Collapsing motions of a diatomic gas whose density depends only on time

Abstract

One submodel of gas motion with a linear velocity field is considered in the paper. Namely, a submodel that defines the movements of a polytropic gas with a density that depends only on time. A polytropic gas is a gas for which the internal energy is a function linear in temperature. The submodel under consideration is given by a system of ordinary differential equations of the 22nd order for unknown functions. These functions characterize the movements of gas particles and determine the type of density, pressure and entropy functions. The exact solution is sought for a special case, namely for a diagonal linearity matrix. Two new exact solutions have been found. The type of vector functions of velocity, density and pressure are determined. By the form of the velocity function, the world lines of motion of gas particles are recorded. In the three-dimensional space of coordinates x, y, z, the trajectories of gas particles for various initial data are constructed. A qualitative analysis of the movement was carried out. The Jacobi matrix is constructed. The moments of collapse of gas particles are determined by the value of the Jacobian. Both solutions have collapse: collapse on a straight line and collapse at a point.