We consider a system of gas dynamics equations with the state equation of a monatomic gas. The equations admit a group of transformations with a 14-dimensional Lie algebra. We consider 4-dimensional subalgebras containing the projective operator from the optimal system of subalgebras. The invariants of the basis operators are computed. Eight simple invariant solutions of rank 0 are obtained. Of these, four physical solutions specify a gas motion with a linear velocity field and one physical solution specifies a motion with a linear dependence of components of the velocity vector on two space coordinates. All these solutions except one have variable entropy. The motion of gas particles as a whole is constructed for the isentropic solution. The solutions obtained have a density singularity on a constant or moving plane, which is a boundary with vacuum or a wall.
Keywords: gas dynamics equations, projective operator, invariant solution
Received March 3, 2023
Revised April 14, 2023
Accepted April 17, 2023
Renata Fuatovna Nikonorova, Cand. Sci. (Phys.-Math.), Mavlyutov Institute of Mechanics – Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, 450054 Russia, e-mail: renatanikon@gmail.com
REFERENCES
1. Ovsyannikov L. V. “Simple” solutions of the equations of dynamics for a polytropic gas. J. Appl. Mech. Tech. Phys., 1999, vol. 40, no. 2, pp. 191–197. doi: 10.1007/BF02468514
2. Panov A.V. Rank 0 invariant solutions of dynamics of two-phase medium. In: Internat. Conf. Anal. Appl. Math. (ICAAM 2016): AIP Conf. Proc. 2016. Vol. 1759, no. 1. P. 020083-1–020083-5. doi: 10.1063/1.4959697
3. Pavlenko A.S. Symmetries and solutions of equations of two-dimensional motions of polytropic gas. Sib. Elektron. Mat. Izv., 2005, vol. 2, pp. 291–307 (in Russian).
4. Golovin S.V. Podmodeli dinamiki politropnogo gaza [Submodels of polytropic gas dynamics]: dissertation. Novosibirsk, Novosibirsk State University Publ., 2000, 116 p.
5. Cherevko A. A. Optimal system of subalgebras for Lie algebra of operators admitted by an equation system of gas dynamics with equation of state $p=f(S)\,\rho^{5/3}$. Preprint no. 4-96. Novosibirsk, Inst. of Hydrodynamics of the Syberian Branch of the Russian Academy of Science, 1996, 39 p.
6. Shayakhmetova R.F. Inserted invariant submodels for motion of monatomic gas. Sib. Elektron. Mat. Izv., 2014, vol. 11, pp. 605–625 (in Russian).
7. Shayakhmetova R.F. Invariant submodels of rank 3 and rank 2 monatomic gas with the projective operator. Trudy Inst. Mekh. Ufim. Nauch. Tsentra RAN, 2016, Vol. 11, no. 1, pp. 127–135 (in Russian).
8. Nikonorova R.F. The lowest-rank monatomic gas submodels constructed on the basis of three-dimensional symmetry subalgebras. Sib. Elektron. Mat. Izv., 2018, vol. 15, pp. 1216–1226 (in Russian). doi: 10.17377/semi.2018.15.098
9. Sidorov A.F. Two classes of solutions of the gasdynamical equations. J. Appl. Mech. Tech. Phys., 1980, vol. 21, pp. 588–594. doi: 10.1007/BF00910159
10. Ovsyannikov L.V. New solution of hydrodynamic equations. Dokl. AN SSSR, 1956, vol. 111, no. 1, pp. 47–49 (in Russian).
11. Ovsyannikov L.V. The “podmodeli” program. Gas dynamics. J. Appl. Math. Mech., 1994, vol. 58, no. 4, pp. 601–627. doi: 10.1016/0021-8928(94)90137-6
12. Khabirov S.V. A hierarchy of submodels of differential equations. Sib. Math. J., 2013, vol. 54, no. 6, pp. 1110–1119. doi: 10.1134/S0037446613060189
13. Siraeva D.T., Yulmukhametova Yu.V. Transformations of gas dynamics equations and basis operators of a admitted 11-dimensional Lie algebra. Mnogofazovye sistemy, 2020, vol. 15, no. 3–4, pp. 217–222 (in Russian). doi: 10.21662/mfs2020.3.133
14. Ovsyannikov L.V., Chupakhin A.P. Regular partially invariant submodels of the equations of gas dynamics. J. Appl. Math. Mech., 1996, vol. 60, no. 6, pp. 969–978. doi: 10.1016/S0021-8928(96)00119-0
15. Khabirov S.V. Simple partially invariant solutions. Ufa Math. J., 2019, vol. 11, no. 1, pp. 90–99. doi: 10.13108/2019-11-1-90
16. Pavlenko A. S. Projective submodel of the Ovsyannikov vortex. J. Appl. Mech. Tech. Phys., 2005, vol. 46, no. 4, pp. 459–470. doi: 10.1007/s10808-005-0097-2
17. Shayakhmetova R.F. Vortexed expansion of monatomic gas. Trudy Inst. Mekh. Ufim. Nauch. Tsentra RAN, 2014, vol. 10, pp. 110–113 (in Russian). doi: 10.21662/uim2014.1.021
18. Fellner K., Schmeiser C. Classification of equilibrium solutions of the cometary flow equation and explicit solutions of the Euler equations for monatomic ideal gases. J. Stat. Phys., 2007, vol. 129, no. 3, pp. 493—507. doi: 10.1007/s10955-007-9396-8
Cite this article as: R.F. Nikonorova. Simple invariant solutions of the dynamic equation for a monatomic gas. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2023, vol. 29, no. 2, pp. 115–132; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S186–S203.
