Similarity analysis of a nonlinear fin equation. (English) Zbl 1114.80003

Pakdemirli, M.; Sahin, A. Z.

Appl. Math. Lett. 19, No. 4, 378-384 (2006).

The similarity analysis of a nonlinear fin equation is done by determining the scaling, translational and spiral group symmetries. The authors perform a classification of special Lie group symmetries with respect to the conductivity and heat transfer coefficient. In view of the similarity transformation the fin equation can be converted into an ordinary differential equation which is much easier to solve than the original partial differential equation.

Reviewer: Vladimir Sládek (Bratislava)

MSC:

80A20	Heat and mass transfer, heat flow (MSC2010)
22E70	Applications of Lie groups to the sciences; explicit representations

Keywords:

heat transfer; similarity transformation; scaling; translation; spiral symmetry

References:

[1]	Aziz, A.; Na, T. Y., Periodic heat transfer in fins with variable thermal parameters, International Journal of Heat and Mass Transfer, 24, 8, 1397-1404 (1981) · Zbl 0458.76076
[2]	Chung, B.; Iyer, J., Optimum design of longitudinal rectangular fins and cylindrical spines with variable heat-transfer coefficient, Heat Transfer Engineering, 14, 1, 31-42 (1993)
[3]	Bobisud, L. E.; Calvert, J. E., A problem in cooling fin design, Quarterly of Applied Mathematics, 57, 2, 369-380 (1999) · Zbl 1031.34013
[4]	Dulkin, I. N.; Garasko, G. I., Analytical solutions of the 1-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient—II. Recurrent direct solution, International Journal of Heat and Mass Transfer, 45, 9, 1905-1914 (2002) · Zbl 1023.76047
[5]	Bert, C. W., Application of differential transform method to heat conduction in tapered fins, ASME Journal of Heat Transfer, 124, 1, 208-209 (2002)
[6]	Hansen, A. G.; Na, T. Y., Similarity solutions of laminar, incompressible, boundary layer equations of non-Newtonian fluids, ASME Journal of Basic Engineering, 71 (1968) · Zbl 0153.28402
[7]	Timol, M. G.; Kalthia, N. L., Similarity solutions of three dimensional boundary layer equations of non-Newtonian fluids, International Journal of Non-Linear Mechanics, 21, 475 (1986) · Zbl 0605.76004
[8]	Pakdemirli, M., Similarity analysis of boundary layer equations of a class of non-Newtonian fluids, International Journal of Non-Linear Mechanics, 29, 2, 187-196 (1994) · Zbl 0795.76008
[9]	Pakdemirli, M.; Yürüsoy, M., Similarity transformations for partial differential equations, SIAM Review, 40, 1, 96-101 (1998) · Zbl 0911.35096
[10]	Pakdemirli, M., The boundary layer equations of third grade fluids, International Journal of Non-Linear Mechanics, 27, 5, 785-793 (1992) · Zbl 0764.76004
[11]	Bluman, G. W.; Kumei, S., Symmetries and Differential Equations (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0718.35003