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.

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

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
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.

Classical Kinetic Theory

2018 ◽  
Author(s):  
Klaus Morawetz
Keyword(s):  
Kinetic Equation ◽  
Equation Of State ◽  
Mean Field ◽  
Ideal Gas ◽  
Hydrodynamic Equations ◽  
Free Particles ◽  
Internal Mechanism ◽  
The Mean ◽  
Energy Gains ◽  
Non Locality

The classical non-ideal gas shows that the two original concepts of the pressure based of the motion and the forces have eventually developed into drift and dissipation contributions. Collisions of realistic particles are nonlocal and non-instant. A collision delay characterizes the effective duration of collisions, and three displacements, describe its effective non-locality. Consequently, the scattering integral of kinetic equation is nonlocal and non-instant. The non-instant and nonlocal corrections to the scattering integral directly result in the virial corrections to the equation of state. The interaction of particles via long-range potential tails is approximated by a mean field which acts as an external field. The effect of the mean field on free particles is covered by the momentum drift. The effect of the mean field on the colliding pairs causes the momentum and the energy gains which enter the scattering integral and lead to an internal mechanism of energy conversion. The entropy production is shown and the nonequilibrium hydrodynamic equations are derived. Two concepts of quasiparticle, the spectral and the variational one, are explored with the help of the virial of forces.

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.

Invariant submodels of a rank 2 with rotation for the equations of gas dynamics

2014 ◽  
Vol 10 ◽  
pp. 114-115
Author(s):  
Yu.V. Yulmukhametova
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Equation Of State ◽  
Gas Dynamics ◽  
Second Order ◽  
Equations Of Gas Dynamics ◽  
Rank 2

The various invariant submodels of a rank 2 allowing the operator of rotation for the equations of gas dynamics with the arbitrariest equation of state are considered. It is shown that such submodels come down to an ordinary differential equation of the second order.

scholarly journals THE REST-FRAME INSTANT FORM OF RELATIVISTIC PERFECT FLUIDS WITH EQUATION OF STATE ρ=ρ(n, s) AND OF NONDISSIPATIVE ELASTIC MATERIALS

2000 ◽  
Vol 15 (31) ◽  
pp. 4943-5015 ◽  
Author(s):  
LUCA LUSANNA ◽  
DOBROMILA NOWAK-SZCZEPANIAK
Keyword(s):  
Equation Of State ◽  
Rest Frame ◽  
Lagrangian Coordinates ◽  
Elastic Materials ◽  
Instant Form ◽  
Perfect Fluids ◽  
Minkowski Space Time ◽  
Spacelike Hypersurfaces ◽  
Photon Gas ◽  
Tetrad Gravity

For perfect fluids with equation of state ρ=ρ(n, s), Brown1 gave an action principle depending only on their Lagrange coordinates αi(x) without Clebsch potentials. After a reformulation on arbitrary spacelike hypersurfaces in Minkowski space–time, the Wigner-covariant rest-frame instant form of these perfect fluids is given. Their Hamiltonian invariant mass can be given in closed form for the dust and the photon gas. The action for the coupling to tetrad gravity is given. Dixon's multipoles for the perfect fluids are studied on the rest-frame Wigner hyperplane. It is also shown that the same formalism can be applied to nondissipative relativistic elastic materials described in terms of Lagrangian coordinates.

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.

scholarly journals Emptiness formation in polytropic quantum liquids

2022 ◽  
Author(s):  
Hsiu-Chung Yeh ◽  
Dimitri M Gangardt ◽  
A Kamenev
Keyword(s):  
Equation Of State ◽  
Large Deviations ◽  
Infinite Sequence ◽  
Analytic Solutions ◽  
Hydrodynamic Equations ◽  
Imaginary Time ◽  
Instanton Solution ◽  
Instanton Action ◽  
Quantum Liquids ◽  
Polytropic Equation

Abstract We study large deviations in interacting quantum liquids with the polytropic equation of state P (ρ) ∼ ργ, where ρ is density and P is pressure. By solving hydrodynamic equations in imaginary time we evaluate the instanton action and calculate the emptiness formation probability (EFP), the probability that no particle resides in a macroscopic interval of a given size. Analytic solutions are found for a certain infinite sequence of rational polytropic indexes γ and the result can be analytically continued to any value of γ ≥ 1. Our findings agree with (and significantly expand on) previously known analytical and numerical results for EFP in quantum liquids. We also discuss interesting universal spacetime features of the instanton solution.

Perturbations on fluid surfaces

1965 ◽  
Vol 284 (1398) ◽  
pp. 397-401 ◽  
Keyword(s):  
Equation Of State ◽  
Unsteady Flow ◽  
Spherical Shell ◽  
Numerical Integration ◽  
Analytic Solution ◽  
Two Dimensional ◽  
Hydrodynamic Equations ◽  
Fluid Surface ◽  
Material Equation ◽  
Moving Fluid

Perturbations on a moving fluid surface are considered by means of two techniques. The first involves full numerical integration of the hydrodynamic equations of two-dimensional unsteady flow, while the second makes a number of assumptions to give an analytic solution. The predictions of the second method are compared with the results of the first method for the case of a thick collapsing spherical shell, and in particular the effect of the material equation of state on the growth of the perturbations is investigated.

scholarly journals Hydrodynamic equations for microscopic phase densities

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Viktor Gerasimenko ◽  
Vyacheslav Shtyk ◽  
Anatoly Zagorodny
Keyword(s):  
Initial Value Problem ◽  
Evolution Equations ◽  
Hydrodynamic Type ◽  
Hydrodynamic Equations ◽  
Initial Value ◽  
Type Hierarchy

AbstractEvolution equations for marginal generalized microscopic phase densities are introduced. The evolution equations for average values of microscopic phase densities are derived and a solution of the initial-value problem for the hydrodynamic type hierarchy obtained is constructed.

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