Dmitrii Tykmakov

For the professional education of specialists who are able to carry out scientific and research activities, an important task is to adapt the programs of higher professional education to the modern condition of science. Methods of mathematical modeling are often used in studies of many technological processes and natural phenomena. One of the varieties of physical processes is the flow of liquid or gaseous media. At the same time, there are often flows of inhomogeneous media in applied research, and there is no methodology for modeling the dynamics of inhomogeneous "complex" media in the classical course "Fluid and gas mechanics". For specialists in mathematical modeling, it is necessary to know the methods of mathematical modeling of the inhomogeneous media dynamics in addition to the classical methods of hydrodynamics. The purpose of this work is to develop a methodology for teaching the section of hydrodynamics related to the flows of inhomogeneous media. The section of the course "Hydrodynamics" is designed from the standpoint of the development of mathematical models and methods for their analytical solution. Students of the specialty "Applied Mathematics" are shown the main approaches to the development of mathematical models of "complex" flows, as well as the exact solutions of systems of equations are shown using simple examples. As part of the inclusion of inhomogeneous media dynamics elements in the course of "hydrodynamics", various concepts of developing models of inhomogeneous media dynamics are demonstrated. The models are presented in the form of systems of partial differential equations, including the equations of conservation of mass, momentum and energy. Taking into account the specifics of the specialty "Applied Mathematics", this work considers the use of methods of both the theory of partial differential equations and the theory of ordinary differential equations. For specialists in applied mathematics, it is important to understand the basic methodologies for modeling flows of inhomogeneous media. The techniques differ in approaches, which is important for modeling various types of inhomogeneous media, depending on the composition of the inhomogeneous medium and different volumetric contents of the components in the total volume of the mixture. The paper presents various types of mathematical models on the example of flows that allow simple analytical solutions. The material presented in the work can be useful in compiling the course "Hydrodynamics" for the specialty "Applied Mathematics", which would include elements of the theory of inhomogeneous media dynamics. Such a section could give a general idea of the mathematical models of multiphase and multicomponent media, and the understanding of methods for simplifying models and integrating systems of differential equations is also important.