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 {T_s} (or time-dependent heat flux H = h t − 1 / 2 H = h{t^{ - 1/2}} ) imposed at the surface of a halfspace initially at uniform temperature T 0 {T_0} is considered. A temperature-dependent thermal conductivity variation, k ( T ) = k 0 exp ⁡ [ λ ( T − T 0 ) / T 0 ] k\left ( T \right ) = {k_0}\exp \left [ {\lambda (T - {T_0})/{T_0}} \right ] , and a constant product of density and specific heat, ρ C \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 ϕ = k ( T ) / k 0 \phi = k\left ( T \right )/{k_0} . Attention is then focused on the singular problem resulting from the limits ϕ s = ϕ ( T s ) ↓ 0 {\phi _s} = \phi \left ( {{T_s}} \right ) \downarrow 0 and ϕ s → ∞ {\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 \[ lim ϕ s ↓ 0 h = − ( 1.182754 ⋅ ⋅ ⋅ ) ( T 0 / λ ) [ ρ C k 0 / 2 ] 1 / 2 + O ( ϕ s l n ϕ s ) \lim \limits _{{\phi _s} \downarrow 0} h = - (1.182754 \cdot \cdot \cdot )({T_0}/\lambda ){[\rho C{k_0}/2]^{1/2}} + O({\phi _s}ln{\phi _s}) \]