Summary
The problem of solidification of two semi-infinite materials with arbitrarily prescribed initial conditions is studied. This is different from the classical Stefan problem; there are no prescribed boundary conditions. It is found that there are, depending on the prescribed initial conditions, four different possibilities: (i) solidification starts immediately and the interfacial boundary is of the order oft 1/2 ast→0, (ii) solidification also starts immediately but the interfacial boundary is of an order other thant 1/2, (iii) no solidification will ever occur, and (iv) there is a pre-solidification period during which the temperatures of both materials are redistributed and solidification will occur only after the surface temperature has reached the freezing point. Conditions for the occurrence of these cases are examined and established. The exact solutions for each of these cases are derived and discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- a n b n c n d n :
-
series coefficients of the temperature solutions, Eq. (2.7)
- k :
-
coefficient of thermal conductivity
- s(t) :
-
interfacial position
- x, t :
-
space and time variables
- u n v r :
-
series coefficients of the initial data, Eq. (2.4)
- A m n B m n :
-
constants defined by Eq. (2.12)
- C :
-
specific heat
- E n(z)F n(z):
-
functions in the error integral family, Eq. (2.5)
- G n(z):
-
polynomial in the error integral family, Eq. (2.5)
- I n(z)J n(z):
-
functions in the error integral family, Eq. (3.5)
- K(x, t; α):
-
fundamental solution of the heat equation, Eq. (3.6)
- L :
-
latent heat
- T(x, t) :
-
temperature
- U(x), V(x) :
-
initial data
- α:
-
coefficient of thermal diffusibility
- β r :
-
multinomial coefficient
- λ n :
-
series coefficient of the interfacial position
- µ i :
-
(k i /k II)ω i ,i=I,III
- ϱ:
-
density
- η:
-
\( = s/\sqrt {4\alpha t} \)
- ξ:
-
\( = x/\sqrt {4\alpha t} \)
- τ:
-
t 1/2
- ω i :
-
\( = \sqrt {\alpha _{II} /\alpha _i ,} = I,III\)
- Δ, Ω:
-
constants defined by Eq. (3.1)
- Φ n (z):
-
iterated derivative of the error integral
- f :
-
freezing point
- q :
-
initial state
- p :
-
pre-solidification period
- s :
-
surface atx=0
- I:
-
solid phase
- II:
-
newly formed solid phase
- III:
-
liquid phase
- ∼:
-
pre-solidification state
- *:
-
solidification state after pre-solidification
References
Riemann, G. F. B., Weber, H.: Die partiellen Differentialgleichungen der mathematischen Physik, Bd. 2, 5. Aufl. Braunschweig: 1912.
Stefan, J.: Über einige Probleme der Theorie der Wärmeleitung. S.-B. Wien. Akad. Mat. Natur.98, 473–484 (1889).
Rubinstein, L. I.: The Stefan problem (English translation). (Transl. Math. Monog. 27.) Providence: Am. Math. Soc. 1971.
Carslaw, H. S., Jaeger, J. C.: Conduction of heat in solids, 2nd ed. Oxford: Clarendon Press 1959.
Crank, J.: The mathematics of diffusion, 2nd ed. Oxford: Clarendon Press 1975.
Ruddle, R. W.: The solidification of castings. London: Inst. Metals 1957.
Bankoff, S. G.: Heat conduction of diffusion with change of phase. Adv. Chem. Eng.5, 75–150 (1964).
Muehlbauer, J. C., Sunderland, J. E.: Heat conduction with freezing or melting. Appl. Mech. Rev.18, 951–959 (1965).
Tiller, W. A.: Principles of solidification. In: Arts and sciences of growing crystals (Gilman, J. J., ed.). New York: Wiley 1963.
Boley, B. A.: Survey of recent developments in the fields of heat conduction in solids and thermo-elasticity. Nuclear Eng. Des.18, 377–399 (1972).
Ockendon, J. R., Hodgkins, W. R. (eds.). Moving boundary problems in heat flow and diffusion. Oxford: Clarendon Press 1975.
Wilson, D. G., Solomon, A. D., Boggs, P. T. (eds.): Moving boundary problems. New York: Academic Press 1978.
Tao, L. N.: The Stefan problem with arbitrary initial and boundary conditions. Quart. Appl. Math.36, 223–233 (1978).
Tao, L. N.: The analyticity of solutions of the Stefan problem. Arch. Rat. Mech. Anal.72, 285–301 (1980).
Tao, L. N.: On free boundary problems with arbitrary initial and flux conditions. Z. Angew. Math. Phys.30, 416–426 (1979).
Tao, L. N.: Free boundary problems with radiation boundary conditions. Quart. Appl. Math.37, 1–10 (1979).
Tao, L. N.: The exact solutions of some Stefan problems with arbitrary heat flux and initial conditions. J. Appl. Mech.48, 732–736 (1981).
Widder, D. V.: The heat equation. New York: Academic Press 1975.
Tao, L. N.: On the material time derivative of arbitrary order. Quart. Appl. Math.36, 323–324 (1978).
Abramowitz, M., Stegun, I. A. (eds.): Handbook of mathematical functions, NBS AMS 55. Washington D.C.: U.S. Government Printing Office 1964.
Tao, L. N.: On solidification problems including the density jump at the moving boundary. Quart. J. Mech. Appl. Math.32, 175–185 (1979).
Tao, L. N.: Dynamics of growth or dissolution of a gas bubble. J. Chem. Phys.69, 4189–4194 (1978).
Friedman, A.: Partial differential equations of parabolic type. Englewood Cliffs, N. J.: Prentice-Hall 1964.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tao, L.N. The Cauchy-Stefan problem. Acta Mechanica 45, 49–64 (1982). https://doi.org/10.1007/BF01295570
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01295570