|

Evolution of spatially localized thermal perturbations

Authors: Martinson L.K., Chigiryova O.Yu. Published: 24.12.2015
Published in issue: #6(63)/2015  
DOI: 10.18698/1812-3368-2015-6-16-24

 
Category: Mathematics | Chapter: Mathematical Physics  
Keywords: boundary problem of nonlinear heat conduction, effect of spatial localization of thermal perturbations

The article considers a one-dimensional boundary problem of the nonlinear heat conduction equation in a two-dimensional layer filled with a medium with bulk heat absorption. Numerical solutions to this problem for different characteristic values, while using a standard difference scheme, confirm the theoretical conclusions about the propagation mode of thermal perturbations. Thermal perturbations from the heated walls prove to propagate through the nonlinear medium with bulk heat absorption at the finite rate of the front propagation. An effect of spatial localization of thermal perturbations, which can even reach the finite depth for nonterminating time, is also observed at the defined values of the problem characteristics.

References

[1] Martinson L.K., Malov Yu.I. Differentsial’nye uravneniya matematicheskoy fiziki [Differential equations of mathematical physics]. Moscow, MGTU im. N.E. Baumana Publ., 2002. 368 p.

[2] Martinson L.K. Issledovanie matematicheskoy modeli protsessa nelineynoy teploprovodnosti v sredakh s ob’emnym pogloshcheniem. V kn.: Matematicheskoe modelirovanie. Protsessy v nelineynykh sredakh [Research into the process mathematical model of nonlinear thermal conductivity in spatial absorption media. In the book: Mathematic Simulation. Processes in Nonlinear Media]. Moscow, Nauka Publ., 1986, pp. 279-309.

[3] Maslov V.P., Danilov V.G., Volosov K.A. Matematicheskoe modelirovanie protsessov teplomassoperenosa [Mathematic simulation of heat and mass transfer processes]. Moscow, Nauka Publ., 1987. 362 p.

[4] Martinson L.K., Chigireva O.Yu. Spatial Localization of Thermal Perturbations in Nonlinear Process of Heat Conduction. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2013, no. 4, pp. 27-33 (in Russ.).

[5] Martinson L.K., Chigireva O.Yu. Boundary-value problems for quasilinear equations of the parabolic type. Irreversible processes in nature and engineering: Proceedings of the Seventh Russian National Conference. Neobratimye protsessy v prirode i tekhnike: Tr. Sed’moy Vseross. konf. V 3 ch. Moscow, 2013. Part II, pp. 32-33 (in Russ.).

[6] Samarskiy A.A., Galaktionov V.A., Kurdyumov S.P., Mikhaylov A.P. Rezhimy s obostreniem v zadachakh dlya kvazilineynykh parabolicheskikh uravneniy [Blowup regimes in problems for quasilinear parabolic equations]. Moscow, Nauka Publ., 1987. 480 p.

[7] Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, MGU Publ., 2004. 798 p.

[8] Samarskiy A.A. Teoriya raznostnykh skhem [Difference scheme theory]. Moscow, Nauka Publ., 1977. 656 p.

[9] Amosov A.A., Dubinskiy Yu.A., Kopchenova N.V. Vychislitel’nye metody dlya inzhenerov [Computational approaches for engineers]. Moscow, Vyssh. shk. Publ., 1994. 544 p.

[10] Matus P.P. Correctness of difference schemes for the semilinear parabolic equation with generalized solutions. Zh. Vychisl. Mat. Mat. Fiz. [Comput. Math. Math. Phys.], 2010, vol. 50, no. 12, pp. 2155-2175 (in Russ.).