Solution of radiative transfer equation in finite medium involving Fresnel reflection
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Solution of radiative transfer equation in finite medium involving Fresnel reflection Dalia A. Gharbieaa Department of Basic Science, Misr Higher Institute for Engineering and Technology, Mansoura, Egypt Received: 20 March 2020 / Accepted: 19 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Radiative transfer problem in an absorbing, scattering and homogeneous finite medium with Rayleigh scattering subjected to externally incident radiation is considered. The solution is evaluated when the case of Fresnel and diffuse reflections is present at the boundary. Galerkin approximation is introduced to calculate some functionals arising in radiative transfer problems such as reflection and transmission coefficients. Numerical results are reported and compared with the available published data which show very good agreement.
1 Introduction Rayleigh scattering occurs when the incident wavelength is greater than the size of the scattering particles. Lord Rayleigh explained why the sky is blue and sunset is red by using this scattering function. This scattering function can be used in different areas in physics, for example, astrophysics, oceanography and tissue optics. The first work on the transfer of the polarized radiation has carried out by Gans [1], Chandrasekhar [2], and Sobolev [3]. Pomraning [4, 5] has delivered a variational expression for the albedo for a half-space in the case of isotropic phase function, Rayleigh phase function over polarization and Rayleigh scattering property accounting for polarization. Radiative transfer equation for Rayleigh scattering was solved for different media using different methods. Bicer and Kaskas [6] solved this equation in infinite medium using Green’s function. Degheidy and Abdel-Krim [7] represent the effect of Fresnel and diffuse reflectivities on light transport in half space. Abdel Krim et al. [8] presented a solution of the radiative transfer equation in a plane parallel medium involving Rayleigh scattering phase function with specular and diffuse reflecting boundaries. A vast amount of work is available in the literature on the solution of radiative transfer equation with Rayleigh scattering and diffuse reflection in finite medium (Bicer and Kaskas [6], Abdel-Krim et al. [8], etc.). But it appears that little work has been reported of the problem of radiation transfer involving Fresnel reflection (Sieweter [9], Degheidy and Abdel-Krim [7] and Williams [10]). In this work, we solve the time-independent, monoenergetic, radiative transfer equation for Rayleigh scattering at unpolarized case in finite medium for Fresnel case using Galerkin technique.
a e-mail: [email protected] (corresponding author)
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Eur. Phys. J. Plus
(2020) 135:812
2 Formulation of the problem The unpolarized radiative transfer equation for an absorbing, gray, anisotropic scattering, plane parallel medium of optical thickness (b) can be written as [11] ∂ψ(x, μ) ω μ + ψ(x, μ) ∂x 2
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