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

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 compression of gas followed by expansion

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.

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

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.

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

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 solution of the hydrodynamic submodels of rank 2 with rotation

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.

Reduction of differential and invariant submodels to invariant

Classification of differential and invariant submodels of the 3-h-parametrical group consisting of two transfers and transfer with stretching is given in work. The reduction of four submodels to an invariant submodel on some subgroup is proved.

The established screw flows of gas

A number of properties of an invariant submodel of the established screw flow of gas is noticed. Group classification is carried out. Invariant and some differentially invariant decisions are considered. The condition of existence of the decision without feature is removed