Self-similar solutions to isothermal shock problems

TL;DR

This paper derives and verifies analytical and numerical solutions for isothermal shock problems in various geometries, highlighting new solutions for implosion and explosion scenarios with specific initial conditions.

Contribution

It presents the first exact solutions for isothermal implosion and explosion problems using similarity methods and explores the effects of initial density parameters.

Findings

Exact solutions for isothermal implosion and explosion problems are obtained.

Numerical solutions are validated against analytical expressions.

Initial density parameters influence the feasibility of solutions.

Abstract

We investigate exact solutions for isothermal shock problems in different one-dimensional geometries. These solutions are given as analytical expressions if possible, or are computed using standard numerical methods for solving ordinary differential equations. We test the numerical solutions against the analytical expressions to verify the correctness of all numerical algorithms. We use similarity methods to derive a system of ordinary differential equations (ODE) yielding exact solutions for power law density distributions as initial conditions. Further, the system of ODEs accounts for implosion problems (IP) as well as explosion problems (EP) by changing the initial or boundary conditions, respectively. Taking genuinely isothermal approximations into account leads to additional insights of EPs in contrast to earlier models. We neglect a constant initial energy contribution but introduce a parameter to adjust the initial mass distribution of the system. Moreover, we show that due to this parameter a constant initial density is not allowed for isothermal EPs. Reasonable restrictions for this parameter are given. Both, the (genuinely) isothermal implosion as well as the explosion problem are solved for the first time.