### Bublik V.

## Differential invariants in nonclassical models of hydrodynamics

**DIFFERENTIAL INVARIANTS IN NONCLASSICAL MODELS OF HYDRODYNAMICS
V.V. Bublik
**

*Khristianovich institute of theotetical and applied mechanics SB RAS, 630090, Novosibirsk, Russia*

In this paper, differential invariants are used to construct solutions for the 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 re-sults of mathematical modeling more accurate and reliable.

REFERENCES

1. **Andreev V.K., Bublik V.V., Bytev V.O.** Symmetries of Nonclassical Models of Hydrodynamics. Novosibirsk: Nauka, 2003 (in Russian)

2. **Sidorov A.F., Shapeev V.P., Yanenko N.N.** Method of Differential Constraints and Its Applications in Gas Dynamics. Novosibirsk: Nauka, 1984 (in Russian)

3. **Ovsiannikov L.V.** Group Analysis of Differential Equations. Moscow: Nauka, 1978 (English transl. published by Academic Press, New York, 1982)

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