Heat flow near the triple junction of an evaporating meniscus and a substrate

In the title problem, the temperature satisfies Laplac's equation within the solid and liquid, and Newton's law of cooling at the liquid–vapour phase interface. That boundary condition defines a length L ∼ 10nm on which the interface changes from being, in effect, adiabatic at the contact line to isothermal at infinity. We give an exact solution showing how this change affects the temperature field when the solid occupies a half–space, and the liquid a quarter–space (so the liquid–solid contact angle θ=π/2); the liquid–solid conductivity ratio k is arbitrary. We use this solution to verify the predictions of existing analysis of the limit ? → 0 with θ fixed but arbitrary. In the limit r/L → 0 of vanishing distance from the contact line, the new solution reduces to an existing local solution.