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Ultra-relativistic geometrical shock dynamics and vorticity

Published online by Cambridge University Press:  14 May 2008

JEREMY GOODMAN  and
ANDREW MACFADYEN
JEREMY GOODMAN
Affiliation:
Institute for Advanced Study, Princeton, NJ 08544, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
ANDREW MACFADYEN
Affiliation:
Institute for Advanced Study, Princeton, NJ 08544, USA Department of Physics, New York University, New York, NY 10003, USA
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Abstract

Geometrical shock dynamics, also called CCW theory, yields approximate equations for shock propagation in which only the conditions at the shock appear explicitly; the post-shock flow is presumed approximately uniform and enters implicitly via a Riemann invariant. The non-relativistic theory, formulated by G. B. Whitham and others, matches many experimental results surprisingly well. Motivated by astrophysical applications, we adapt the theory to ultra-relativistic shocks advancing into an ideal fluid whose pressure is negligible ahead of the shock, but is one third of its proper energy density behind the shock. Exact results are recovered for some self-similar cylindrical and spherical shocks with power-law pre-shock density profiles. Comparison is made with numerical solutions of the full hydrodynamic equations. We review relativistic vorticity and circulation. In an ultra-relativistic ideal fluid, circulation can be defined so that it changes only at shocks, notwithstanding entropy gradients in smooth parts of the flow.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 604 , 10 June 2008 , pp. 325 - 338
Copyright
Copyright © Cambridge University Press 2008

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