Differential invariants in nonclassical models of hydrodynamics

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Abstract

In this paper, differential invariants are used to construct solutions for equations of the dynamics of a viscous heat-conducting gas and the dynamics of a viscous incompressible fluid modified by nanopowder inoculators. To describe the dynamics of a viscous heat-conducting gas, we use the complete system of Navier-Stokes equations with allowance for heat fluxes. Mathematical description of the dynamics of liquid metals under high-energy external influences (laser radiation or plasma flow) includes, in addition to the Navier-Stokes system of an incompressible viscous fluid, also heat fluxes and processes of nonequilibrium crystallization of a deformable fluid. Differentially invariant solutions are a generalization of partially invariant solutions, and their active study for various models of continuous medium mechanics is just beginning. Differentially invariant solutions can also be considered as solutions with differential constraints; therefore, when developing them, the approaches and methods developed by the science schools of academicians N. N. Yanenko and A. F. Sidorov will be actively used. In the construction of partially invariant and differentially invariant solutions, there are overdetermined systems of differential equations that require a compatibility analysis. The algorithms for reducing such systems to involution in a finite number of steps are described by Cartan, Finikov, Kuranishi, and other authors. However, the difficultly foreseeable volume of intermediate calculations complicates their practical application. Therefore, the methods of computer algebra are actively used here, which largely helps in solving this difficult problem. It is proposed to use the constructed exact solutions as tests for formulas, algorithms and their software implementations when developing and creating numerical methods and computational program complexes. This combination of effective numerical methods, capable of solving a wide class of problems, with analytical methods makes it possible to make the results of mathematical modeling more accurate and reliable.

Original languageEnglish
Title of host publicationProceedings of the XXV Conference on High-Energy Processes in Condensed Matter, HEPCM 2017
Subtitle of host publicationDedicated to the 60th Anniversary of the Khristianovich Institute of Theoretical and Applied Mechanics SB RAS
Editors Fomin
PublisherAmerican Institute of Physics Inc.
Number of pages14
Volume1893
ISBN (Electronic)9780735415782
DOIs
Publication statusPublished - 26 Oct 2017
Event25th Conference on High-Energy Processes in Condensed Matter, HEPCM 2017 - Novosibirsk, Russian Federation
Duration: 5 Jun 20179 Jun 2017

Publication series

NameAIP Conference Proceedings
PublisherAMER INST PHYSICS
Volume1893
ISSN (Print)0094-243X

Conference

Conference25th Conference on High-Energy Processes in Condensed Matter, HEPCM 2017
CountryRussian Federation
CityNovosibirsk
Period05.06.201709.06.2017

Keywords

  • EQUATIONS
  • GAS
  • DYNAMICS
  • STATE

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