Conductive-radiative heat transfer in grey materials
Authors:
Mika Laitinen and Timo Tiihonen
Journal:
Quart. Appl. Math. 59 (2001), 737-768
MSC:
Primary 35J65; Secondary 35K60, 80A20
DOI:
https://doi.org/10.1090/qam/1866555
MathSciNet review:
MR1866555
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Abstract: We study the well-posedness of a class of models describing heat transfer by conduction and radiation. For that purpose we propose an abstract mathematical framework that allows us to prove existence, uniqueness and the comparison principle for the weak solution with minimal or almost minimal a priori assumptions for the data. The theory covers different types of grey materials, that is, both semitransparent and opaque bodies as well as isotropic or nonisotropic scattering/reflection provided that the material properties do not depend on the wavelength of the radiation. To demonstrate the use of the abstract theory we consider in detail two examples, heat transfer between opaque bodies with diffuse-grey surfaces and a model with semitransparent material and specularly reflecting surfaces.
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Herbert Gajewski, Konrad Gröger, and Klaus Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974
David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983
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E. Hennebach, P. Junghanns, and G. Vainikko, Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems, Integral Equations Operator Theory 22, 37–64 (1995)
C. T. Kelley, Existence and uniqueness of solutions of nonlinear systems of conductive-radiative heat transfer equations, Transport Theory Statist. Phys. 25, 249–260 (1996)
Michal Křížek and Liping Liu, On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type, Applicationes Mathematicae 24, 97–107 (1996)
Mika Laitinen and Timo Tiihonen, Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials , Math. Meth. Appl. Sci. 21, 375–392 (1998)
Michael Metzger, Existence for a time-dependent heat equation with non-local radiation terms, Math. Meth. Appl. Sci. 22, 1101–1119 (1999)
M. F. Modest, Radiative Heat Transfer, McGraw-Hill, New York, 1993
Jindr̆ich Nec̆as, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967
C. Perret and P. Witoiriski, Équation de la chaleur et réflections multiples, Ann. Inst. Henri Poincaré Anal. Non Linéaire 8, 677–689 (1991)
Rogerio Martins Saldanha da Gama, Existence, uniqueness and construction of the solution of the energy transfer problem in a rigid and nonconvex black body, Z. Angew. Math. Phys. 42, 334–347 (1991)
Rogerio Martins Saldanha da Gama, An alternative mathematical modelling for coupled conduction/radiation energy transfer phenomenon in a system of N gray bodies surrounded by a vacuum, Internat. J. Non-Linear Mechanics 30, 433–447 (1995)
Robert Siegel and John R. Howell, Thermal Radiation Heat Transfer, Hemisphere, third edition, 1992
E. M. Sparrow and R. D. Cess, Radiation Heat Transfer, Hemisphere, augmented edition, 1978
Timo Tiihonen, Stefan-Boltzmann radiation on non-convex surfaces, Math. Meth. Appl. Sci. 20, 47–57 (1997)
Timo Tiihonen, A nonlocal problem arising from heat radiation on non-convex surfaces, European J. Appl. Math. 8, 403–416 (1998)
Lutz Weis, Integral operators and changes of density, Indiana Univ. Math. J. 31, 83–96 (1982)
Eberhard Zeidler, Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990
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© Copyright 2001
American Mathematical Society