×

zbMATH — the first resource for mathematics

Analytical solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient. (English) Zbl 1221.74083
Summary: The homotopy analysis method (HAM) is used to evaluate the analytical approximate solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient. The fin efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity is obtained as a function of thermo-geometric fin parameter. It is shown that the thermal conductivity parameter has a strong influence over the fin efficiency. The analytic solution of the problem is obtained by using the HAM. The HAM contains the auxiliary parameter \(\hbar\), which adjusts and controls the convergence region of the solution series in a simple way. By choosing the auxiliary parameter \(\hbar\) in a suitable way, we can obtain reasonable solution for large values of \(M\) and \(\beta \).

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74F05 Thermal effects in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aziz, A.; Enamul Hug, S.M., Perturbation solution for convecting fin with variable thermal conductivity, J heat transfer-trans ASME, 97, 300-301, (1975)
[2] Aziz, A.; Benzies, J.Y., Application of perturbation techniques to heat-transfer problems with variable thermal properties, Int J heat mass transfer, 19, 271-276, (1976)
[3] Pakdemirli, M.; Sahin, A.Z., Group classification of fin equation with variable thermal properties, Int J eng sci, 42, 1875-1889, (2004) · Zbl 1211.35141
[4] Bokhari, A.H.; Kara, A.H.; Zaman, F.D., A note on a symmetry analysis and exact solutions of a nonlinear fin equation, Appl math lett, 19, 1356-1360, (2006) · Zbl 1143.35311
[5] Kim, S.; Huang, C.-H., A series solution of the non-linear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient, J phys D: appl phys, 40, 2979-2987, (2007) · Zbl 1144.82346
[6] Liao SJ, The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University; 1992.
[7] Liao, S.J., An explicit, totally analytic approximate solution for Blasius-viscous flow problems, Int J non-linear mech, 34, 759-778, (1999) · Zbl 1342.74180
[8] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton (FL)
[9] Liao, S.J., A general approach to get series solution of non-similarity boundary-layer flows, Commun nonlinear sci numer simulat, 14, 2144-2159, (2009) · Zbl 1221.76068
[10] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl math comput, 147, 499-513, (2004) · Zbl 1086.35005
[11] Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transfer, 48, 2529-2539, (2005) · Zbl 1189.76142
[12] Domairry, G.; Fazeli, M., Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Commun nonlinear sci numer simulat, 14, 489-499, (2009)
[13] Inc, M., Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Math comput simulat, (2008) · Zbl 1153.65100
[14] Chowdhury, M.S.H.; Hashim, I.; Abdulaziz, O., Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems, Commun nonlinear sci numer simulat, 14, 371-378, (2009) · Zbl 1221.80021
[15] Tan, Y.; Abbasbandy, S., Homotopy analysis method for quadratic Riccati differential equation, Commun nonlinear sci numer simulat, 13, 3, 539-546, (2008) · Zbl 1132.34305
[16] Molabahrami, A.; Khani, F., The homotopy analysis method to solve the burgers – huxley equation, Nonlinear anal real world appl, 10, 2, 589-600, (2009) · Zbl 1167.35483
[17] Khani, F.; Ahmadzadeh Raji, M.; Hamedi-Nezhad, S., A series solution of the fin problem with a temperature-dependent 3 thermal conductivity, Commun nonlinear sci numer simulat, 14, 3007-3017, (2009)
[18] Sajid, M.; Ahmad, I.; Hayat, T.; Ayub, M., Unsteady flow and heat transfer of a second grade fluid over a stretching sheet, Commun nonlinear sci numer simulat, 14, 1, 96-108, (2009) · Zbl 1221.76022
[19] Abbasbandy, S., Soliton solutions for the 5th-order KdV equation with the homotopy analysis method, Nonlinear dyn, 51, 83-87, (2008) · Zbl 1170.76317
[20] Song, L.; Zhang, H.Q., Application of homotopy analysis method to fractional KdV-burgers – kuramoto equation, Phys lett A, 367, 88-94, (2007) · Zbl 1209.65115
[21] Liao, S.J., Notes on the homotopy analysis method: some definitions and theorems, Commun nonlinear sci numer simulat, 14, 4, 983-997, (2009) · Zbl 1221.65126
[22] Watson, L.T., Globally convergent homotopy methods: a tutorial, Appl math comput, 31, 369-396, (1989) · Zbl 0689.65033
[23] Watson, L.T., Engineering applications of the Chow-Yorke algorithm, Appl math comput, 9, 111-133, (1981) · Zbl 0481.65029
[24] Liao, S.J., Comparison between the homotopy analysis method and homotopy perturbation method, Appl math comput, 169, 1186-1194, (2005) · Zbl 1082.65534
[25] He, J.H., Comparison of homotopy perturbation method and homotopy analysis method, Appl math comput, 156, 527-539, (2004) · Zbl 1062.65074
[26] Allan, F.M., Derivation of the Adomian decomposition method using the homotopy analysis method, Appl math comput, 190, 6-14, (2007) · Zbl 1125.65063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.