Self-similarity of the Third Type in the Strong Explosion Problem

TL;DR

This paper discovers a new self-similar solution of the third type for strong explosion problems in power-law atmospheres, filling the gap between known solutions for specific density profiles.

Contribution

It identifies and provides numerical evidence for a novel self-similar solution of the third type in the intermediate density index range, previously unknown.

Findings

Found a self-similar solution for 3<α<3.26

Provided numerical proof of the solution as an asymptotic state

Demonstrated the solution is neither of type I nor type II

Abstract

Propagation of a blast wave due to strong explosion in the center of a power-law-density ($\rho \propto r^{-\alpha}$) spherically symmetric atmosphere is studied. For adiabatic index of 5/3, the solution was known to be self-similar, (of type I) for $\alpha <3$, self-similar (of type II) for $\alpha >3.26$, and unknown in between. We find a self-similar solution for $3<\alpha <3.26$, and give a (tentative) numerical proof that this solution is indeed an asymptotic of the strong explosion. This self-similar solution is neither of type I (dimensional analysis does not work), nor of type II (the index of the solution is known without solving an eigenvalue problem).