M. S. Abdel Krim

Modulational instability of a weakly relativistic ion acoustic wave in a warm plasma with nonthermal electrons

An investigation has been made of modulational instability of a nonlinear ion acoustic wave in a weakly relativistic warm unmagnetized nonthermal plasma whose constituents are an inertial ion fluid and nonthermally distributed electrons. Up to the second order of the perturbation theory, a nonlinear Schrodinger type (NST) equation for the complex amplitude of the perturbed ion density is obtained. The coefficients of this equation show that the relativistic effect, the finite ion temperature and the nonthermal electrons modify the condition of the modulational stability. The association between the small-wavenumber limit of the NST equation and the oscillatory solution of the Korteweg-de Varies equation, obtained by a reductive perturbation theory, is satisfied.

EFFECTS OF FRESNEL AND DIFFUSED REFLECTIVITIES ON LIGHT TRANSPORT IN A HALF-SPACE MEDIUM

Abstract Radiation transfer problem for isotropically scattering, non-absorbing half-space medium with angular-dependent (specular) and constant diffuse reflecting boundary is considered. The angular-dependent reflectivity of the boundary is considered as Fresnel’s reflection probability function. Some physical and engineering quantities of interest such as angular distribution of radiation, total density of radiation and extrapolated endpoint are computed for various values of refractive index and diffuse reflecting coefficient using variational method. For sake of comparisons, results with average and effective reflectivities are carried out for different values of refractive indices. Our results for non-diffuse reflectivity are compared with the available data given for specular Fresnel’s reflectivity which show excellent agreement.

Variational approach to the milne problem with a specular and diffuse reflecting boundary

Abstract An approximate solution of the Milne integral equation for specular and diffuse reflecting boundary is deduced by using variational calculus. An approximate, (four variable parameters) expression, for the particle density and emergent angular distribution has been proposed. Numerical results for extrapolated endpoint, particle density and angular distribution in a region close to the boundary are calculated. The percentage difference for extrapolated endpoint, taking the exact values as a reference, is approximately 1.4 × 10−10 for a non-reflecting medium and does not exceed 0.01% for all degrees of blackness in the diffusion reflection. The results for a combination of specular and diffuse reflections are also given.

Milne problem in stochastic plane medium with linear anisotropic scattering

Abstract The Milne problem of radiative transfer in planar stochastic medium, with linear anisotropic scattering is considered. The medium is assumed to be consisting of two randomly mixed immiscible fluids, with the mixing statistics described as two state homogeneous Markov process. Pomraning–Eddington approach is used to obtain an explicit form for the radiation energy density in deterministic case. A formalism, developed to treat radiative transfer in binary statistical mixtures, is used to obtain the ensemble averaged energy density in stochastic case for the problem under consideration. It is shown that the statistical nature of the medium leads to two contradictory different definitions for the linear extrapolated distance. In the case of isotropic scattering, numerical results are given with comparisons. Results are also given in the case of linear anisotropic scattering at different values of anisotropic coefficient.

Radiation transfer for linearly anisotropic phase functions

Radiative transfer equation in a plane-parallel medium with isotropic boundary conditions for linearly anisotropic scattering phase function is considered. Two coupled integral equations for total density of radiation and total radiation flux are obtained. The Galerkin method is used to solve these equations. Numerical results for the radiative fluxes at the boundaries show that the Galerkin method yields accurate results compared well with other exact methods.

Stochastic radiative transfer in a finite plane with anisotropic scattering in a binary Markovian mixture

Abstract The time-independent linear transport problem in a stochastic finite-plane medium with linear anisotropic scattering is considered. The medium is assumed to consist of two randomly mixed immiscible fluids, with the mixing statistics described as a two-state homogeneous Markov process. The Pomraning–Eddington approach is used to obtain an explicit solution to the problem in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution for the problem under consideration. In the case of isotropic scattering, explicit analytic results for reflectivity and transmissivity, which show a good agreement with Monte Carlo benchmark results, are given. Results for reflectivity and transmissivity in the case of linear anisotropic scattering are also given.

Radiation transfer problems with reflective boundary conditions and internal sources

Abstract Exact relations for the radiation heat flux at the boundaries of a slab with reflecting boundary conditions and internal sources are obtained in terms of the albedos of a source-free slab with isotropic boundary conditions. The advantage of this equation is that the internal sources appear in the expression for radiation heat flux. A particular solution of the Boltzmann equation is not needed. Available exact values for albedos give exact values for the radiation heat flux. In this study, we calculate the albedos by using the bivariational technique. Homogeneous as well as inhomogeneous media may be considered.

Angular distribution of radiation in an isotropically scattering slab

The equation of radiative transfer in an isotropically scattering slab subject to general boundary conditions is solved. The Padé approximation technique is used to calculate the reflected and transmitted angular distributions. Numerical results for angular distributions through and at the boundaries of a finite slab are calculated by the Padé approximation technique. The results for a Padé approximation of [0/1] are compared with results obtained by the Galerkin method.

Radiation transfer in a diffuse and specular reflecting slab with Rayleigh scattering

A method of analysis is presented for solving the radiative transfer problem in an absorbing, emitting, inhomogeneous, and anisotropically scattering plane-parallel medium with specular and diffuse reflecting boundaries and internal source (problem 1). Exact relations for the radiation heat flux at the boundaries of problem 1 are obtained in terms of the radiation density and albedos of the corresponding source-free medium with specular reflecting boundaries (problem 2). Two coupled integral equations for the radiation density and the second moment of the radiation intensity for problem 2 with Rayleigh phase functions are obtained. The Galerkin method is used to solve these equations. Albedos of problem 2 are compared with theFn method. Numerical results for radiation heat fluxes at the boundaries of problem 1 are tabulated for different forms of the internal source.

Energy albedo for multiregion slabs

An asymptotic solution for the equation of radiative transfer in an inhomogeneous medium was obtained on the basis of the corresponding solutions for homogeneous sub-layers in the slowing down region. Function relations between the reflection and transmission coefficients for the whole slab and those of the sublayers are given. The invariant embedding concepts are used to get the reflection and transmission coefficients for the sub-layers. We assumed different models for the slowing-down kernels. Laplace transform was used to transform the Boltzmann equation to one velocity approximation with re-scaled mean-free path and single-scattering albedo. Numerical results are given for energy albedo as a function of the mass number of the host medium.