Invariant submodels of rank 3 and rank 2 monatomic gas with the projective operator

2016 ◽  
Vol 11 (1) ◽  
pp. 127-135
Author(s):  
R.F. Shayakhmetova
Keyword(s):  
Canonical Form ◽  
State Equation ◽  
Gas Dynamics Equations ◽  
Flow Lines ◽  
Rank 2 ◽  
Invariant Submodel ◽  
Stationary Type ◽  
Projective Operator ◽  
Bernoulli Integral ◽  
Monatomic Gas

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.

The compression of gas followed by expansion

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.

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

Mathematics ◽  
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.

The motion of the particles volume corresponding to invariant solution of rank 2 submodel of hydrodynamic type

2016 ◽  
Vol 11 (2) ◽  
pp. 205-209
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Invariant Solution ◽  
Hydrodynamic Type ◽  
Lagrangian Coordinates ◽  
Hydrodynamic Equations ◽  
Rank 2 ◽  
Invariant Submodel ◽  
Entropy Functions

Invariant submodel of rank 2 on the subalgebra consisting of the sum of transfers for hydrodynamic equations with the equation of state in the form of pressure as the sum of density and entropy functions, is presented. In terms of the Lagrangian coordinates from condition of nonhyperbolic submodel solutions depending on the four essential constants are obtained. For simplicity, we consider the solution depending on two constants. The trajectory of particles motion, the motion of parallelepiped of the same particles are studied using the Maple.

scholarly journals Sign Characteristics of Regular Hermitian Matrix Pencils under Generic Rank-1 and Rank-2 Perturbations

2015 ◽  
Vol 30 ◽  
pp. 760-794 ◽  
Author(s):  
Leonhard Batzke
Keyword(s):  
Canonical Form ◽  
Hermitian Matrix ◽  
Structure Preserving ◽  
Spectral Behavior ◽  
Generic Rank ◽  
Jordan Structure ◽  
Matrix Pencils ◽  
Rank 2 ◽  
Infinite Blocks

The spectral behavior of regular Hermitian matrix pencils is examined under certain structure-preserving rank-1 and rank-2 perturbations. Since Hermitian pencils have signs attached to real (and infinite) blocks in canonical form, it is not only the Jordan structure but also this so-called sign characteristic that needs to be examined under perturbation. The observed effects are as follows: Under a rank-1 or rank-2 perturbation, generically the largest one or two, respectively, Jordan blocks at each eigenvalue lambda are destroyed, and if lambda is an eigenvalue of the perturbation, also one new block of size one is created at lambda. If lambda is real (or infinite), additionally all signs at lambda but one or two, respectively, that correspond to the destroyed blocks, are preserved under perturbation. Also, if the potential new block of size one is real, its sign is in most cases prescribed to be the sign that is attached to the eigenvalue lambda in the perturbation.

scholarly journals On invariant finite-difference schemes for equations of one-dimensional flows of a polytropic gas for problems with spatial symmetries

2021 ◽  
pp. 1-34
Author(s):  
Aleksander Alekseevich Russkov ◽  
Evgeny Igorevich Kaptsov
Keyword(s):  
Conservation Laws ◽  
Gas Dynamics ◽  
State Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Local Conservation ◽  
Gas Dynamics Equations ◽  
One Dimensional ◽  
Polytropic Gas ◽  
Invariant Properties

One-dimensional polytropic gas dynamics equations for plane, radially symmetric, and spherically symmetric flows are considered. Invariant properties of equations are discussed, local conservation laws are derived. Additional conservation laws are written, which take place only in case of special values of adiabatic exponent. Classical difference scheme of Samarsky-Popov for gas dynamics has all difference analogs of conservation laws, except for additional ones. In difference schemes additional conservative laws take place in case of special state equation approximation. Scheme of Samarsky-Popov with special state equation was initially suggested by V.A. Korobitsyn. He described it as ‘thermodynamically consistend’ In current paper group properties, and conservation laws of thermodynamically consistent schemes are discussed, and numerical implementation for plane, cylinder, and spherical flows is perfomed.

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

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.

The solution of the hydrodynamic submodels of rank 2 with rotation

2016 ◽  
Vol 11 (1) ◽  
pp. 20-23
Author(s):  
Yu.V. Yulmukhametova
Keyword(s):  
Linear Dependence ◽  
Radial Component ◽  
Polytropic Gas ◽  
All Solutions ◽  
Spatial Coordinates ◽  
Rank 2 ◽  
Invariant Submodel ◽  
Evolution Type

Found all solutions invariant submodel of grade 2 evolution type for the polytropic gas with the assumption of a linear dependence of the radial component of the velocity from the spatial coordinates.

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