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Physics of Fluids / Volume 18 / Issue 2 / ARTICLES / Others

Phys. Fluids 18, 027106 (2006); doi:10.1063/1.2174567 (8 pages)

First and second type self-similar solutions of implosions and explosions containing ultrarelativistic shocks

Re’em Sari

Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, California 91125

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(Received 27 December 2005; accepted 12 January 2006; published online 24 February 2006)

We derive self-similar solutions for ultrarelativistic shock waves propagating into cold material of power law density profile in radius ρ∝r−k. We treat both implosions and explosions in three geometries: planar, cylindrical, and spherical. For spherical explosions these are the first type solutions of Blandford and McKee for k<4; they are the second type solutions found by Best and Sari for k>5−math. In addition we find new, hollow (with evacuated interior), first type solutions that may be applicable for 4<k<17/4. This “sequence” with increasing k of first type solutions, hollow first type solutions, and then second type solutions is reminiscent of the nonrelativistic sequence. However, although in the nonrelativistic case there is a range of k which corresponds to a “gap”—a range in k with neither first nor second type solution which separates the hollow first type solutions and the second type solutions, here there is an “overlap”: a range of k for which current considerations allow for both hollow first and second type solutions. Further understanding is needed to determine which of the two solutions apply in this overlap regime. We provide similar exploration for the other geometries and for imploding configurations. Interestingly, we find a gap for imploding spherical shocks and exploding planar shocks and an overlap for imploding planar solutions. Cylindrical configurations have no hollow solutions and exhibit a direct transition from first type to second type solutions, without a gap or an overlap region.

© 2006 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. THE SELF-SIMILAR EQUATIONS IN ARBITRARY DIMENSIONS
  3. FIRST TYPE SOLUTIONS
  4. SECOND TYPE SOLUTIONS
  5. THE TYPE-I TO TYPE-II TRANSITION
  6. DISCUSSION

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KEYWORDS and PACS

Keywords

explosions, chemically reactive flow, shock waves, fractals, relativistic fluid dynamics

PACS

  • 47.40.Rs

    Detonation waves

  • 47.40.Nm

    Shock wave interactions and shock effects

  • 47.70.Fw

    Chemically reactive flows

  • 47.53.+n

    Fractals in fluid dynamics

  • 47.75.+f

    Relativistic fluid dynamics

  • 82.33.Vx

    Reactions in flames, combustion, and explosions

ARTICLE DATA

Permalink
http://link.aip.org/link/doi/10.1063/1.2174567

PUBLICATION DATA

ISSN:

1070-6631 (print)  
1089-7666 (online)

Publisher:

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    E. Waxman and D. Shvarts, "Second-type self-similar solutions to the strong explosion problem," Phys. Fluids A 5, 1035 (1993)PFADEB000005000004001035000001.

    R. D. Blandford and C. F. McKee, "Fluid dynamics of relativistic blast waves," Phys. Fluids 19, 1130 (1976)PFLDAS000019000008001130000001.

    P. Best and R. Sari, "Second-type self-similar solutions to the ultrarelativistic strong explosion problem," Phys. Fluids 12, 3029 (2000)PHFLE6000012000011003029000001.

    J. C. Hidalgo and S. Mendoza, "Self-similar imploding relativistic shock waves," Phys. Fluids 17, 96101 (2005)PHFLE6000017000009096101000001.


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