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 ${T_s}$ (or time-dependent heat flux $H = h{t^{ - 1/2}}$) imposed at the surface of a halfspace initially at uniform temperature ${T_0}$ is considered. A temperature-dependent thermal conductivity variation, $k\left ( T \right ) = {k_0}\exp \left [ {\lambda (T - {T_0})/{T_0}} \right ]$, and a constant product of density and specific heat, $\rho C$, 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 $\phi = k\left ( T \right )/{k_0}$. Attention is then focused on the singular problem resulting from the limits ${\phi _s} = \phi \left ( {{T_s}} \right ) \downarrow 0$ and ${\phi _s} \to \infty$. 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 \[ \lim \limits _{{\phi _s} \downarrow 0} h = - (1.182754 \cdot \cdot \cdot )({T_0}/\lambda ){[\rho C{k_0}/2]^{1/2}} + O({\phi _s}ln{\phi _s})\]
L. Y. Cooper, Constant temperature at the surface of an initially uniform temperature, variable conductivity half space, J. Heat Trans. 93, 55–60 (1971)
C. Wagner, On the solution to diffusion problems involving concentration-dependent diffusion coefficients, J. Metals 4, 91–96 (1952)
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- 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
L. Y. Cooper, Constant temperature at the surface of an initially uniform temperature, variable conductivity half space, J. Heat Trans. 93, 55–60 (1971)
C. Wagner, On the solution to diffusion problems involving concentration-dependent diffusion coefficients, J. Metals 4, 91–96 (1952)
J. Crank, Mathematics of diffusion, Oxford (1957)
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)
L. E. Fraenkel, On the method of matched asymptotic expansion—Part I: A matching principle, Proc-Camb. Phil. Soc. 65, 209–231 (1969)
M. Van Dyke, Perturbation methods in fluid mechanics, Academic Press (1964)
G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a complex variable, McGraw-Hill (1966)
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Article copyright:
© Copyright 1975
American Mathematical Society