A singular perturbation solution to a problem of extreme temperatures imposed at the surface of a variable-conductivity halfspace: small surface conductivity
The transient temperature field resulting from a constant and uniform temperature T, (or time-dependent heat flux H = ht~1/2) imposed at the surface of a halfspace initially at uniform temperature T0 is considered. A temperature-dependent thermal conductivity variation, k(T) = k0 exp [X(T — T0)/To], and a constant product of density and specific heat, pC, 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 4> = k(T)/k0 . Attention is then focused on the singular problem resulting from the limits <£s = 4>(T,) J, 0 and <p, —* °°. 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 h = -(1.182754 • • ■)(T0A)[pCfc0/2]1/2 + 0(4>, In *.). ■MO