Slightly two- or three-dimensional self-similar solutions

Re'em Sari, Nate Bode*, Almog Yalinewich, Andrew MacFadyen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

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

Original languageAmerican English
Article number087102
JournalPhysics of Fluids
Volume24
Issue number8
DOIs
StatePublished - 16 Aug 2012

Bibliographical note

Funding Information:
We thank Yonatan Oren for helpful discussions. This research was partially supported by ERC and IRG Grants and by a Packard fellowship. R.S. is a Guggenheim fellow and a Radcliffe fellow. A.I.M. acknowledges support from NSF Grant No. AST-1009863 and NASA Grant No. NNX10AF62G.

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