Abstract
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.
Highlights
Group analysis of differential equations is a powerful tool for obtaining exact solutions to nonlinear differential equations [1,2]
The gas dynamics equations are considered in the case of the special state equation and in the case of the state equation of a monatomic gas. 12-dimensional Lie algebra L12 and the 14-dimensional Lie algebra L14 correspond to the transformation groups admitted by the gas dynamics equations with specified state equations
The gas dynamics equations have been considered in the case of the state equation of a monatomic gas and the special state equation
Summary
Group analysis of differential equations is a powerful tool for obtaining exact solutions to nonlinear differential equations [1,2]. The general form of the solution to the gas dynamics equations with an arbitrary state equation with a linear velocity field with pressure and density depending on time is determined by an autonomous system of ordinary differential equations in [29]. Exact solutions of the gas dynamics equations with a linear velocity field are obtained from the submodels. For the gas dynamics equations with an arbitrary state equation, a representation of the solution with a linear velocity field is specified, a classification of submodels according to the state equations is obtained. In [14], a complete classification of submodels with a solution in the form of a linear velocity field was carried out according to the types of state equations
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