Surface heat flux of a cooling plate ($q_s(t)=k\Delta T[1+2{\sum }_{n\ge 1}{e}^{-n^{2}t/{\tau }_0}]/a$)

For a hot initial plate cooled from above, Fourier's law gives the displayed upward heat flux. At old ages only the steady value $k\Delta T/a$ remains. At young ages the sum approaches a Gaussian integral, giving $q_s\sim k\Delta T/\sqrt{\pi \kappa t}$. This divergence is an idealized sudden surface cooling, not an infinite observed ridge heat flux.