Reduction of an invariant submodel of gas dynamics to canonical form

The system of gas dynamics equations with the state equation of the monatomic gas admits a group of transformations with a 14-dimensional Lie algebra. A projective operator is specific to this algebra. We consider all one-dimensional subalgebras containing the projective operator. Invariants are calculated and invariant submodel of rank 3 is constructed for each of subalgebras. All submodels are stationary type. They are reduced to the canonical form. Area hyperbolicity of obtained system were specified. Integral entropy is obtained along the flow lines. An ordinary differential equation to the invariant functions is obtained along the flow lines (analogue of a Bernoulli integral for stationary motions). We consider all two-dimensional subalgebras containing projective operator. Invariant submodel of rank 2 stationary type is constructed for each of subalgebras. Submodels are reduced to the canonical form.

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Invariant submodel of rank 1 and two families of exact solutions of gas dynamics equations with an equation of state of the special form

Journal of Physics Conference Series ◽

10.1088/1742-6596/2099/1/012017 ◽

2021 ◽

Vol 2099 (1) ◽

pp. 012017

Author(s):

D Siraeva

Keyword(s):

Equation Of State ◽

Exact Solutions ◽

Special Form ◽

Gas Dynamics ◽

Three Dimensional ◽

Inhomogeneous Deformation ◽

System Of Equations ◽

Gas Dynamics Equations ◽

Linear Velocity Field ◽

Invariant Submodel

Abstract In this article, the gas dynamics equations with an equation of state of the special form are considered.The equation of state is the pressure which is equal to the sum of two functions, with one being a function of a density, and the other one being a function of an entropy. The system of equations is invariant under the action of 12-parameter transformations group. For three-dimensional subalgebra 3.32 of the 12-dimensional Lie algebra invariants are calculated, an invariant submodel of rank 1 is constructed, and two families of exact solutions are obtained. The obtained solutions specify the motion of particles in space with a linear velocity field with inhomogeneous deformation. The first family of solutions has two moments of time of particles collapse. The second family of solutions has one moment of time of particles collapse on the plane. In the simplest case of second family of solutions, a surface consisting of particle trajectories is constructed.

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The compression of gas followed by expansion

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2017.2.029 ◽

2017 ◽

Vol 12 (2) ◽

pp. 195-198

Author(s):

R.F. Shayakhmetova

Keyword(s):

Gas Dynamics ◽

State Equation ◽

Two Dimensional ◽

Overdetermined System ◽

Gas Dynamics Equations ◽

Gas Compression ◽

Particular Solution ◽

Invariant Submodel ◽

Projective Operator ◽

Monatomic Gas

We consider the system of gas dynamics equations with the state equation of the monatomic gas. The system admits a group of transformations with a 14-dimensional Lie algebra. A projective operator is specific to this algebra. We consider the invariant submodel on two-dimensional subalgebra containing the projective operator. For the vorticity-free motions, the submodel is reduced to an overdetermined system of three equations. A particular solution is found for it, physical interpretation is given, and trajectories of gas particles are depicted. The solution gives the gas compression followed by expansion.

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Supernovae and interstellar gas dynamics

Symposium - International Astronomical Union ◽

10.1017/s0074180900100774 ◽

1967 ◽

Vol 31 ◽

pp. 117-119

Author(s):

F. D. Kahn ◽

L. Woltjer

Keyword(s):

Lower Limit ◽

High Velocity ◽

Gas Dynamics ◽

Interstellar Cloud ◽

Interstellar Gas ◽

Random Motions ◽

Relativistic Particles

The efficiency of the transfer of energy from supernovae into interstellar cloud motions is investigated. A lower limit of about 0·002 is obtained, but values near 0·01 are more likely. Taking all uncertainties in the theory and observations into account, the energy per supernova, in the form of relativistic particles or high-velocity matter, needed to maintain the random motions in the interstellar gas is estimated as 1051·4±1ergs.

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