THE COMBINED-OPERATIONAL METHOD IN RADIATIVE TRANSFER

Abstract

The study of Radiative Transfer in media with curvature has received much attention in recent years. Attempt at solving such problems can be divided into three distinct groups, namely a) those aiming at obtaining approximate solutions of integro-differential equations for transfer (Sen [54) , Sobolev [61] , Lenoble and Sekera [52], Barkov [23], Smith [55), Wilson and Sen (77-80] etc.), b) those using the method of transforms (Smith [56, 57J, Cassell [44], Hunt [48] etc") and c) those based directly or indirectly on Ambartzumian' s mathematical and physical techniques. The third catagory of methods can again be subdivided into three groups, namely i) the Combined Operations method, ii) the Probabilistic method, and iii) the Invariant Imbedding method. In the present thesis, our main concern is the adaptation of the Combined Operations method, developed by Busbridge for the slab geometry, to solve transfer problems in spherical and cylindrical geometries" The thesis has been divided into two parts. Part 1 consists of two Chapters. In Chapter I, a brief account of the basic notions of radiation field and equation of radiative transfer has been given, integral equations for source function for the slab, spherical and cylindrical geometrics have been derived, and outlines of Ambartzumian’s mathematical method and the Combined Operations method of Busbridge for transfer problems in a semi-infinite plane-parallel medium have been considered. In Chapter II, the existing works based on Ambartzumian techniques have been reviewed. They comprised the critical appreciations of the works on the Invariant Imbedding method, the Probabilistic method in both plane parallel and curved geometry, and of those on the Combined Operations method in plane parallel geometry. In Part II of the thesis, the possibilities of utilising the Combined 0perations method to solve transfer problems in media with curvature have been examined. The problem of diffuse reflection by a homogeneous, isotropically scattering medium, having internal sources in addition to the incident flux at the boundary surface, has been considered. Chapter III is devoted to the case of a spherical medium and Chapter IV to the case of an infinite cylindrical medium. In each case, the scattering function has been defined in terms of the Neumann series solution of an auxiliary integral equation. Integra-differential equations for the scattering function and the emergent intensity have been established. These equations are considered suitable for numerial calculations. (Bellman and Kalaba (31), Bellman, Kagiwada and Kalaba [32], Rybicki [53]). Once the scattering function and the emergent intensity are calculated, the source function can be evaluated. The results contained in this part of the thesis are new and are being proposed for the first time.