Abstract
Fin materials can be observed in a variety of engineering applications. They are used to ease the dissipation of heat from a heated wall to the surrounding environment. In this work, we consider a nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient. The equation(s) under study are highly nonlinear. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. Firstly, we consider the Lie group analysis for different cases of thermal conductivity and the heat transfer coefficients. These classifications are obtained from the Lie group analysis. Then, the first integrals of the nonlinear straight fin problem are constructed by three methods, namely, Noether’s classical method, partial Noether approach and Ibragimov’s nonlocal conservation method. Some exact analytical solutions are also constructed. The obtained result is also compared with the result obtained by other methods.
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References
X Xin, arXiv:1501.03580 (2015)
R Naz, I L Freire and I Naeem, Abs. Appl. Anal. 2014, 978636 (2014)
S Kim, J Moon, and C H Huang, J. Phys. D: Appl. Phys. 40(14), 4382 (2007)
R J Moitsheki and M D Mhlongo, J. Appl. Math. (2012)
A Aziz and S M Enamul Hug, J. Heat Transfer-Trans. ASME 97, 300 (1975)
C-H Chiu and C-K Chen, Int. J. Heat Mass Transfer (2002)
H Tari, D D Ganji, and H Babazadeh , Phys. Lett. A 363, 213 (2007)
G G Polat, Ö Orhan and T Özer, Adv. Math. Phys. 2014, 695408 (2014)
Ö Orhan, G G Polat, and T Özer, J. Inequalities Appl. 2013, 147 (2013)
A Aziz, Int. J. Heat Mass Transfer 20, 1253 (1977)
M N Bouaziz and S Hanini, Heat and Mass Transfer 44, 1 (2007)
E Noether, Transport Theor. Stat. Phys. 1(3), 186 (1971)
A H Kara, F M Mahomed, I Naeem, and C WafoSoh, Math. Meth. Appl. Sci. 30(16), 2079 (2007)
N H Ibragimov, J. Math. Anal. Appl. 333, 311 (2007)
N H Ibragimov, Similarity and Symmetry Methods Lecture Notes in Applied and Computational Mechanics 73, 61 (2014)
R K Gazizov, N H Ibragimov, and S Y Lukashchuk, Commun. Nonlinear Sci. Numer. Simul. 19, 404 (2014)
N H Ibragimov, J. Phys. A 44(43), 432002 (2011)
J L Cieśliński and T Nikiciuk, J. Phys. A 43 (2010)
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YAŞAR, E., YILDIRIM, Y. & GİRESUNLU, İ.B. First integrals and analytical solutions of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient. Pramana - J Phys 87, 18 (2016). https://doi.org/10.1007/s12043-016-1227-5
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DOI: https://doi.org/10.1007/s12043-016-1227-5