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Equilibrium and Non-Equilibrium diffusion approximation for the radiative transfer equation

Elena Demattè, Juan J. L. Velázquez

TL;DR

The paper analyzes the diffusion limit of the radiative transfer equation with emission-absorption and scattering, coupling to a temperature equation on a convex domain. Using matched asymptotic expansions, it derives a spectrum of limit problems across multiple scaling regimes, distinguishing equilibrium diffusion (Planck-distribution leading) from non-equilibrium diffusion (departure from Planck) and detailing the associated Milne and thermalization boundary layers as well as initial layers in time. The work provides a structured classification of stationary and time-dependent diffusion limits under varying relations among Milne, thermalization, and macroscopic lengths, and clarifies how boundary data and energy balance determine the limit systems. The results have implications for understanding when radiative heat transfer can be accurately modeled by diffusion in different physical regimes and highlight the roles of boundary/initial-layer corrections in practical computations.

Abstract

In this paper we study the distribution of the temperature within a body where the heat is transported only by radiation. Specifically, we consider the situation where both emission-absorption and scattering processes take place. We study the initial boundary value problem given by the coupling of the radiative transfer equation with the energy balance equation on a convex domain $ Ω\subset \mathbb{R}^3 $ in the diffusion approximation regime, i.e. when the mean free path of the photons tends to zero. Using the method of matched asymptotic expansions we will derive the limit initial boundary value problems for all different possible scaling limit regimes and we will classify them as equilibrium or non-equilibrium diffusion approximation. Moreover, we will observe the formation of boundary and initial layers for which suitable equations are obtained. We will consider both stationary and time dependent problems as well as different situations in which the light is assumed to propagate either instantaneously or with finite speed.

Equilibrium and Non-Equilibrium diffusion approximation for the radiative transfer equation

TL;DR

The paper analyzes the diffusion limit of the radiative transfer equation with emission-absorption and scattering, coupling to a temperature equation on a convex domain. Using matched asymptotic expansions, it derives a spectrum of limit problems across multiple scaling regimes, distinguishing equilibrium diffusion (Planck-distribution leading) from non-equilibrium diffusion (departure from Planck) and detailing the associated Milne and thermalization boundary layers as well as initial layers in time. The work provides a structured classification of stationary and time-dependent diffusion limits under varying relations among Milne, thermalization, and macroscopic lengths, and clarifies how boundary data and energy balance determine the limit systems. The results have implications for understanding when radiative heat transfer can be accurately modeled by diffusion in different physical regimes and highlight the roles of boundary/initial-layer corrections in practical computations.

Abstract

In this paper we study the distribution of the temperature within a body where the heat is transported only by radiation. Specifically, we consider the situation where both emission-absorption and scattering processes take place. We study the initial boundary value problem given by the coupling of the radiative transfer equation with the energy balance equation on a convex domain in the diffusion approximation regime, i.e. when the mean free path of the photons tends to zero. Using the method of matched asymptotic expansions we will derive the limit initial boundary value problems for all different possible scaling limit regimes and we will classify them as equilibrium or non-equilibrium diffusion approximation. Moreover, we will observe the formation of boundary and initial layers for which suitable equations are obtained. We will consider both stationary and time dependent problems as well as different situations in which the light is assumed to propagate either instantaneously or with finite speed.
Paper Structure (30 sections, 3 theorems, 129 equations, 1 figure, 1 table)

This paper contains 30 sections, 3 theorems, 129 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Scaling lengths and results
  3. Revision of the literature
  4. Structure of the paper
  5. Preliminary results
  6. Properties of the scattering operator
  7. Relation between the temperature and the radiation intensity
  8. The stationary diffusion approximation: different scales
  9. Case 1.1: $\ell_M=\ell_T\ll\ell_S$ and $L=1$. Equilibrium approximation
  10. Case 1.2: $\ell_M=\ell_T=\ell_S\ll L$. Equilibrium approximation
  11. Case 2: $\ell_M\ll\ell_T\ll L$. Equilibrium approximation
  12. Case 3: $\ell_M\ll \ell_T=L$. Transition from equilibrium to non-equilibrium.
  13. Case 4: $\ell_M\ll L\ll\ell_T$. Non-Equilibrium approximation
  14. Time dependent diffusion approximation. The case of infinite speed of light ($c=\infty$)
  15. Outer problems
  16. ...and 15 more sections

Key Result

Proposition 2.1

Let $K\in C\left({\mathbb{S}^2}\times{\mathbb{S}^2}\right)$, invariant under rotations, non-negative and satisfying Assume $\varphi\in L^\infty\left({\mathbb{S}^2}\right)$ satisfies $H[\varphi]=\varphi$. Then

Figures (1)

  • Figure 1: Representation of the change of variables.

Theorems & Definitions (9)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark
  • Remark
  • Remark
  • Lemma A.1
  • proof
  • proof : Proof of Proposition \ref{['prop.constant']}