Numerical analysis of the mathematical model of the turbulent flow dynamics of a multiphase medium in network-like objects

This paper presents methods of mathematical analysis used to solve the applied problems of the theory of transport of solid media – thermal flows and viscous liquids in network-like objects. The initial-boundary problem for the Navier-Stokes system, which lies at the basis of the mathematical description of the so-called turbulent transport processes of Newtonian liquids with a given viscosity, is defined and studied. It is assumed that the liquid has a complex internal rheology and is a multi-phase continuous medium. The distinctive feature of the process under consideration is the absence of a classical differential equation at the node points of the network-like area (the surfaces of mutual adhesion of subdomains). Sufficient conditions for the unique weak solvability of the initial-boundary problem are presented, which are obtained by the classical analysis of approximations of the exact solution by means of a priori estimates derived from the energy inequality for norms of solutions of the Navier-Stokes equation. An optimization problem, which is natural in the analysis of transport processes of continuous media on a network-like carrier, is considered. The state spaces of the Navier-Stokes system, spaces of controls and observations, for which the uniqueness of the solution of the optimization problem is proved, are indicated. The suggested approach and corresponding methods are equipped with the necessary algorithm and illustrated by the examples of numerical analysis of test problems. The basis of the analysis lies in the classical approach to studying mathematical models of transport processes of continuous media. The paper is aimed at developing qualitative and approximate methods for investigating mathematical models of various types of continuous media transport.

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