Homogenized Models of Radiation Transfer in Multiphase Media

Abstract

The physical model of an absorbing scattering medium (ASM) and the corresponding mathematical model of the radiation transfer equation (RTE) were originally formulated to study dilute dispersed systems like fog [SiHo02]. Such media contain well-separated small particles (droplets), and so they are essentially heterogeneous. Therefore, even the derivation of the conventional RTE can be considered as a problem of homogenization. Nevertheless, homogenization of radiation transfer is often meant as a problem for the conventional RTE with oscillating coefficients [Pa05]. Oscillations with a period much smaller than the characteristic length scale of the problem can be averaged to obtain the effective coefficients of the homogenized RTE. Thus, from a physical point of view, a heterogeneous ASM with short-scale variations of the radiative properties is replaced by an equivalent homogeneous ASM with the effective radiative properties. Such a homogenization problem contains two small length scales of different size. The smallest scale is the size of the scattering inhomogeneity (particle) and the intermediate scale is the period of oscillations of the radiative properties.

The radiative properties of dilute dispersed media can be obtained from the scattering properties of a single particle [SiHo02], but considerable difficulties arise in dense dispersed systems where the volume fraction of the dispersed phases is comparable with the volume fraction of the matrix. The distances between the scatterers (particles) become comparable with their sizes in such systems, so that a mutual influence of the scatterers should be taken into account. This is referred to as dependent scattering. The current mathematical approach to dependent scattering is the RTE with modified radiative properties [BaSa00]. However, the applicability of the RTE to dense dispersed systems has never been rigorously proved.