Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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

$\displaystyle \mathop {\lim }\limits_{{\phi _s} \downarrow 0} h = - (1.182754 \... ... \cdot \cdot )({T_0}/\lambda ){[\rho C{k_0}/2]^{1/2}} + O({\phi _s}ln{\phi _s})$


References [Enhancements On Off] (What's this?)

  • [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
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  • [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

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