Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 332 / 2 / v / Solution

Define the dimensionless stagnant-layer depth and time byso $H^2/\kappa$ is the thermal diffusion time. The maximizing interface temperature givesEquating conductive and convective heat fluxes therefore gives the algebraic relation

Since $k=\rho c_p\kappa$, where $c_p$ is the specific heat capacity, the bulk heat balance in the convecting depth $H-h$ becomesRetaining the heat capacity of the thin stagnant layer changes this only by relative order $\eta$.

For $\mathcal{F}\gg 1$ and $\theta =O(1)$, the flux relation gives $\eta =O({\mathcal{F}}^{-1})$. The leading equation is consequentlySeparating variables and using $\theta (0)=1$ giveshence