2023. Vol. 18. Issue 1, Pp. 9–16

URL: http://mfs.uimech.org/mfs2023.1.002

DOI: 10.21662/mfs2023.1.002

URL: http://mfs.uimech.org/mfs2023.1.002

DOI: 10.21662/mfs2023.1.002

Collapsing motions of a diatomic gas whose density depends only on time

Gumerov I.I.^{∗}, Katashova A.A.^{∗}, Yulmukhametova Y.V.^{∗∗}

One submodel of gas motion with a linear velocity field is considered in the paper. Namely, a submodel that defines
the movements of a polytropic gas with a density that depends only on time. A polytropic gas is a gas for which
the internal energy is a function linear in temperature. The submodel under consideration is given by a system of
ordinary differential equations of the 22nd order for unknown functions. These functions characterize the movements
of gas particles and determine the type of density, pressure and entropy functions. The exact solution is sought for a
special case, namely for a diagonal linearity matrix. Two new exact solutions have been found. The type of vector
functions of velocity, density and pressure are determined. By the form of the velocity function, the world lines of
motion of gas particles are recorded. In the three-dimensional space of coordinates

polytropic gas,

diatomic gas,

linear velocity field,

inhomogeneous deformation,

trajectories of gas particles,

world lines of particle motion,

Jacobian

Objective: to find and study new exact solutions to the equations of gas dynamics with a linear velocity field and time-dependent density.

Methods: The following approach was chosen to find new exact solutions. The linearity matrix is selected in a diagonal form with several identical elements. After substituting such a solution into the ordinary differential equations of the model, the form of these matrix elements is determined.

Result: The type of the linearity matrix allowed us to determine the type of all gas-dynamic functions. Velocity, density, pressure and entropy.

- Dirichlet G.L. Untersuchunger uber eih Problem der Gydrodynamik // J. Reine Angew. Math. 1860. Vol. 58. P. 181.
- Riman B. [Essays] Sochinenia. M.-L.: GITTL, 1948. Pp. 339-366 (In Russian).
- Ovsannikov L.V. [New solution of hydrodynamic equations] Novoe reshenia uravnenia gidrodinamili // DAN USSR. 1956. Vol. 111, No 1. Pp. 47–49 (In Russian).
- Dyson J.F. Dynamics of a spinning gas cloud // J. Math. Mech. 1968. Vol. 18, No. 1. Pp. 91–101.

DOI: 10.1512/iumj.1969.18.18009 - Giron J.F., Ramsey S.D., Baty R.S. Nemchinov-Dyson Solutions of the Two-Dimensional Axisymmetric Inviscid Compressible Flow
Equations // Phys. Fluids. 2020. Vol. 32. 127116.

DOI: 10.1063/5.0032170 - Guo Y., Hadžić M., Jang J. Continued Gravitational Collapse for Newtonian Stars // Arch. Rational Mech. Anal. 2021. Vol. 239. Pp. 431-–552.

DOI: 10.1007/s00205-020-01580-w - Nikonorova R., Siraeva D., Yulmukhametova Y. New Exact Solutions with a Linear Velocity Field for the Gas Dynamics Equations for Two
Types of State Equations // Mathematics. 2022. Vol. 10, Issue 1. 123.

DOI: 10.3390/math10010123 - Yulmukhametova Yu.V. Solution of ideal gas equations describing Galilean invariant motions with helical level lines, with collapse on
a helicoid // Vest. Sam. state tehn. univ. Ser. phys.-mat. science. 2019. Vol. 23, No 4. Pp. 797–808 (in Russian).

DOI: 10.14498/vsgtu1703 - Yulmukhametova Yu.V. [Submodels of gas dynamics with a linear velocity field]Podmodeli gazovoi dinamiki s lineinim polem scorostey : dissertation ... Candidate of Physical and Mathematical Sciences : 01.01.02 / Yulmukhametova Yulia Valeryevna; [Place of defense: Ufa Scientific Center of the Russian Academy of Sciences]. Ufa, 2011. 118 p.
- Ovsyannikov L.V. [Lectures on the fundamentals of gas dynamics] Lektsii po osnovam gazovoy dinamiki. M.-Izhevsk: komp’yuternykh issledovaniy. 2003. 336 p. (in Russian).