Asymptotic theory of the linear transport equation in anisotropic media

Abstract

We consider linear transport in an anisotropic medium with velocity dependent cross sections σ(r,v,t) and scattering kernel P(r,v′→v,t). We introduce a scaling in terms of a small parameter ϵ, where the leading-order term describes an equilibrium in velocity space between collisions with a cross section that is an even function of v and scattering modes even-even and odd-odd in v and v′. We show that the asymptotic solution of the transport equation leads to a diffusion equation with a drift term with an error in ϵ2 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 P(r,Ω,t).

The Asymptotic Drift-Diffusion Limit of Thermal Neutrons

“…The results of our analysis are that (1) the leading-order scalar-flux amplitude satisfies a drift-diffusion equation, (2) the leading-order and next-order scalar fluxes have Maxwell-Boltzmann energy distributions at the local material tem perature, (3) the drift velocity is proportional to the local temperature gradient in the scattering medium, and (4) the angular flux is isotropic to leading order and linearly anisotropic to the next order. The first and fourth results have appeared in previous analyses (Sanchez et al, 2008), but to our knowledge the connections with the Maxwell-Boltzmann energy distribution and mate rial temperature gradients have not been previously published.…”

“…Perhaps the best-known example is the asymptotic limit of large scat tering cross sections and small absorption cross sections. In this situation the leading-order neutron transport solution satisfies a diffusion or drift-diffusion equation (Larsen and Keller, 1974;Habetler and Matkowsky, 1975;Sanchez, Ragusa, and Masiello, 2008). Other transport applications have their own use ful asymptotic limits.…”

Diffusion asymptotics for linear transport with low regularity

“…The conditions on the parameters σ ε and γ ε can be relaxed as well, e.g., the parameters may depend on v in a certain form; see e.g. [24]. As can be seen from the proofs of our results, also the conditions on the data can be relaxed to some extent.…”

Asymptotic solution of light transport problems in optically thick luminescent media

“…Refs. [36][37][38][39][40]. Many studies have addressed this limitation using various other methods.…”

“…Substituting (42) into (40) and collecting the O(1) and O( ) terms, gives the following two equations in the half-space ζ * > 0:…”