On the self-similar power-driven expansion into vacuum problem

The problem of hydrodynamic expansion of material into vacuum under the influence of continuous, in-depth energy deposition is analyzed. The technique described in the present paper differs from previous approximations in that a functional relation is assumed for one of the unknowns in the set of ODE. Strong physical arguments are presented concerning the validity of this choice. The set of ODE which results from a similarity transformation of the PDE is used to find the other unknowns and to provide a criterion of applicability. Further, the set of nonlinear PDE which describes the expansion of material into a vacuum is presented. It is shown that this set is transformed to a set of nonlinear ODE using the similarity procedure. Finally, approximate solutions are obtained with the known numerical solutions. It is concluded that using the approximate solutions that are obtained, it is possible to replace similarity variables and to be able to follow the temporal evolution in space of the density, velocity and energy.