Existence theory for a one-dimensional problem arising from the boundary layer analysis of radiative flows

• Published in: Progress in nuclear energy (New series) Vol. 53; no. 8; pp. 1105 - 1113
• Main Authors: de Azevedo, F.S., Thompson, M., Sauter, E., Vilhena, M.T.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier Ltd 01.11.2011; Elsevier
• Subjects: Anisotropy; Applied sciences; Asymptotic methods; Asymptotic properties; Conservation; Differential equations; Energy; Energy. Thermal use of fuels; Exact sciences and technology; Fission nuclear power plants; Fuels; Function space; Installations for energy generation and conversion: thermal and electrical energy; Mathematical analysis; Mathematical models; Nuclear fuels; Nuclear reactor components; Radiative transfer; Second order nonlinear integro-differential parabolic equations; Singular perturbations; Singular perturbations; Second order nonlinear integro-differential parabolic equations; Radiative transfer; Asymptotic methods; Transport equation; Differential equation; Perturbation; Optically thick medium; Modeling; Nuclear reactor; Equation system; Numerical simulation; Anisotropic scattering; Comparative study; Boundary layer; Second order nonlinear integro-differential; parabolic equations
• ISSN: 0149-1970
• DOI: 10.1016/j.pnucene.2011.06.005

We consider a simplified system of equations which models the transfer of energy with conductive, convective and radiative effects inside a convex region filled with a compressible fluid whose velocity field is known. The asymptotic analysis for positive but small distance from an optically thick medium leads to a one-dimensional system of differential equation which couples the temperature and the radiative intensity. We show that this system obeys a conservation law and this feature is explored in order to reduce the problem to a single one-dimension transport equation with anisotropic scattering. This equation admits a formulation in terms of integral operators in a suitable function space which allows us to establish the existence of a solution and infer its behavior far from the boundary. We also provide numerical simulations and comparison with the theoretical results which we have shown in order to validate our methodology. ► We establish the existence of solutions for a boundary layer problem. ► We study the asymptotics of the convective-radiative transport equation. ► We establish the existence of solutions of a one-dimensional transport equation with signed kernel.