Slightly two- or three-dimensional self-similar solutions

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 ρ(r, θ, ϕ) = r^−ω[1 + σF(θ, ϕ)], where ω > 3 and σ ≪ 1. Analytical solutions are obtained by expanding the arbitrary function F(θ, ϕ) in spherical harmonics. We compare our results with two-dimensional numerical simulations, and find good agreement.