We consider linear transport in an anisotropic medium with velocity dependent cross sections and scattering kernel . We introduce a scaling in terms of a small parameter , where the leading-order term describes an equilibrium in velocity space between collisions with a cross section that is an even function of and scattering modes even-even and odd-odd in and . We show that the asymptotic solution of the transport equation leads to a diffusion equation with a drift term with an error in and derive consistent initial and boundary conditions from the analysis of the initial and boundary layers. The analysis of the drift terms shows that they result from anisotropic interactions with the medium and also from streaming between neighboring but different equilibria. The restriction of our results to isotropic media yields back the Larsen–Keller diffusion equation, while the one-speed form reduces to the result obtained by Pomraning and Prinja [Ann. Nucl. Energy 22, 159 (1995)] for the particular case of isotropic cross sections with an “output” scattering kernel .
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August 2008
Research Article|
August 13 2008
Asymptotic theory of the linear transport equation in anisotropic media
Richard Sanchez;
Richard Sanchez
a)
1Commissariat à l’Energie Atomique, DEN/DANS/DM2S, Service d’Etudes de Réacteurs et de Mathématiques Appliquées,
CEA de Saclay
, 91191 Gif-sur-Yvette Cedex, France
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Jean Ragusa;
Jean Ragusa
b)
2Department of Nuclear Engineering,
Texas A&M University
, 3133 TAMU, College Station, Texas 77843-3133, USA
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Emiliano Masiello
Emiliano Masiello
c)
1Commissariat à l’Energie Atomique, DEN/DANS/DM2S, Service d’Etudes de Réacteurs et de Mathématiques Appliquées,
CEA de Saclay
, 91191 Gif-sur-Yvette Cedex, France
Search for other works by this author on:
Richard Sanchez
1,a)
Jean Ragusa
2,b)
Emiliano Masiello
1,c)
1Commissariat à l’Energie Atomique, DEN/DANS/DM2S, Service d’Etudes de Réacteurs et de Mathématiques Appliquées,
CEA de Saclay
, 91191 Gif-sur-Yvette Cedex, France
2Department of Nuclear Engineering,
Texas A&M University
, 3133 TAMU, College Station, Texas 77843-3133, USA
a)
Electronic mail: [email protected].
b)
Electronic mail: [email protected].
c)
Electronic mail: [email protected].
J. Math. Phys. 49, 083504 (2008)
Article history
Received:
May 11 2008
Accepted:
July 04 2008
Citation
Richard Sanchez, Jean Ragusa, Emiliano Masiello; Asymptotic theory of the linear transport equation in anisotropic media. J. Math. Phys. 1 August 2008; 49 (8): 083504. https://doi.org/10.1063/1.2966094
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