THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

AN APPROXIMATE ANALYTICAL (INTEGRAL-BALANCE) SOLUTION TO A NONLINEAR HEAT DIFFUSION EQUATION

ABSTRACT
The communication presents a closed form approximate solution of the nonlinear diffusion equation of a power-law nonlinearity of the diffusivity developed by the heat-balance integral method. The main step in the initial transformation of the governing equation avoiding the Kirchhoff transformation is demonstrated. The consequent application of the integral method is exemplified by a solution of a Dirichlet problem with an approximate parabolic profile. Cases with predetermined positive integer and optimized non-integer exponents have been analyzed.
KEYWORDS
PAPER SUBMITTED: 2014-03-26
PAPER REVISED: 2014-04-19
PAPER ACCEPTED: 2014-06-13
PUBLISHED ONLINE: 2014-06-21
DOI REFERENCE: https://doi.org/10.2298/TSCI140326074H
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Issue 2, PAGES [723 - 733]
REFERENCES
  1. Hill, J.M., Similarity solutions for nonlinear diffusion- a new integration procedure, J. Eng, Math, 23 (1989), 1, pp.141-155;
  2. Prasad , S.N., and Salomon, J.B., A new method for analytical solution of a degenerate diffusion equation, Adv. Water Research, 28 (2005),10, pp. 1091-1101.
  3. N. F. Smyth, N.F., Hill,J.M., High-Order Nonlinear Diffusion, IMA Journal of Applied Mathematics, 40 (1988),1, pp. 73-86.
  4. Zel'dovich, Ya.B., Kompaneets, A. S. , On the theory of heat propagation for temperature dependent thermal conductivity, in: Collection Commemorating the 70th Anniversary of A. F. Joffe, Izv. Akad. Nauk SSSR, 1950, pp. 61-71.
  5. Buckmaster, J., Viscous sheets advancing over dry beds, J. Fluid Mech., 81 (1977),4, pp. 735-756.
  6. Marino, B. M., Thomas, L. P., Gratton, R., Diez, J. A., Betelu, S., Waiting-time solutions of a nonlinear diffusion equation: Experimental study of a creeping flow near a waiting front, Physical Review E, 54 (1996),3, pp, 2628-2636.
  7. Lonngren, K.E., Ames, W.F., Hirose, A., Thomas J., Field penetration into plasma with nonlinear conductivity, The Physics of Fluids, 17 (1974), 10., pp. 19191-1920.
  8. Khan, Z.H., Gul, R., Khan, W.A., Effect of variable thermal conductivity on heat transfer from a hollow sphere with heat generation using homotopy perturbation method, In Proc. ASME Summer Heat Transfer Conf, 2008, August 1-14, 2008, Article HT2008-56448, Jacksonville, Florida, USA.
  9. Pattle, R. E. Diffusion from an instantaneous point source with concentration-dependent coefficient, Quart.J. Mech. Appl. Math., 12 (1959), 4, pp. 407-409.
  10. Muskat, M., The Flow of Homogeneous Fluids through Porous Media, McGraw-Hill, New York,1937.
  11. Peletier, L.A., Asymptotic behavior of solutions of the porous media equations, SIAM J. Appl. Math, 21 (1971), 4, pp.542-551.
  12. Aronson, D.G., The porous medium equation, in Nonlinear Diffusion Problems, Lecture Notes in Math. 1224, A. Fasano and M. Primicerio, eds., Springer, Berlin, 1986, pp. 1-46.
  13. King, J.R., The isolation oxidation of silicon: The reaction-controlled case, SIAM J. Appl. Math., 49 (1989),4, pp.1064-1080.
  14. Lacey, A. A., Ockendon, J. R., Tayler, A. B. 1982, Waiting-time solutions of a nonlinear diffusion equation. SIAM J. appl. Math. 42 (1982),6, pp. 1252-1264.
  15. Atkinson, C., Reuter, G.E.H., Ridler-Rowe, C.J., Traveling wave solutions for some nonlinear diffusion equations, SIAM J. Appl. Math, 12 (1981), 6, 880- 892.
  16. Strunin, D.V., Attractors in confined source problems for coupled nonlinear diffusion, SIAM J. Appl. Math., 67 (2007) ,6, pp. 1654-1674,
  17. Nasseri, M.,Daneshbod, Y., Pirouz, M.D., Rakhshandehroo, Gh.R., Shirzad, A., New analytical solution to water content simulation in prorous media, J. Irrigation and Drainage Eng., ??, 2012, April, ??, 328-335.
  18. Hill, D.L., Hill, J.M., Similarity solutions for nonlinear diffusion-further exact solutions, J. Eng, Math, 24 (1990), 1, pp.109-124;
  19. Hill, J.M., Hill, D.L., On the derivation of first integrals for similarity solutions, J. Eng, Math, 25 (1991), 2, pp.287-299.
  20. Tomatis, D. Heat Conduction in nuclear fuel by the Kirchhoff transformation, Ann. Nucl. Energy., 57 (2013),July, pp.100-105,
  21. Knight, J.H., Philip, J.R., Exact solutions in nonlinear diffusion, J. Eng. Math. , 8 (1974), 3,pp. 219-227.
  22. Kolpakov, V.A., Novomelski, D.N., Novozhenin, M.P., Determination of the surface temperature of sample in region of its interaction with a nonelectrode plasma flow using the Kirchhoff transformation of a quadratic function, Technical Physics, 58 (2013), 11, pp.1554-1557.
  23. King, J.R., Integral results for nonlinear diffusion equations, J. Eng, Math, 25 (1991), 2, pp.191-205.
  24. Paripour, M., Babolian, E., Saeidian, J., Analytic solutions to diffusion equations, Math. Comp. Model., 51 (2010),5-6, pp.649-657.
  25. Walker, G.G., Scott, E.P., Evaluation of estimation methods for high unsteady heat fluxes from surface measurements, J. Thermophys. Heat Transfer, 12 (1998), 4, 543-551.
  26. Goodman T.R., Application of Integral Methods to Transient Nonlinear Heat Transfer, Advances in Heat Transfer, T. F. Irvine and J. P. Hartnett, eds., 1 (1964), Academic Press, San Diego, CA, pp. 51-122.
  27. Hristov, J. The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and Benchmark Exercises, Thermal Science, 13(2009), 2, pp.27-48.
  28. Mitchell, S. L., Myers, T. G., Application of standard and refined heat balance integral methods to onedimensional Stefan problems," SIAM Review, 52 (2010), 1, pp. 57-86.
  29. Langford , D., The heat balance integral method, Int. J. Heat Mass Transfer,16 (1973),12, pp.2424- 2428.
  30. Myers, J.G., Optimizing the exponent in the heat balance and refined integral methods, Int. Comm. Heat Mass Transfer, 36 (2009), 2, pp. 143-147.
  31. Sadighi, A., Ganji, D.D., Exact solutions of nonlinear diffusion equations by variational iteration method, Comp. Math Appl., 54 (2007), 7-8, pp. 1112-1121.
  32. Hristov, J., Integral-Balance Solution to a Nonlinear Heat Equation: Power-law temperature-dependent thermal diffusivity and fixed boundary conditions, Heat Mass Transfer (2014) in press.

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence