# Ideal Gas Outflow from a Cylindrical or Spherical Source into a Vacuum

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## Abstract

The solutions of initial and boundary value problems of the outflow of an ideal (inviscid and non-heat-conducting) gas from cylindrical and spherical sources into a vacuum are obtained. Time is measured from the moment, when the source is turned on; at this moment the source is surrounded by a vacuum. The entropy, flow rate, and the Mach number of the gas outflowing from the source are given, together with the source radius; the Mach number can be greater of or equal to unity. If the source radius is greater than zero, then the flow domain in the “radial coordinate–time” plane consists of the stationary source flow and adjoining non-self-similar centered expansion wave consisting of *C*^{−}-characteristics. The stationary flow is described by the known formulas, while the expansion wave is calculated by the method of characteristics. The calculations by this method confirm the earlier obtained laws for large values of the radial coordinate. The interface between the vacuum and the expansion wave is the straight trajectory of particles and, at the same time, a unique rectilinear *C*^{−}-characteristic. For the source of zero radius (“pointwise” source) the velocity, density, and speed of sound of the outflowing gas are infinite. The gas velocity remains infinite everywhere, while the density and speed of sound become zero for any non-zero values of the radial coordinate. For the pointwise source the problem of outflow into a vacuum is self-similar. In the plane of the “self-similar” velocity and speed of sound its solution is given by three singular points of a differential equation in these variables. At one of these points the self-similar velocity is infinite, the self-similar speed of sound is zero, and the self-similar independent variable varies from zero to infinity, with the exception of the extreme values.

## Keywords

initial and boundary value problems cylindrical and spherical sources outflow into a vacuum flow structures for nonzero and zero source radii self-similar solution for the pointwise source## Preview

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