Fundamentals of Mathematical Modeling of One-Dimensional Flows of Fluid and Gas in Pipelines

Examination of phenomena is carried out with the help ofmodels. Each model represents a definite schematization of the phenomenon taking into account not all the characteristic factors but some of them governing the phenomena and characterizing it from some area of interest to the researcher. For example, to examine the motion of a body the material point model is often used. In such a model the dimensions of the body are assumed to be equal to zero and the whole mass to be concentrated at a point. In other words we ignore a lot of factors associated with body size and shape, the material from which the body is made and so on. The question is: to what extent would such a schematization be efficient in examining the phenomenon? As we all know such a body does not exist in nature. Nevertheless, when examining the motion of planets around the sun or satellites around the earth, and in many other cases, the material point model gives brilliant results in the calculation of the trajectories of a body under consideration. In the examination of oscillations of a small load on an elastic spring we meet with greater schematization of the phenomenon. First the load is taken as a point mass m, that is we use the material point model, ignoring body size and shape and the physical and chemical properties of the body material. Secondly, the elastic string is also schematized by replacing it by the so-called restoring force F = −k · x, where x(t) is the deviation of the material point modeling the load under consideration from the equilibrium position and k is the factor characterizing the elasticity of the string. Here we do not take into account the physical-chemical properties of the string, its construction and material properties and so on. Further schematization could be done by taking into account the drag arising from the air flow around the moving load and the rubbing of the load during its motion along the guide. The use of the differential equation