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,*Mathematics of diffusion*, Oxford (1957) 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 expansion--Part I*:*A matching principle*, Proc-Camb. Phil. Soc.**65**, 209-231 (1969) MR**0237898****[6]**M. Van Dyke,*Perturbation methods in fluid mechanics*, Academic Press (1964) MR**0176702****[7]**G. F. Carrier, M. Krook, and C. E. Pearson,*Functions of a complex variable*, McGraw-Hill (1966) MR**0222256**

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

Article copyright:
© Copyright 1975
American Mathematical Society