Second-type self-similar solutions to the strong explosion problem

The flow resulting from a strong explosion at the center of an ideal gas sphere, whose density drops with the distance r from the origin as r ^-ω, is assumed to approach asymptotically the self-similar solutions by Sedov and Taylor. It is shown that the Sedov-Taylor (ST) solutions that exist only for ω<5 and are probably the most familiar example for self-similar solutions of the first type fail to describe the asymptotic flow obtained for 3≤ω<5. New second-type self-similar solutions that describe the asymptotic flow for 3<ω<5, as well as for ω≥5, are presented and analyzed. The shock waves described by these solutions are accelerating while the shock waves described by the ST solutions for ω<3 are decelerating. The new solutions are related to a new singular point in Guderley's map. They exist only for ω values smaller than some ω_c that depends upon the adiabatic index of the gas. The asymptotic flow obtained for ω≥ω_c is discussed in a subsequent paper.