Slightly Two or Three Dimensional Self-Similar Solutions

TL;DR

This paper develops analytical methods for slightly two- or three-dimensional self-similar hydrodynamic solutions, extending traditional one-dimensional solutions using perturbation theory and spherical harmonics, with validation against numerical simulations.

Contribution

It introduces a novel approach to derive slightly multi-dimensional self-similar solutions in hydrodynamics using linear perturbation theory around one-dimensional solutions.

Findings

Analytical solutions match well with numerical simulations.

Method effectively extends one-dimensional solutions to higher dimensions.

Applicable to strong spherical explosions with small angular deviations.

Abstract

Self similarity allows for analytic or semi-analytic solutions to many hydrodynamics problems. Most of these solutions are one dimensional. Using linear perturbation theory, expanded around such a one-dimensional solution, we find self-similar hydrodynamic solutions that are two- or three-dimensional. Since the deviation from a one-dimensional solution is small, we call these slightly two-dimensional and slightly three-dimensional self-similar solutions, respectively. As an example, we treat strong spherical explosions of the second type. A strong explosion propagates into an ideal gas with negligible temperature and density profile of the form rho(r,theta,phi)=r^{-omega}[1+sigma*F(theta,phi)], where omega>3 and sigma << 1. Analytical solutions are obtained by expanding the arbitrary function F(theta,phi) in spherical harmonics. We compare our results with two dimensional numerical simulations, and find good agreement.