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

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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New Exact Solutions with a Linear Velocity Field for the Gas Dynamics Equations for Two Types of State Equations

Mathematics ◽

10.3390/math10010123 ◽

2022 ◽

Vol 10 (1) ◽

pp. 123

Author(s):

Renata Nikonorova ◽

Dilara Siraeva ◽

Yulia Yulmukhametova

Keyword(s):

Velocity Field ◽

Exact Solutions ◽

Gas Dynamics ◽

Three Dimensional ◽

State Equation ◽

Linear Velocity ◽

State Equations ◽

Gas Dynamics Equations ◽

Linear Velocity Field ◽

Monatomic Gas

In this paper, exact solutions with a linear velocity field are sought for the gas dynamics equations in the case of the special state equation and the state equation of a monatomic gas. These state equations extend the transformation group admitted by the system to 12 and 14 parameters, respectively. Invariant submodels of rank one are constructed from two three-dimensional subalgebras of the corresponding Lie algebras, and exact solutions with a linear velocity field with inhomogeneous deformation are obtained. On the one hand of the special state equation, the submodel describes an isochoric vortex motion of particles, isobaric along each world line and restricted by a moving plane. The motions of particles occur along parabolas and along rays in parallel planes. The spherical volume of particles turns into an ellipsoid at finite moments of time, and as time tends to infinity, the particles end up on an infinite strip of finite width. On the other hand of the state equation of a monatomic gas, the submodel describes vortex compaction to the origin and the subsequent expansion of gas particles in half-spaces. The motion of any allocated volume of gas retains a spherical shape. It is shown that for any positive moment of time, it is possible to choose the radius of a spherical volume such that the characteristic conoid beginning from its center never reaches particles outside this volume. As a result of the generalization of the solutions with a linear velocity field, exact solutions of a wider class are obtained without conditions of invariance of density and pressure with respect to the selected three-dimensional subalgebras.

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Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

Multiphase Systems ◽

10.21662/mfs2018.3.009 ◽

2018 ◽

Vol 13 (3) ◽

pp. 59-63 ◽

Cited By ~ 1

Author(s):

D.T. Siraeva

Keyword(s):

Equation Of State ◽

Lie Algebra ◽

Lie Algebras ◽

Gas Dynamics ◽

Hydrodynamic Type ◽

System Of Equations ◽

Two Dimensional ◽

Equations Of Gas Dynamics ◽

Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

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Investigation of the properties of a quasi-gas-dynamic system of equations based on the solution of the Riemann problem for a mixture of gases

Keldysh Institute Preprints ◽

10.20948/prepr-2021-52 ◽

2021 ◽

pp. 1-12

Author(s):

Ismatolo Ramazanovich Khaytaliev ◽

Evgeny Vladimirovich Shilnikov

Keyword(s):

Dynamic System ◽

Riemann Problem ◽

Gas Dynamics ◽

Contact Discontinuity ◽

System Of Equations ◽

Gas Dynamics Equations ◽

Gas Dynamic ◽

Volume Method ◽

Different Gases ◽

Accuracy And Stability

The accuracy and stability of an explicit numerical algorithm for modeling the flows of a mixture of compressible gases in the transonic regime are investigated by the example of solving the Riemann problem on the decay of a gas-dynamic discontinuity between different gases. The algorithm is constructed using the finite volume method based on the regularized gas dynamics equations for a mixture of gases. A method for suppressing nonphysical oscillations occurring behind the contact discontinuity is found.

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ON THE PROBLEM OF RESTORING A FUNCTION IN THREE DIMENSIONAL SPACE BY THE FAMILY OF CONES

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.10.11.10 ◽

2021 ◽

Vol 10 (11) ◽

pp. 3505-3513

Author(s):

Z.Kh. Ochilov ◽

M.I. Muminov

Keyword(s):

Weight Function ◽

Exact Solutions ◽

Special Form ◽

Dimensional Space ◽

Three Dimensional ◽

Weight Functions ◽

The Family ◽

The Given ◽

Three Dimensional Space

In this paper, we consider the problem of recovering a function in three-dimensional space from a family of cones with a weight function of a special form. Exact solutions of the problem are obtained for the given weight functions. A class of parameters for the problem that has no solution is constructed.

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ANALYTICAL MODELS FOR QUARK STARS

International Journal of Modern Physics D ◽

10.1142/s0218271807011103 ◽

2007 ◽

Vol 16 (11) ◽

pp. 1803-1811 ◽

Cited By ~ 61

Author(s):

K. KOMATHIRAJ ◽

S. D. MAHARAJ

Keyword(s):

Electromagnetic Field ◽

Equation Of State ◽

Exact Solutions ◽

Quark Matter ◽

Matter Content ◽

Analytical Models ◽

System Of Equations ◽

Maxwell System ◽

Elementary Functions ◽

Quark Stars

We find two new classes of exact solutions to the Einstein–Maxwell system of equations. The matter content satisfies a linear equation of state consistent with quark matter; a particular form of one of the gravitational potentials is specified to generate solutions. The exact solutions can be written in terms of elementary functions, and these can be related to quark matter in the presence of an electromagnetic field. The first class of solutions generalizes the Mak–Harko model. The second class of solutions does not admit any singularities in the matter and gravitational potentials at the center.

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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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