Homogeneous singular vortex

An analytical description is given to the spherical partially invariant solution of the gas-dynamics equations in the case of additional symmetry – the homogeneous singular vortex. The solution was specified by a generalized potential – an auxiliary function satisfying the inhomogeneous Schwarz equation. It is proved that the part of the factor system of the homogeneous singular vortex in a Lagrangian representation that describes the kinematics of a gas particle is a system of linear equations with the potential defined by the solution of the Schwarz equation. For particular values of the adiabatic exponent equal to 1, 4/3, and 5/3, the solution of the Schwarz equation is written in terms of lower-order equations. The isothermal gas flow in the homogeneous singular vortex isdescribed. It is shown that a periodic geometrical trajectory configuration can exist but the gas density in this case has a singularity. A physically definite solution exists on time intervals that do not contain singularity points. Examples of motion obtained by implementation of analytical formulas on a computer are given.