2019. Vol. 14. Issue 4, Pp. 274–278

URL: http://mfs.uimech.org/mfs2019.4.035,en

DOI: 10.21662/mfs2019.4.035

URL: http://mfs.uimech.org/mfs2019.4.035,en

DOI: 10.21662/mfs2019.4.035

Barochronous shear gas motion

Yulmukhametova Yu.V.

Mavlyutov Institute of Mechanics UFRC RAS, Ufa , Russia

The equations of ideal gas dynamics admit an 11-dimensional Lie algebra of first-order differentiation operators. All subalgebras of this algebra are listed. Khabirov S.V. for all 48 types of 4-dimensional subalgebras, the bases of point invariants are calculated and three 4-dimensional subalgebras are considered that produce regular partially invariant solutions in Cartesian, cylindrical and spherical coordinates, respectively. In this paper, we pose the problem of finding the solution of 3-dimensional equations of gas dynamics in a Cartesian coordinate system with an arbitrary equation of state, built on invariants of a 4-dimensional subalgebra. The basic operators of the considered subalgebra are combinations of translations and Galilean transfers. The invariants of this subalgebra define a representation of the solution for unknown hydrodynamic functions. Speed components are linear functions in terms of spatial variables. Moreover, density and pressure depend only on time. After substituting the solution representation, we studied the compatibility of the resulting system of differential equations. The system is collaborative and has an exact solution. Such a solution describes the isentropic barochronous shear motion of a gas. The equations of the world lines of motion of gas particles are found. The moments of particle collapse are established. There were two of them. The equations of collapse surfaces are found and written. For the flat case, several statements about the nature of the motion of gas particles are proved.

equations of gas dynamics,

invariants of a subalgebra,

collapse,

exact solution,

Jacobian,

world lines

Purpose: finding the exact solution to the equations of ideal gas dynamics in a Cartesian coordinate system. Investigation of the obtained solution for the presence of a collapse of gas particles. Determining the type of particle collapse.

Methods: various methods for solving differential equations and group analysis methods are used.

A brief message contains a detailed description of the process of finding the exact solution of the differential equations of 3-dimensional gas dynamics in a Cartesian coordinate system. To find it, invariants of the 4-dimensional subalgebra are used, which allow us to simplify the system of differential equations and lower its order. A solution is found that describes the isentropic barochronous shear motion of the gas. From the found velocity functions, the world lines of motion of the gas particles and the moments of collapse are determined. Equations of collapse surfaces are also found. In the planar case, it has been analytically shown that two different gas particles cannot be at the same point in the plane at the same time.

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