= Quasistatic temperature of a conductively cooled film
{title2=$T_s=T_0-\Delta T\alpha h/(\kappa+\alpha h)$}
For a film with fixed lower temperature $T_0$ and upper condition $-\kappa T_z=\alpha(T_s-T_0+\Delta T)$, negligible horizontal thermal <diffusion> and thermal <advection> reduce the <heat equation> to $T_{zz}=0$. Thus $T=T_0-\alpha\Delta T z/(\kappa+\alpha h)$. The local approximation requires $(h/L)^2\ll1$ and $Uh^2/(\kappa L)\ll1$, together with slow time variation relative to $h^2/\kappa$. For $\alpha h/\kappa\ll1$, $T_s\simeq T_0-\Delta T\alpha h/\kappa$. The coefficient $\kappa$ is <thermal diffusivity>, and the boundary coefficient $\alpha$ is normalized accordingly.
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