Conservation laws and associated Lie point symmetries admitted by the transient heat conduction problem for heat transfer in straight fins

• Authors: Partner Ndlovu , Rasselo Moitsheki
• Languages of publication: EN

Abstracts

EN

Some new conservation laws for the transient heat conduction problem for heat transfer in a straight fin are constructed. The thermal conductivity is given by a power law in one case and by a linear function of temperature in the other. Conservation laws are derived using the direct method when thermal conductivity is given by the power law and the multiplier method when thermal conductivity is given as a linear function of temperature. The heat transfer coefficient is assumed to be given by the power law function of temperature. Furthermore, we determine the Lie point symmetries associated with the conserved vectors for the model with power law thermal conductivity.

Keywords

EN

fins; heat transfer; conservation laws; multiplier method; direct method; Lie point symmetries; exact solutions

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 8
• Pages: 984-994

Dates

published

1 - 8 - 2013

online

23 - 10 - 2013

Contributors

author

Partner Ndlovu

• Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, Johannesburg, University of the Witwatersrand, Private Bag 3, Wits, 2050, South Africa

author

Rasselo Moitsheki

• Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, Johannesburg, University of the Witwatersrand, Private Bag 3, Wits, 2050, South Africa

References

• [1] P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-NSF Regional Conference in Applied Math., SIAM, 1973 http://dx.doi.org/10.1137/1.9781611970562[Crossref]
• [2] R.J. LeVeque, Numerical Methods for Conservation Laws, Second edition (Springer, 1992) http://dx.doi.org/10.1007/978-3-0348-8629-1[Crossref]
• [3] B. Cockburn, S. Hou, C.-W Shu. Math. Comput. 54, 545 (1990)
• [4] R. Naz, F.M. Mahomed, D.P. Mason, Appl. Math. Comput. 205, 212 (2008) http://dx.doi.org/10.1016/j.amc.2008.06.042[Crossref]
• [5] E. Noether, Transport Theor. Stat. Phys. 1, 183 (1971) http://dx.doi.org/10.1080/00411457108231445[Crossref]
• [6] A.D. Kraus, A. Aziz, J. Welty, Extended Surface Heat Transfer (Wiley, New York, 2001)
• [7] A.H. Bokhari, A.H. Kara, F.D. Zaman, Appl. Math. Lett. 19, 1356 (2006) http://dx.doi.org/10.1016/j.aml.2006.02.003[Crossref]
• [8] M. Pakdemirli, A.Z. Sahin, Int. J. Eng. Sci. 42, 1875 (2004) http://dx.doi.org/10.1016/j.ijengsci.2004.04.005[Crossref]
• [9] M. Pakdemirli, A.Z. Sahin, Appl. Math. Lett. 19, 378 (2006) http://dx.doi.org/10.1016/j.aml.2005.04.017[Crossref]
• [10] O.O. Vaneeva, A.G. Johnpillai, R.O. Popovych, C. Sophocleous, Appl. Math. Lett. 21, 248 (2008) http://dx.doi.org/10.1016/j.aml.2007.02.023[Crossref]
• [11] R.O. Popovych, C. Sophocleous, O.O. Vaneeva, Appl. Math. Lett. 21, 209 (2008) http://dx.doi.org/10.1016/j.aml.2007.03.009[Crossref]
• [12] A. Moradi, Int. J. Eng. Appl. Sci. 3, 1 (2011) http://dx.doi.org/10.1504/IJSE.2011.037717[Crossref]
• [13] R.J. Moitsheki, T. Hayat, M.Y. Malik, Nonlinear Anal. Real 11, 3287 (2010) http://dx.doi.org/10.1016/j.nonrwa.2009.11.021[Crossref]
• [14] M.H. Chowdhury, I. Hashim, On decomposition solutions of fins with temperature dependent surface heat flux: multi-boiling heat transfer, IMT-GT (University of Sains Malaysia, Penang, 2006)
• [15] S.C. Anco, G.W. Bluman, Eur. J. Appl. Math. 13, 545 (2002)
• [16] S.C. Anco, G.W. Bluman, Eur. J. Appl. Math. 13, 567 (2002)
• [17] G.W. Bluman, A.F. Cheviakov, S.C. Anco, Applications of symmetry methods to partial differential equations (Springer-Verlag, New York, 2010) http://dx.doi.org/10.1007/978-0-387-68028-6[Crossref]
• [18] D.Q. Kern, A.D. Kraus, Extended Surface Heat Transfer (McGraw-Hill, New York, 1972)
• [19] H.C. Ünal, Int. J. Heat Mass Trans. 31, 1483 (1988) http://dx.doi.org/10.1016/0017-9310(88)90257-8[Crossref]
• [20] S. Vitanov, V. Palankovski, S. Maroldt, R. Quay, Solid State Electronin. 54, 1105 (2010) http://dx.doi.org/10.1016/j.sse.2010.05.026[Crossref]
• [21] A. Jezowski, B.A. Danilchenko, M. Bockowski, I. Grzegory, S. Krukowski, T. Suski, T. Paszkiewicz, Solid State Commun. 128, 69 (2003) http://dx.doi.org/10.1016/S0038-1098(03)00629-X[Crossref]
• [22] M.D. Kamatagi, N.S. Sankeshwar, B.G. Mulimani, Diam. Relat. Mater. 16, 98 (2007) http://dx.doi.org/10.1016/j.diamond.2006.04.004[Crossref]
• [23] M.D. Kamatagi, R.G. Vaidya, N.S. Sankeshwar, B.G. Mulimani, Int. J. Heat Mass Trans. 52, 2885 (2009) http://dx.doi.org/10.1016/j.ijheatmasstransfer.2008.10.032[Crossref]
• [24] M.D. Mhlongo, R.J. Moitsheki, O.D. Makinde, Int. J. Heat Mass Trans. 57, 117 (2013) http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.012[Crossref]
• [25] A.F. Mills, basic heat and mass transfer (Prentice Hall, 1999)
• [26] A.H. Kara, F.M. Mahomed, Int. J. Theor. Phys. 39, 23 (2000) http://dx.doi.org/10.1023/A:1003686831523[Crossref]
• [27] A.H. Kara, F.M. Mahomed, J. Nonlinear Math. Phy. 9, 60 (2002) http://dx.doi.org/10.2991/jnmp.2002.9.s2.6[Crossref]
• [28] O.O. Vaneeva, A.G. Johnpillai, R.O. Popovych, C. Sophocleous, J. Math. Anal. Appl. 330, 1363 (2007) http://dx.doi.org/10.1016/j.jmaa.2006.08.056[Crossref]
• [29] R.O. Popovych, M. Ivanova, J. Math. Phys. 46, 043502 (2005) http://dx.doi.org/10.1063/1.1865813[Crossref]
• [30] N.M. Ivanova, R.O. Popovych, C. Sophocleous, O.O. Vaneeva, Physica A. 388 343 (2009) http://dx.doi.org/10.1016/j.physa.2008.10.018[Crossref]
• [31] G.W. Bluman, S.C. Anco, Symmetry and integration methods for differential equations (Springer-Verlag, New York, 2002)
• [32] G.W. Bluman, S. Kumei, Symmetries and differential equations (Springer-Verlag, New York, 1989) http://dx.doi.org/10.1007/978-1-4757-4307-4[Crossref]
• [33] P.J. Olver, Applications of Lie groups of differential equations (Springer-Verlag, New York, 1986) http://dx.doi.org/10.1007/978-1-4684-0274-2[Crossref]
• [34] A.V. Dorondnitsyn, USSR Comput. Math. Math. Phys. 22, 115 (1982) http://dx.doi.org/10.1016/0041-5553(82)90102-1[Crossref]
• [35] N.H. Ibragimov (Editor), Lie group analysis of differential equations - symmetries, exact solutions and conservation laws, Volume 1, Boca raton, FL, Chemical Rubber Company, 1994
• [36] A. Sjöberg, Nonlinear Anal. Real 10, 3472 (2009) http://dx.doi.org/10.1016/j.nonrwa.2008.09.029[Crossref]
• [37] A.H. Bokhari, A.Y. Dweik, F.D. Zaman, A.H. Kara, F.M. Mahomed, Nonlinear Anal. Real 11, 3763 (2010) http://dx.doi.org/10.1016/j.nonrwa.2010.02.006[Crossref]
• [38] J.G. Kingston, C. Sophocleous, J. Phys. A. Math. Gen. 31, 1597 (1998) http://dx.doi.org/10.1088/0305-4470/31/6/010[Crossref]
• [39] R.O. Popovych, A.M. Samoilenko, J. Phys. A. 41, 362002 (2008) http://dx.doi.org/10.1088/1751-8113/41/36/362002[Crossref]

Identifiers

DOI

10.2478/s11534-013-0306-1