Motion of gas particles based on the Galilei group

S. V. Khabirov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 173-187

Résumé

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.

Détail

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

Bibliographie

[1] Ovsyannikov L.V., “Programma “Podmodeli”. Gazovaya dinamika”, Prikl. matematika i mekhanika, 58:4 (1994), 30–55 | MR | Zbl

[2] Ovsyannikov L.V., “Nekotorye itogi vypolneniya programmy “Podmodeli” dlya uravnenii gazovoi dinamiki”, Prikl. matematika i mekhanika, 63:3 (1999), 362–372 | MR | Zbl

[3] Khabirov S.V., “Optimalnye sistemy summy dvukh idealov, dopuskaemykh uravneniyami gidrodinamicheskogo tipa”, Ufim. mat. zhurn, 6:2 (2014), 99–103

[4] Khabirov S.V., Mukminov T.F., “Graf vlozhennykh podalgebr 11-mernoi algebry simmetrii sploshnoi sredy”, Sib. elektron. mat. izv., 16 (2019), 121–143 | DOI | MR | Zbl

[5] Ovsyannikov L.V., Lektsii po osnovam gazovoi dinamiki, In-t kompyuternykh issledovanii, Moskva; Izhevsk, 2003, 336 pp. | MR

[6] Ovsyannikov L.V., Chupakhin A.P., “Regulyarnye chastichno invariantnye podmodeli uravnenii gazovoi dinamiki”, Prikl. matematika i mekhanika, 60:6 (1996), 990–999 | MR | Zbl

[7] Ovsyannikov L.V., “Regulyarnye i neregulyarnye chastichno invariantnye resheniya”, Dokl. RAN, 343:2 (1995), 156–159 | MR | Zbl

[8] Ovsyannikov L.V., “Novye resheniya uravnenii gidrodinamiki”, Dokl. AN SSSR, 111:1 (1956), 47–49 | Zbl

[9] Tarasova Yu.V., “Klassifikatsiya podmodelei s lineinym polem skorostei v gazovoi dinamike”, Sib. zhurn. industr. matematiki, 12:4 (2009), 128–136 | Zbl

[10] Yulmukhametova Yu.V., “Podmodeli dvizheniya gaza s lineinym polem skorostei v vyrozhdennom sluchae”, Sib. zhurn. industr. matematiki, 14:2 (2011), 139–150 | Zbl

[11] Ovsyannikov L.V., “Izobaricheskie dvizheniya gaza”, Differents. uravneniya, 30:10 (1994), 1792–1799 | MR | Zbl

[12] Chupakhin A.P., Barokhronnye dvizheniya gaza: obschie svoistva i podmodeli tipov $(1,2)$ i $(1,1)$, Preprint IGiL SO RAN No4-98, Novosibirsk, 1998, 67 pp.

[13] Chupakhin A.P., Nebarokhronnye podmodeli tipov $(1.2)$ i $(1.1)$ uravnenii gazovoi dinamiki, Preprint IGiL SO RAN No1-99, Novosibirsk, 1999, 40 pp.

[14] Ovsyannikov L.V., “Osobyi vikhr”, Prikl. mekhanika i tekhnich. fizika, 36:3 (1995), 45–52 | MR | Zbl

[15] Ovsyannikov L.V., “O “prostykh” resheniyakh uravnenii dinamiki politropnogo gaza”, Prikl. mekhanika i tekhnich. fizika, 40:2 (1999), 5–12 | MR | Zbl

[16] Khabirov S.V., “Zadacha Gursa o nepreryvnom sopryazhenii radialnykh pryamolineinykh dvizhenii gaza”, Mat. zametki, 79:4 (2006), 601–606 | Zbl