Radiative Heating of a Glass Plate
MathematicS In Action, Tome 5 (2012) no. 1, pp. 1-30.

This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary conditions with the exact radiative transfer equation, assuming the absorption coefficient κ(λ) to be piecewise constant and null for small values of the wavelength λ as in the paper of N. Siedow, T. Grosan, D. Lochegnies, E. Romero, “Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering”, J. Am. Ceram. Soc., 88(8):2181-2187 (2005). An important observation is that for a fixed value of the wavelength λ, Planck function is a Lipschitz function with respect to the temperature. Using this fact, we deduce that the solution is at most unique. To prove existence of a solution, we define a fixed point problem related to our initial boundary value problem to which we apply Schauder theorem in a closed convex subset of the Banach separable space L 2 (0,t f ;C([0,l])). We use also Stampacchia truncation method to derive lower and upper bounds on the solution.

Publié le :
DOI : https://doi.org/10.5802/msia.6
Classification : 35K20,  35K55,  35K58,  35K90,  35Q20,  35Q60,  35Q80
Mots clés : elementary pencil of rays, Planck function, radiative transfer equation, glass plate, nonlinear heat-conduction equation, Stampacchia truncation method, Schauder theorem, Vitali theorem.
     author = {Luc Paquet and Raouf El Cheikh and Dominique Lochegnies and Norbert Siedow},
     title = {Radiative {Heating} of a {Glass} {Plate}},
     journal = {MathematicS In Action},
     pages = {1--30},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {5},
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     year = {2012},
     doi = {10.5802/msia.6},
     mrnumber = {3015737},
     zbl = {1328.35240},
     language = {en},
     url = {https://msia.centre-mersenne.org/articles/10.5802/msia.6/}
AU  - Luc Paquet
AU  - Raouf El Cheikh
AU  - Dominique Lochegnies
AU  - Norbert Siedow
TI  - Radiative Heating of a Glass Plate
JO  - MathematicS In Action
PY  - 2012
DA  - 2012///
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VL  - 5
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PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://msia.centre-mersenne.org/articles/10.5802/msia.6/
UR  - https://www.ams.org/mathscinet-getitem?mr=3015737
UR  - https://zbmath.org/?q=an%3A1328.35240
UR  - https://doi.org/10.5802/msia.6
DO  - 10.5802/msia.6
LA  - en
ID  - MSIA_2012__5_1_1_0
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Luc Paquet; Raouf El Cheikh; Dominique Lochegnies; Norbert Siedow. Radiative Heating of a Glass Plate. MathematicS In Action, Tome 5 (2012) no. 1, pp. 1-30. doi : 10.5802/msia.6. https://msia.centre-mersenne.org/articles/10.5802/msia.6/

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