion through straight round pipe, with one perturbation parameter is given. The perturbation is introduced into the equation through the disturbed viscosity, depending on the temperature. The solution of the corresponding differential equation is sought in the form of a series in the small parameter. Thus the flow reaction to the internal disturbance is studied. The solution, presented in this article, gives an opportunity to further studying the temperature impact on a viscous weakly compressible liquid motion through a pipe. This research may be of interest to the problems of hydrocarbons control flow through main pipelines.

In the work a mathematical model for the non-stationary motion of liquid homogeneous viscous mixtures through pipelines is constructed. The corresponding integral equations expressing the laws of conservation of mass, momentum and energy are deriving, from which, in turn, we get the corresponding differential equations. The formulation of the initial and boundary conditions is given; the necessary additional relations that close the corresponding system of differential equations are indicated. A numerical algorithm for solving such a system is proposed. The corresponding numerical examples are given.