Invariant submodels of a special vortex

Abstract

Two invariant submodels of a spherically partially invariant model of gas dynamics called a special vortex are investigated: a steady-state model and a homogeneous submodel. A complete analytical description of them is given: all the invariant functions specifying the solution have a representation in terms of an auxiliary function and its derivatives. This function is the solution of the first-order ordinary differential equation for the steady-state special vortex and the Schwarz equation for the homogeneous special vortex. A qualitative description of the gas in the homogeneous special vortex is given. The characteristic feature of this motion is the formation of a gas cloud from the rarefied medium accompanying its motion towards the observer and the subsequent dispersion again to a rarefied state (in the limit to a vacuum) at infinity. A barochronic homogeneous special vortex is fully described. It is proved that the special vortex is generated by special initial data: the algebraic invariants of the Jacobian matrices of the velocity vector field depend solely on the invariant independent variables of time and the radial coordinate. A representation of the invariants in terms of the parameters of the singular vortex is obtained.