A singular perturbation solution to a problem of extreme temperatures imposed at the surface of a variable-conductivity halfspace: small surface conductivity

Author:
Leonard Y. Cooper

Journal:
Quart. Appl. Math. **32** (1975), 427-444

MSC:
Primary 65N99

DOI:
https://doi.org/10.1090/qam/451787

MathSciNet review:
451787

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Abstract: The transient temperature field resulting from a constant and uniform temperature (or time-dependent heat flux ) imposed at the surface of a halfspace initially at uniform temperature is considered. A temperature-dependent thermal conductivity variation, , and a constant product of density and specific heat, , are assumed to be accurate models for the halfspace for some useful temperature range. The problem is initially formulated in terms of the dimensionless conductivity . Attention is then focused on the singular problem resulting from the limits and . This work considers the use of matched asymptotic expansions to solve the problem under the first of these limits. In particular, Fraenkel's interpretation [5] of Van Dyke's method of inner and outer expansions [6] is carefully applied to the problem under consideration. Besides obtaining a uniformly valid solution to the problem, a particularly interesting explicit result is deduced, namely

**[1]**L. Y. Cooper,*Constant temperature at the surface of an initially uniform temperature, variable conductivity half space*, J. Heat Trans.**93**, 55-60 (1971)**[2]**C. Wagner,*On the solution to diffusion problems involving concentration-dependent diffusion coefficients*, J. Metals**4**, 91-96 (1952)**[3]**J. Crank,*The mathematics of diffusion*, Oxford, at the Clarendon Press, 1956. MR**0082827****[4]**W. R. Gardner and M. S. Mayhugh,*Solutions and tests of the diffusion equation for the movement of water in soil*, Soil Sci. Soc. Am. Proc.**22**, 197-201 (1958)**[5]**L. E. Fraenkel,*On the method of matched asymptotic expansions. I. A matching principle*, Proc. Cambridge Philos. Soc.**65**(1969), 209–231. MR**0237898**

L. E. Fraenkel,*On the method of matched asymptotic expansions. II. Some applications of the composite series*, Proc. Cambridge Philos. Soc.**65**(1969), 233–261. MR**0237899**

L. E. Fraenkel,*On the method of matched asymptotic expansions. III. Two boundary-value problems*, Proc. Cambridge Philos. Soc.**65**(1969), 263–284. MR**0237900****[6]**Milton Van Dyke,*Perturbation methods in fluid mechanics*, Applied Mathematics and Mechanics, Vol. 8, Academic Press, New York-London, 1964. MR**0176702****[7]**George F. Carrier, Max Krook, and Carl E. Pearson,*Functions of a complex variable: Theory and technique*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0222256**

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Additional Information

DOI:
https://doi.org/10.1090/qam/451787

Article copyright:
© Copyright 1975
American Mathematical Society