Flow, Turbulence and Combustion

, Volume 62, Issue 1, pp 1–27 | Cite as

Solidification or Melting Initiation over a Limited Portion of an Edge in a Plate

  • S.C. Gupta


This paper proposes a method for obtaining short-time analytical solutions to problems in pure heat conduction, and heat conduction with phase change. The method employs the notion of ‘fictitious initial temperatures’ in some fictitious extensions of the original phase region. Analytical results obtained in the case of a heat conduction problem in a rectangular plate are presented first and compared with numerical solutions. This analytical solution is also required later in the determination of liquid temperature in the phase change problem. The method is then extended to a two-phase solidification problem in which solidification starts over a limited portion of one of the vertical edges of the rectangular plate. The freezing front in this case consists of spread along the vertical edge and growth towards the interior. The spread along the edge can have asymptotic behaviour not commonly found. The method is applicable to other geometries, e.g. inside and outside of a long cylinder, a three-dimensional slab, etc.

phase change problems freezing front Green's function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, 2nd edn. Clarendon Press, Oxford (1959).Google Scholar
  2. 2.
    Lederman, J.M. and Boley, B.A., Axisymmetric melting or solidification of circular cylinders. Internat. J. Heat Mass Transfer 13 (1970) 413–427.Google Scholar
  3. 3.
    Ozisik, M. N., Boundary Value Problems of Heat Conduction. International Textbook Company, Scranton (1968).Google Scholar
  4. 4.
    Gupta, S.C., Axisymmetric solidification in a long cylindrical mold. Appl. Sci. Res. 42 (1985) 229–244.Google Scholar
  5. 5.
    Yanenko, N.N., The Methods of Fractional Steps. Springer-Verlag, Berlin (1971).Google Scholar
  6. 6.
    Crank, J., Free and Moving Boundary Problems. Clarendon Press, Oxford (1984).Google Scholar
  7. 7.
    Sikarskie, D.L. and Boley, B.A., The solution of a class of two-dimensional melting and solidification problems. Internat. J. Solids and Structures I (1965) 207–234.Google Scholar
  8. 8.
    Boley, B.A., A method of heat conduction analysis of melting and solidification problems. J. Math. Phys. 40 (1960) 300–313.Google Scholar
  9. 9.
    Fraster, J., Cryogenic techniques in surgery. Cryogenics 7 (1979) 375–381.Google Scholar
  10. 10.
    Na, S.J. and Park, S.W., A theoretical study on electrical and thermal response in resistance spot welding. Welding J. Suppl. 75(8) (1996) 233s-241s.Google Scholar
  11. 11.
    Domey, J., Aidun, D.K., Ahmadi, G., Regel, L.L. and Wilcox, W.R., Numerical simulation of the effect of gravity on weld pool shape. Welding J. Suppl. 74(8) (1995) 263s-268s.Google Scholar
  12. 12.
    Crowley, A.B. and Ockendon, J.R., A Stefan problem with a non-monotone boundary. J. Inst. Math. Appl. 20 (1977) 269–281.Google Scholar
  13. 13.
    Gupta, S.C., Temperature and moving boundary in two-phase freezing due to an axisymmetric cold spot. Quart. Appl. Math. 45 (1987) 205–222.Google Scholar
  14. 14.
    Gupta, S.C., Numerical and analytical solutions of one dimensional freezing of dilute binary alloys with coupled heat and mass transfer. Internat. J. Heat Mass Transfer 33 (1990) 393–602.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • S.C. Gupta
    • 1
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations