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