S.V. Khabirov. Motion of gas particles based on the Galilei group ... P. 173-187

Invariants of the Galilei group determine the invariant and partially invariant solutions of continuum mechanics equations. Invariant motions have a point density collapse with straight world lines. The invariant characteristics of the equations of gas dynamics, which can be used to construct weak solutions with a discontinuity of the derivatives, are considered. Partially invariant solutions with a linear velocity field are investigated for special gas equations; such solutions are regular. There are possible solutions with a point collapse at an infinitely distant point. A classification of such solutions is given for the state equations from the group classification of the gas dynamics equations. The motion of gas particles for such solutions occurs along curvilinear trajectories to a point collapse or from a point source. The classification uses the method of separation of variables in the equation for functions of different independent variables. The same motion of gas particles is possible for different equations of state.

Keywords: gas dynamics, Galilei group, partially invariant solutions, linear field of velocities, point collapse, state equation, method of separation of variables

Received December 25, 2020

Revised February 8, 2021

Accepted February 15, 2021

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-29-10071) and within the State Assignment no. 0246-2019-0052.

Salavat Valeevich Khabirov, Dr. Phys.-Math. Sci., Prof., Mavlyutov Institute of Mechanics – Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, 450054 Russia,  e-mail: habirov@anrb.ru

REFERENCES

1.   Ovsyannikov L.V. The “podmodeli” program. Gas dynamics. Journal of Applied Mathematics and Mechanics, 1994, vol. 58, no. 4, pp. 601–627. doi: 10.1016/0021-8928(94)90137-6 

2.   Ovsyannikov L.V. Some results of the implementation of the “podmodeli” program for the gas dynamics equations. Journal of Applied Mathematics and Mechanics, 1999, vol. 63, no. 3, pp. 349–358. doi: 10.1016/S0021-8928(99)00046-5 

3.   Khabirov S.V. Optimal system for sum of two ideals admitted by hydrodynamic type equations. Ufa Mathematical Journal, 2014, vol. 6, no. 2, pp. 97–101. doi: 10.13108/2014-6-2-97 

4.   Khabirov S.V., Mukminov T.F. Graf of embedded subalgebras of 11–dimensional symmetry algebra for continuous medium. Sib. Elektron. Mat. Izv., 2019, vol. 16, pp. 121–143. doi: 10.33048/semi.2019.16.006 . (in Russian)

5.   Ovsyannikov L.V. Lektsii po osnovam gazovoi dinamiki (Lectures on the fundamentals of gas dynamics). Moscow; Izhevsk: Institut komp’yuternykh issledovanii, 2003, 336 p. ISBN: 5-93972-201-6 .

6.   Ovsyannikov L.V. Regular partially invariant submodels of the equations of gas dynamics. Journal of Applied Mathematics and Mechanics, 1996, vol. 60, no. 6, pp. 969–978. doi: 10.1016/S0021-8928(96)00119-0 

7.   Ovsyannikov L.V. The regular and nonregular partially invariant solutions. Doklady Akademii nauk, 1995, vol. 343, no. 2, pp. 156–159. (in Russian)

8.   Ovsyannikov L.V. New solutions of hydrodynamics equations. Doklady Akademii nauk SSSR, 1956, vol. 111, no. 1, pp. 47–49. (in Russian)

9.   Tarasova Yu.V. Classification of submodels with a linear velocity field in gas dynamics. Journal of Applied and Industrial Mathematics, 2010, vol. 4, no. 4, pp. 570–577. doi: 10.1134/S1990478910040125 

10.   Yulmukhametova Yu.V. Submodels of gas motion with a linear velocity field in the degenerate case. Journal of Applied and Industrial Mathematics, 2012, vol. 6, no. 1, pp. 123–133. doi: 10.1134/S1990478912010139 

11.   Ovsyannikov L.V. Isobaric gas flows. Differential Equations, 1994, vol. 30, no. 10, pp. 1656–1662.

12.   Chupakhin A.P. Barokhronnye dvizheniya gaza: obshchie svoistva i podmodeli tipov (1,2) i (1,1) (Barochronous gas motions: general properties and submodels of types (1,2) and (1,1)). Novosibirsk: Preprint IGiL SO RAN, 1998, no. 4–98, 67 p.

13.   Chupakhin A.P. Nebarokhronnyye podmodeli tipov (1.2) i (1.1) uravneniy gazovoy dinamiki (Non-barochronous submodels of types (1.2) and (1.1) of equations of gas dynamics). Novosibirsk: Preprint IGiL SO RAN, 1999, no. 1–99, 40 p.

14.   Ovsyannikov L.V. Singular vortex. Journal of Applied Mechanics and Technical Physics, 1995, vol. 36, no. 3, pp. 360–366. doi: 10.1007/BF02369772 

15.   Ovsyannikov L.V. “Simple” solutions of the equations of dynamics for a polytropic gas. Journal of Applied Mechanics and Technical Physics, 1999, vol. 40, no. 2, pp. 191–197. doi: 10.1007/BF02468514 

16.   Khabirov S.V. Goursat problem of the continuous conjugation of radial rectilinear motions of a gas. Mathematical Notes, 2006, vol. 79, no. 4, pp. 555–560. doi: 10.1007/s11006-006-0062-2 

Cite this article as: S.V. Khabirov. Motion of gas particles based on the Galilei group, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 1, pp. 173–187.