Self-similar problems of the compression of an ideal gas and its dispersion from a point

Self-similar solutions describing one-dimensional unsteady flows of an ideal (inviscid and non-heat-conducting) perfect gas are considered. Whereas, in the well-known problem of isentropically compressing a gas to a plane, axis or centre of symmetry (henceforth, to a centre of symmetry CS) with a unit self-similarity index, the result of compression is a uniform flow moving to the CS, the problem of the retardation of such a flow of a continuous centred wave and the shock wave adjacent to it (the one shock wave in the plane case) subsequently arises. The gas is at rest behind the shock wave travelling from the CS. The change in the signs of the time and velocity in the solutions describing the finite isentropic compression of the gas gives a representation of the evolution of the flow in the case of uniform dispersion of the gas from the CS. Other known self-similar solutions with unit self-similarity exponent give an unbounded isentropic compression of a finite mass of gas to the CS (“compression into a point”). For such a compression, the density, pressure, internal energy and velocity are infinite but the entropy is finite. The entropy is also finite after the gas has been arrested by the shock wave travelling from the CS. The new self-similar problem concerning the “dispersion from a point” (plane or CS) of a finite mass of “hot” gas with an infinite initial energy, zero velocity and finite entropy is solved. In the new solutions (with and without a void in the neighbourhood of the CS), by virtue of the “mass integral” (its role is similar to the role of the energy integral in the strong explosion problem), all the trajectories of the hot gas particles are isolines of the self-similar variable with a self-similarity index found by dimensional analysis. The effect of a finite initial density of the cold gas surrounding the compressed gas on the solutions found, the self-similar solution arising here, and the occasionally paradoxical features of the self-similar solutions in the case of dispersion into a vacuum are discussed.